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A comprehensive model of stress generation and relief processes in thin films deposited with energetic ions
Surface & Coatings Technology 200 (2006) 4345 – 4354 www.elsevier.com/locate/surfcoat
A comprehensive model of stress generation and relief processes in thin films deposited with energetic ions M.M.M. Bilek*, D.R. McKenzie School of Physics, A28, University of Sydney, NSW 2006, Australia Received 31 January 2005; accepted in revised form 9 February 2005 Available online 17 June 2005
Abstract Levels of intrinsic stress in thin films deposited using energetic condensation of ions and/or atoms are known to depend strongly on the energy of the depositing species. At low energies (up to a few eV) thin films are often observed to have a large number of voids and show tensile stress. As the energy increases, the stress is observed to rise to a maximum at an intermediate energy—usually at a few tens to a few hundreds of eV. Thereafter as the energy is increased further, the intrinsic stress is found to decrease. Variations on this ‘‘universal’’ behaviour occur when species of two quite different energies are deposited whereupon the higher of the two energies seems to determine the level of intrinsic stress even when the fraction of ions at that energy is comparatively small. This finding has led to the use of the plasma immersion ion implantation (PIII) method during film deposition to produce coatings with low stress. Although 98 – 99% of the ions impacting the substrate have energies in the range which produces very high intrinsic stress, the intrinsic stress in the film is determined predominantly by the 1 – 2% of ions with high energy (kV and above). Recent experiments show that stress relief can be achieved in films deposited with high levels of stress by post-deposition ion implantation with a non-condensing species, such as argon. We carried out in-situ ellipsometric studies during the implantation and relaxation process, and found that the thickness of the film increased. The increase in thickness in carbon was consistent with a transition of material in the treated volume from sp3 rich to sp2 rich chemical bonding. Atomistic simulations also indicate that the film expands by raising the surface when stress is relieved by energetic bombardment. Experiments with a variety of materials show that a phase transition or change in preferred orientation occurs during the stress relief process. This paper will explore the physical processes involved in determining stress levels in thin films using experimental results together with atomistic simulation. Experimental results of intrinsic stress as a function of ion energy when dc bias is applied, so that a high proportion of the depositing species are affected, are compared to results of PIII&D processes where only a small proportion of ions are affected by the applied bias. Mechanisms and simple models of stress generation and relief are proposed and compared with experiment. The carbon system is examined in detail and the effects of thermal annealing (also found to reduce stress but with different effects on microstructure) are compared with that of high energy ion bombardment. The differences are explained in terms of the model presented. D 2005 Published by Elsevier B.V. Keywords: Stress generation; Stress relief; Annealing; Ion implantation; Ion energy; Ion assisted deposition; Model
1. Introduction Early methods of thin film deposition, such as thermal evaporation, involved low energy (less than a few eV) species condensing on substrate surfaces to produce a coating. These methods produced quite porous materials [1] * Corresponding author. Tel.: +61 2 9351 6079; fax: +61 2 9351 7725. E-mail address: [email protected] (M.M.M. Bilek). 0257-8972/$ - see front matter D 2005 Published by Elsevier B.V. doi:10.1016/j.surfcoat.2005.02.161
made up of columnar structures and voids. The use of ions, with substantially higher energies as a component of the condensing flux is found to produce much denser continuous thin films. It was found that only a small component of the flux needs to be energetic to achieve densification of the microstructure, leading to the introduction of ion beam assisted deposition (IBAD). Unfortunately the benefits of energetic ion bombardment during deposition are often accompanied by the appearance of compressive stress in the
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films. Although the densification of structures is desirable, high levels of intrinsic stress are usually not. The stress causes strain energy, which can drive delamination processes, to be stored in the film. The problem is particularly severe in hard materials such as tetrahedrally bonded carbon and cubic boron nitride, which are of interest for applications as protective coatings. Paradoxically, the creation of such desirable hard phases seems to be associated with the formation of high levels of stress. A great deal of research in recent years has been focused on methods of creating hard phases without high levels of intrinsic stress and many reports on the relief of stress both during and after the deposition process have appeared in the literature. In this paper, we review this body of work to identify the factors influencing the development of intrinsic stress in thin film coatings and to determine to what extent stress is linked to the development of microstructure. We explore stress generation and relief processes associated with the growth of thin films, as well as those found to be effective in relieving stress after deposition, in order to draw together a mechanistic model of stress generation and relief processes in thin film coatings deposited with energetic ions. The model is formulated as an equation which is fitted to a number of stress versus energy curves reported in the literature. It is found to produce consistently better fits to the data than the commonly used model of Davis [2].
2. Densification and stress generation by ‘‘constrained insertion’’ In the case of amorphous films where stress cannot be generated by interface lattice parameter mismatch, internal compressive stress is generally correlated with the formation of a high density phase. There has been a great deal of debate about whether the high density phase is formed due to ‘‘subplantation’’ of atoms [3] or due to the presence of the internal stress [4]. In this section, we examine all of the factors experimentally known to affect the generation of stress in thin film coatings as well as some results from simulation and show that this debate is really about semantics rather than fundamental processes. A recent simulation by Marks et al. [5] using the Environmental Dependent Interaction Potential (EDIP) to simulate carbon thin film growth by energetic ion impacts has shown that stress is generated when the condensing atoms have enough energy to enter the surface and insert into the surface layer of atoms. If the notion of ‘‘subplantation’’ is defined to mean to bury below the surface, as the word suggests, then insertion into the surface is not subplantation and dense phases with high compressive stress can be generated in the absence of subplantation. However, in the literature, the word ‘‘subplantation’’ is sometimes taken to include insertion into the surface layer of atoms as well as burial under the first few monolayers. In all of these cases stress is generated along with the insertion of the atom. The
direct result of the atom insertion is densification of the local region. This results in the development of stress and the formation of a high density phase if the material is unable to undergo plastic deformation. In the case of some materials, especially metals, the insertion of an atom into the surface or sub-surface does not result in local densification and high levels of stress because the material can undergo some plastic flow. In most covalently bonded materials, plastic flow is strongly resisted and the local density will exceed the normal density. In cases where there are two or more distinct structural phases, a high-density metastable phase characterised by highcoordination bonding states is formed. In these cases both the stress and the metastable phase can be regarded as the direct result of the atomic insertion due to the energetic ion impact. The level of macroscopic stress generated by this process depends on the yield characteristics of the material and on the energy penalty for deviations from optimum bond angles. Reductions in stress due to the incorporation of elements with more flexible bonding states have been reported. For example, stress in amorphous carbon [6,7] and amorphous carbon boron nitride grown by pulsed magnetron sputtering [8] is reduced by the addition of silicon into the structure. The stress in vacuum arc or pulsed laser deposited amorphous carbon has also been reduced when metals, such as Fe [9], Ti or Cu [7], are incorporated. It should be noted that in the case of iron incorporation in amorphous carbon, the structure relaxes to the sp2 phase under conditions of ion bombardment which would otherwise generate the sp3 metastable phase in the absence of iron (Wei et al. [7]). This may indicate that the presence of intrinsic stress during growth is required to form the highly coordinated sp3 material as suggested by McKenzie et al. [4]. Based on the ideas discussed in this section, we now seek to formulate a phenomenological model of the stress generation process in materials which do not flow readily. The process begins with the insertion into the surface or sub-surface monolayers of an ion or atom, with impact energy sufficient to penetrate into the growing film. The requirement that the atom has sufficient energy to insert into the surface layer defines a threshold energy for stress generation. This is consistent with experimental observations which show a transition from tensile stress in films deposited with very low energies, such as used in thermal evaporation, to the generation of compressive stress at higher energies in ion assisted processes for deposition, such as IBAD. The process of surface or sub-surface insertion has as its immediate effect an increase in local density and local stress at the point of insertion. The energy brought into the structure by the ion or atom is shared with the neighbouring atoms and will result in thermal vibrations and some local bond breaking. The time for the energy to be removed from the affected volume scales with the size of the volume, which in turn scales with the energy of the impacting ion.
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Marks [10] has shown that the quench time of a spherical thermal spike is proportional to the square of the radius; i.e., proportional to E 2/3, where E is the energy brought in by the impacting atom and distributed within the thermal spike. For energies below a few hundred eV, where the ion impacts generate stress, the quench times (for carbon) are lower than 0.3 ps [10,11]. It can be shown, by estimating velocities of the heated atoms, that such short quench times do not allow significant atom movement. This means that the heated atoms do not retain energy long enough to substantially rearrange their bonding and to allow the heated volume to expand upward through the free surface and reduce the local density. Thus the high local density remains with highly coordinated bonding. The small volume of mobile atoms and the necessity to form bonds with non-mobile atoms on the periphery of the affected volume, leads to a limited number of energetically favorable configurations that can be accessed in the available time. Thus the material has limited opportunity to increase its volume. The result is the formation of high coordination bonding which minimizes energy in the high local stress field, i.e. bonding coordinations that minimize the volume of the arrangement. This concept, of total system energy minimization under constraints, drives the generation of stress and is illustrated in Fig. 1. The inserted energetic atom is shown in black. The dark grey region surrounding it contains atoms which have gained enough energy from the impact to rearrange their bonding. Atoms in the light grey region of the figure have gained some thermal energy but not enough to significantly rearrange their positions as they are bonded to many other atoms in the bulk film (dotted region). The energy in the small affected volume will quench very quickly, further limiting the accessible bonding arrangements. None of the atomic arrangements available to the system under these constraints allows a volume expansion of the region. The lowest energy configuration available to the mobile atoms
is that with the lowest strain energy under compressive stress, i.e. a highly coordinated minimum volume structure. This model can thus explain the simultaneous generation of compressive stress and highly coordinated metastable phases as a result of local densification caused by surface or shallow sub-surface insertion (subplantation) of an atom. Translating this phenomenological model into a mathematical expression describing stress generation as a function of ion energy is not straightforward without a deeper understanding of the details of atomic level processes. It is, however, possible to formulate some functional dependencies if we make assumptions about the shape of the thermal spikes. If we assume that the thermal spike is roughly hemispherical as shown in Fig. 1 and that the stress only accumulates in the light grey region, we find that the energy dependence of the size of the stressed volume is proportional to E 2/3. This is because the volume of the heated region (thermal spike) is proportional to E E B (i.e. k(E E B) = 2/3pr 3, where k is a material dependent constant). E E B is the energy of the incoming ion less the energy required to overcome the potential barrier to enter the surface. Once in the surface, the remaining energy will be distributed among neighbouring atoms. The volume of the light grey pinned region can then be expressed as: 2=3 3k 2 Vstressed ¼ 2pr dt ¼ 2p ð E EB Þ2=3 dt ð1Þ 2p where r is the radius of the volume heated by the inserted ion and dt is the thickness of the interfacial region where atom movement is restrained by bonds to unaffected atoms in the bulk. The thickness dt is expected to be of the order of one bond length and will remain constant as the thermal spike grows in size with increasingly energetic ion impacts. The number of stressed sites produced by the impact is proportional to the volume, V stressed we, can write the following parameterized expression for the stress generated: rgenerated ¼ C ð E EB Þ2=3
t ∝ r 2 ∝ (E-EB)2/3
Fig. 1. Conceptual model of a stress generating thermal spike. The inserted energetic atom is shown in black. The dark grey region surrounding it contains atoms which have gained enough energy from the impact to reconfigure their bonding. Atoms in the light grey region of the hemisphere have gained some thermal energy but not enough to rearrange their positions as they are bonded to atoms in the bulk film (dotted region), not affected by the impact induced thermal spike [19].
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ð2Þ
where C and E B are constants. A slightly different expression is arrived at if the volume of the region affected by the thermal spike is assumed to be cylindrical following Hoffsass [12]. This is consistent with volumes found by molecular dynamics simulation for ions with energy of 500 eV [13] and also according to the energy dependence of implantation depths of ions of greater energy calculated using TRIM [11]. Assuming a cylindrical stress generation region of volume pr 2 d = k(E E B) we arrive at 2
Vstressed ¼ 2prd þ pr dt ¼
2k ð E EB Þ 2 þ pr dt r
ð3Þ
Assuming r is constant, as suggested by the simulations referred to above, we can derive the following parameterized
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expression for the stress generated (again assumed proportional to the volume containing stressed sites): rgenerated ¼ C ð E EB Þ
ð4Þ
where C and B are constants. Note that the same result follows for any shape of stress generation region, including hemispherical, if its volume is proportional to (E E B). These expressions, as well as the square root dependence reported in the work of Nir [14], formulated by Windischmann [15], and utilized in Davis’ parameterized stress model [2], will be combined with a stress relief term and used to fit experimental data in subsequent sections of this paper. The Davis equation has been widely used to fit experimental data in the literature.
3. Stress relief due to ion bombardment during deposition by ‘‘unconstrained rearrangement’’ As the energies of the incident species are increased beyond the stress generating range discussed in the previous section they are observed to start relieving or reducing [2] the stress in the growing film. To the authors’ knowledge, this phenomenon has been reported in the case of all materials deposited as thin films using IBAD methods. The downward trend in stress is continued as the impact energy of the energetic flux component is increased further. In a form of IBAD known as plasma immersion ion implantation and deposition (PIII&D) [16], where the high energy flux component (usually only 1 – 2% of the total flux) is accelerated to energies of over a keV and often as high as 20 or 50 keV, the downward trend in stress is seen to saturate at very low levels [17,18]. The effectiveness of the stress relief is correlated with the product of the energy and the flux of the very energetic (>keV) ions [11,19]. In all of the cases the authors are familiar with (AlN, TiN and carbon), the reduction in stress is associated with a relaxation of the material to the phase or crystalline preferred orientation which is thermodynamically stable at low levels of biaxial stress [11,18,20,21]. Table 1 summarises the changes in phase and crystalline preferred orientation which occur in amorphous carbon, TiN and AlN. These observations led to the development of a model based on the concept of thermal spikes which have a lifetime that is long enough to allow the atoms involved to access all available bonding configurations and to expand
Table 1 Microstructural changes associated with transitions from high stress to low stress induced by ions with high bombardment energies Thin film material
High stress microstructure
Low stress microstructure
Amorphous carbon Titanium nitride (fcc) Aluminium nitride (hexagonal)
sp3 coordination (111) orientation c-axis in plane
sp2 coordination (200) orientation c-axis normal
their volume by raising the level of the free surface above them [11] i.e., unconstrained rearrangement. The volume of such spikes is large enough that the fixed atoms on the periphery no longer determine the system’s minimum energy structure by constraining the structures available to the molten core [19]. As in the case of stress generation, the process begins with the insertion of an ion into the sub-surface region (subplantation). In this case, the ion goes deeper than in the stress generating insertions described in the previous section. As before, the ion shares its energy with surrounding atoms in the structure. In this case, the volume affected by the ion’s energy is much larger so that many coordination shells of atoms around the ion damage track have sufficient energy to participate in bonding rearrangements. Because of the large volume affected, the time for the energy to dissipate is also significantly extended (t ” r 2 ” (E E B)2/3) [10]. The system can now minimize its energy by relieving the stress created by the inserted atom through a significant rearrangement of bonding and a net expansion of the affected volume, resulting in a rise of the surface. Effectively, the system can access many more configurations, including ones relatively distant from the starting state. This is in contrast to the case of the lower energy, small volume and fast cooling thermal spikes, discussed in the previous section, which generate stress. The atomic bonding most favoured by the system when volume expansion is allowed and a larger range of bonding arrangements can be sampled is that which produces the lowest total energy in a low stress field. Due to the availability of free surface expansion this does not have to be the configuration of lowest volume and may now be the configuration with lowest bonding and strain energies. The assumption that the volume affected is proportional to the energy of the implanting ion is supported by atomistic simulation. Experimental observations [22] of the Raman spectrum of carbon films, subjected to PIII during growth, showed an increase in the size of graphitically bonded clusters with increasing implantation energy. As the energy of implanting ions increases, so too does the size of the thermal spike. Since there are now many atoms involved in each thermal spike and all of their bonds undergo significant stress relaxation as a result of the energy dissipated by a single high energy ion, it is possible to relieve large levels of stress with a small fraction of high energy ions. In fact, with ion impacts of several keV, around 1% of the total depositing flux is sufficient to saturate the stress relief and no further relief is gained by increasing the energy or flux. According to our model such saturation would be expected when the volumes relieved by the energetic ions begin to overlap [11]. The model described above leads naturally to an exponential analytical expression describing the relationship between stress and the product of energy and flux of the high energy ions [23]. Assume that the majority (97.5 –
stress
M.M.M. Bilek, D.R. McKenzie / Surface & Coatings Technology 200 (2006) 4345 – 4354 subplantation increases density and compressive stress while quench time is too short to allow relaxation of over dense increasing quench time allows structures more bonding configurations to be accessed and increasingly surface expansion can occur to relieve stress surface insertion some stress remains from residual first generates damage and strained bonds compressive stress
energy
underdense condensation on surface with shadowing creates voids and tensile stress
Fig. 2. Schematic diagram showing the regions of the typical stress – energy curve with explanations for the main features.
99.5%) of the ions incident to the growth surface arrive with moderate energies in the range 10 – 200 eV and thus generate stress, while the remainder arrive with an energy sufficient for stress relief applied by high voltage pulsing of the substrate as in the PIII process. Consider a slab of depth, d, equal to the depth of each thermal spike created by the high energy, stress relieving ion fraction. Suppose that the volume of the slab is Vo. As the film grows, many such slabs are deposited and all are subject to a number of high energy impacts (N) before they are covered by another slab of thickness d and no further impacts affect them. We assume that the moderate energy deposition creates a film with a particular level of stress, r o, by atomic insertion and densification processes. In addition, the regions treated by the high energy, stress relieving thermal spikes are left with a lower residual value, r residual, of stress. The volume in the slab as yet untreated by thermal spikes is V. The rate at which the untreated volume decreases with each new spike is proportional to both the
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volume of each spike (V spike) and the proportion of the total volume left untreated. Thus dV V Vo ¼ Vspike ½lnV VVo ¼ ½nNo dn Vo Vspike
ð5Þ
Vspike N V ¼ exp Vo Vo
ð6Þ
now V spike = E/qE A and N = qVoD = qVowf where the duty factor, D = w/T = wf with high voltage pulse width w and repetition frequency f. Also q is the number density of atoms in the growing film and E A is the average energy distributed to each of the film atoms involved in the thermal spike. Substituting using these relations gives: V Ef w ¼ exp ð7Þ Vo EA Assuming that the stress varies with the volume fraction treated, from the deposited value, r o, to the relaxed value, r residual, it can be described by: V Ef w r ¼ ro þ rresidual ¼ ro exp ð8Þ þ rresidual Vo EA This expression was fitted to experimental data from a number of materials deposited as thin films using the PIII&D method. In all cases the stress generating ion flux was kept constant (i.e. constant r o) while the flux and energy of the high voltage stress relieving population was varied. Eq. (8) produced a very good fit to the data [23,24]. A schematic diagram combining the models for generation and relief into one picture is shown in Fig. 2.
Table 2 Details of best fit equations for films deposited with a single energetic flux Data and Model
C
EA/D
r residual
EB
R2
v2
Rank
a-C [2] (A) a-C [2] (B) a-C [2] (C) a-C [2] (Davis) AlN [2] (A) AlN [2] (B) AlN [2] (C) AlN [2] (Davis) SiBCN [8] (A) SiBCN [8] (B) SiBCN [8] (C) SiBCN [8] (Davis) a-SiH [36] (A) a-SiH [36] (B) a-SiH [36] (C) a-SiH [36] (Davis) a-C [24] (A) a-C [24] (B) a-C [24] (C) a-C [24] (Davis)
1.4 T 0.1 1.098 T 0.013 2.2 T 7.4 2 T 1149 1.9 T 3.1 0.44 T 0.05 2.4 T 0.1 0.37 T 288 0.15 T 0.06 0.021 T 0.005 0.23 T 0.07 0.25 T 50.38 0.08 T 0.02 0.016 T 0.003 0.17 T 0.03 0.0009 T 0.1424 0.40 T 0.02 0.071 T 0.007 0.90 T 0.04 0.7 T 22
68.1 T 2.4 25.36 T 0.23 66 T 184 1 T 489 38.1 T 29.6 58.1 T 2.2 75.5 T 2.4 0.51 T 398 75.1 T16.6 88.6 T 18.4 92.3 T 23.5 1.6 T 323.2 155 T 50 135 T 32 192 T 68 0.01 T1.65 295 T 17 229.5 T 15.2 365.4 T 20.1 0.8 T 27
1.38 T 0.05 0.48 T 0.05 0.4 T 2.1 0T1 0.7 T 1 1.39 T 0.06 1.37 T 0.04 0.00015 T 0.1 1.20 T 0.06 1.16 T 0.10 1.19 T 0.08 0.0002 T 0.0297 0.09 T 0.26 0.09 T 0.22 0.01 T 0.33 1e6 T 0.00017 0.7 T 0.3 1.04 T 0.30 0.23 T 0.34 0.00007 T 0.002
61 T1 6.38 T 0.09 6.4 T 97 0T9 6.4 T 33.8 52.7 T 1.8 65.4 T 0.7 0.07 T 3.32 68.4 T 5.3 43.2 T 8.5 70.3 T 4.7 0.21 T 0.61 42 T 20 18.4 T 19.0 46.3 T 18.1 0T0 6.6 T 6.6 16 T 11 12.6 T 4.5 3e14 T 0.7
0.99959 0.99995 0.88759 0.80181 0.97735 0.99943 0.9997 0.85657 0.91549 0.9041 0.89961 0.81147 0.85884 0.90403 0.85488 0.64036 0.93195 0.9215 0.93263 0.89937
0.00448 0.00292 4.59068 15.87066 1.06973 0.00621 0.00327 4.18454 0.00368 0.00675 0.00437 0.01327 0.00672 0.00587 0.00691 0.01047 0.35555 0.41013 0.352 0.52577
2 1 3 4 3 2 1 4 1 2 2 4 2 1 3 4 2 3 1 4
Best fit curves are plotted in Figs. 3 – 7.
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4. Comparison of models with experimental data Combining the models outlined above for the stress generation and relief processes to describe the variation of intrinsic stress with ion energy over a wide range of energies, gives ER D Model A : r ¼ C ðEG EB Þ2=3 exp þ rresidual EA ð9Þ Model B :
ER D r ¼ C ðEG EB Þexp þ rresidual EA ð10Þ
Model C :
r ¼ C ðEG EB Þ1=2 exp
ER D EA
þ rresidual
Fig. 4. Aluminium nitride data taken from Davis [2]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
where r o is a constant. Table 2 shows parameters for fits to experimental data from the literature, deposited with only one energetic flux component (i.e. using the models A – C with E = E G = E R as well as the model of Davis [2] to fit the data). The model of
Davis was extended with an additive constant to allow a saturation value of the intrinsic stress and to ensure that the number of parameters was the same across the models. The fitted curves are shown in Figs. 3– 7 and are presented in the same order as the data in Table 2. The last column shows the ranking (1 to 4) of the ‘‘goodness of fit’’ for all four models with each data set. Models A – C fit all of the data well and always rank between 1 and 3. The model of Davis [2] provides an inferior fit in all cases (i.e. always ranks 4). Table 3 shows the parameters for fits of model D to data obtained with PIII&D. The first three rows give parameters of fits published elsewhere [23] and the final row shows parameters from a fit to boron nitride data of Abendroth [25]. The fitted curve is shown with the data in Fig. 8. The fit is very good despite the fact that the data is plotted against damage per atom (dpa) instead of the energy– frequency – pulse length product (Efw) or energy-duty cycle product. We believe that this is because the damage per atom in the energy range used in these experiments is proportional to Efw. Damage per atom is a measure of the
Fig. 3. Amorphous carbon data taken from Davis [2]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
Fig. 5. SiBCN data from Vlcek [8]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
ð11Þ where E G is the energy of the stress generating flux component, E R is the energy of the stress relieving flux component and D is its duty cycle (i.e. the flux of high energy stress relieving impacts as a fraction of the total flux). In cases where there is only one energetic flux component which varies in energy, the energy E is equal to E G = E R. In cases such as PIII with an earthed or floating substrate, where the stress generating ions do not vary in energy E G is constant and all of the above expressions reduce to: ER D ð12Þ Model D : r ¼ ro exp þ rresidual EA
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Table 3 Details of best fit for BN produced by PIII&D [25] using model D Data set/parameters r residual (GPa) r o (GPa) TiN [23] AlN [23] Carbon [23] BN [25]
1.11 T 0.35 0.41 T 0.06 0.53 T 0.08 1.57 T 0.04
E A/D (V/s)
10.84 T 1.73 938 T 190 6.82 T 0.50 184 T 17 7.81 T 0.35 325 T 21 41.24 T 8.28 0.129 T 0.010a
R2 0.939 0.989 0.941 0.999
a The data for BN was plotted [25] in terms of damage per atom (dpa) instead of bias multiplied by frequency (Vf ) so the units for parameter c will be dpa instead of V/s.
rearrangement, the total energy available for rearrangement is that of the incoming ion minus any energy barrier to enter through the surface. Fig. 6. a-SiH data from Yin [36]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
fraction of atoms which are displaced from their lattice positions in a crystal as a result of the collision cascade, i.e. the fraction of atoms that have broken bonds with neighbours. For the purposes of our model this damage represents only one part of the energy available for atomic movements and rearrangements. The other two components are the energy transferred to electrons via electronic stopping and the vibrational energy gained by atoms in the affected region. Electronic stopping increases strongly with energy and can be neglected for impacts less than 500 eV. For higher impact energies, as used in PIII&D, it can be viewed as a contribution to the stress relieving thermal spike because the lifetime to cool is greater than the lifetime of the excited electronic states (i.e. the time it takes for them to decay to phonons and therefore heat the volume under consideration). Atoms that gain vibrational energy from the cascade also contribute to the energy available in the thermal spike. Because all of these dissipation mechanisms contribute energy to the region of material undergoing
Fig. 7. Amorphous carbon data with high energy ion bombardment taken from [24]. The curves represent best fits obtained with models (A) dotdashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
5. Post-deposition stress relief In this section, we review three methods which have successfully been employed to relieve stress in films after deposition. The first of these methods involves the use of high energy non-condensing ions, such as argon, to bombard the surface of the film. This method is based on the mechanisms described in the previous section but because the ions are applied post-deposition, stress relief can only be achieved down to the depth of penetration of the energetic ions. Since this depth varies approximately linearly with ion energy, thicker layers of film can be relieved with higher ion energies. This is in contrast to the case when the method is applied during deposition and the entire film can be treated at any ion energy. The second method, which we will describe in this section, relies on the use of flexible substrates (or in the extreme case removal of the film from the substrate) to relieve stress [26]. The third post-deposition method which we will describe is thermal annealing. The three methods differ substantially in their effects on the microstructure of the relieved material. The ion implantation route results in a change in chemical bonding associated with a relaxation to the thermodynamically favoured phase at low stress (similar to the effect
Fig. 8. Data for boron nitride produced with PIII&D [25]. The solid curve shows the model D fit to the data.
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described for stress relief with ion implantation during deposition in the previous section). In contrast, when the stress is relieved by delamination or use of a flexible substrate, the phase of the as-deposited high stress material is retained. For annealing, in some cases there appears to be relaxation without significant phase change for low temperatures while at higher annealing temperatures a phase change to the thermodynamically favoured phase at low stress (as in the ion implantation route) is observed. In the remainder of this section we will explain the results of these three very different types of stress relief in terms of the model developed in the previous two sections. 5.1. High energy ion bombardment Experiments in which highly stressed carbon thin films deposited by the filtered cathodic vacuum arc technique were ion implanted using the plasma immersion ion implantation (PIII) process after deposition have recently been reported [24,34]. The implanting argon ion energy was 10 keV and the stress, thickness and optical properties were monitored. The stress was found to decrease with implanted argon ion dose and the film thickness was observed to increase [24]. Examination of the film using cross-sectional transmission electron microscopy (X-TEM) showed that the stress relieved implanted layer had been converted to the sp2 phase by the argon ion implantation [34]. This is consistent with the model of the stress relieving thermal spike where the lifetime of the spike is long enough to allow the system to relax to the configuration of lowest energy. This entails the removal of strain energy and relaxation to lowest energy phase at low stress. The volume expansion required to accommodate these changes is facilitated by a net upward expansion of the stress relieved layer. 5.2. Delamination and use of flexible substrates Delamination (removal of the film from the substrate) and deposition onto flexible substrates are the only methods reported to date which allow relief of stress in highly stressed, predominantly sp3 bonded carbon films without producing any changes in the bonding or sp3 fraction [26]. In both cases, the film expands in the lateral direction resulting in an increased surface area. As a result, ripples [26] appear on the surface if the film is still attached to the substrate (as in the case of stress reduction on a flexible substrate). The force required to maintain stress in a thin film is ultimately provided by the substrate which deforms somewhat due to the stress in the film. This elastic deflection (often used to measure stress in the film) provides the force which maintains the stress in the film. If the substrate is flexible like rubber or is capable of plastic flow, it will deform and will not provide a force to maintain the stress built up in the film during deposition. The key reason that films grown on such substrates contain the highly coordi-
nated phases when stress generating energies are used during growth is that the timescales involved in the stress generation mechanism of the short-lived thermal spike are extremely short compared to the time it takes for the substrate to deform and relieve the stress. The low volume phase, stable at high stress, is locked in before the stress relieving deformation occurs. There is no energy available in the system at this stage to facilitate a rearrangement of bonding but the stress is relieved by the reduction of the force applied by the substrate. Because there is no energy available to break bonds, the system does not relax to the lowest energy bonding configuration but maintains the bonding generated under high local stress conditions associated with surface and sub-surface atomic insertion occurring during deposition. 5.3. Post-deposition thermal annealing There are many reports on the use of post-deposition thermal annealing to relieve stress in highly stressed, high sp3 amorphous carbon thin films [27 – 33]. The results show that at temperatures of around 600 -C, annealing does relieve stress in high (¨80%) sp3 carbon films with only very minor microstructural change. Observations show that there is simultaneously an increase in the electrical conductivity [31]. This can be explained by an increase in the conductive pathways caused either by an increase in the fraction of conductive graphite like sp2 material or by an improvement in the quality of conductive pathways in the structure. In the absence of increasing sp2 content, the quality of conductive pathways could be increased by clustering of sp2 sites in sheets or filaments. As the temperature is raised, the structure changes towards the sp2 rich phase and at temperature over 1000 -C there is a rapid transition to sp2 bonding. Recent studies of post-deposition thermal annealing of carbon films containing an intermediate level (<50%) of sp3 sites show that the sp2 fraction increases with annealing at 600 -C. This microstructural change is accompanied by a dramatic increase in stress [35], which can be explained by a conversion of sp3 material to sp2 material of lower density. The increase in volume produces an increase in stress and the graphite like sheets orient normal to the plane of the film to minimize the strain energy. The temperature and the time allowed for annealing are important parameters. The temperature allows for atom motion by increasing the amplitude of thermal vibrations in the network. There is a small increase in the energy available to each atom allowing atoms in higher energy configurations associated with the most highly strained bonds to overcome energy barriers to breaking and reforming bonds and reach lower energy bonding configurations. Many such small probability events can occur given sufficient annealing time. In some materials, particularly those with high Young’s modulus, where small reductions in local strain can translate into large reductions
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in stress, this may be enough to significantly relieve the macroscopic stress.
6. General model for post-deposition stress relief Once a film has been deposited, its total energy can be divided into three contributions as follows: Etotal ¼ Ebonds þ Etopol þ Estrain where E bonds is the energy of all of the existing chemical bonds in their unstrained configuration, E topol is the additional energy resulting from straining the bonds to achieve the topology of the structure in the absence of externally applied stresses. E strain is the energy contribution from externally applied stresses, such as that applied by a substrate. For any post-deposition treatment, the system will move in a direction so as to minimize the total energy E total within constraints. For example, treatments in which the film is detached from the substrate will automatically ensure that E strain vanishes regardless of changes in E bonds or E topol, however when a film is attached to a substrate changes in E topol and E bonds may affect E strain. In the case of high energy ion bombardment as a method of stress relief, the system can minimise all three terms independently in the volume of the thermal spike. The energy available during the spike is sufficient to allow major restructuring of the material to achieve a global energy minimum. The material behaves as if molten for the duration of the spike, and minimizes E bonds and E topol as far as possible during the time allowed. The volume of the final structure can adjust to accommodate the minimum energy bonding arrangement because there is sufficient time available for upward expansion of the surface to occur. For all bonding configurations, the relaxation time allows E strain to be reduced by a net upward expansion. Therefore, we would expect this method to reduce E strain as well as to move the bonding and topological arrangements towards the lowest energy structures that apply in the absence of applied stress. Note that E strain cannot be reduced exactly to zero in the film as a whole because some stress will be generated at the boundaries of the thermal spike volume and by residual defects. In the case of delamination, the externally applied stresses and hence E strain are removed post-deposition by expansion of the material at ordinary temperatures. This does not produce changes in E bonds or E topol because at room temperature the number of bonds and topology are fixed. In the case of flexible substrates, the stress is reduced by a plastic deformation of the substrate without changes to the bonding and network topology. The result in both cases is that the E strain term becomes zero, minimizing the system’s energy. In contrast to the case of the thermal spike there is insufficient vibrational energy to cause any reconfigurations of bonding and therefore E bond and E topol remain unaffected while the film expands in the lateral direction. In the case of a
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flexible substrate, the film’s expansion is evidenced by the development of ripples [26] on the surface caused by expansion of the film’s lateral area. In the case of annealing a film on the substrate after deposition, some vibrational energy is given to all of the structure simultaneously, allowing small changes in E topol and E bond to occur. There are constraints as to the accessible states of the system imposed by the presence of the substrate and by the energy barriers. Allowable changes in structure are those with energy barriers lower than the energy of the thermal vibrations at the annealing temperature. Changes in E bond may cause changes in E topol. Also due to the lateral area constraint of attachment to the substrate, changes in E bond and/or E topol may cause changes in E strain. The system may now trade off increases in any of the three terms for larger energy reductions associated with one or more of the others. Therefore annealing may actually result in stress increases in some cases.
7. Conclusions A new model for stress generation and relief processes occurring during growth with energetic ions has been developed and compared with experimental data from a wide variety of materials. This model has been found to consistently fit the data better than the widely used model of Davis. Post-deposition stress relief by energetic ion impacts, by deposition of material onto flexible substrates and by annealing have been reviewed and the results interpreted in terms of a minimization of the total system energy under constraints. One of the outcomes of energy minimization is that annealing, can create stress whilst minimizing the total energy through structural change. In the case of stress relief with highly energetic ion impacts, the model predicts that the phases and preferred orientations observed will be those which minimise the bonding and topological energy since strain can be removed by upward flow of the material. Thus high energy ions relieve stress and produce phases and orientations favoured in low stress conditions, while flexible substrates relieve stress but retain as-deposited phases.
References [1] J.E. Yehoda, B. Yang, K. Vedam, R. Messier, J. Vac. Sci. Technol., A, Vac. Surf. Films 6 (1988) 1631. [2] C.A. Davis, Thin Solid Films 226 (1993) 30. [3] Y. Lifshitz, S.R. Kasi, J.W. Rabalais, Phys. Rev. Lett. 62 (1989) 1290. [4] D.R. McKenzie, D. Muller, B.A. Pailthorpe, Phys. Rev. Lett. 67 (1991) 773. [5] N.A. Marks, J.M. Bell, G.K. Pearce, D.R. McKenzie, M.M.M. Bilek, Diamond Relat. Mater. 12 (2003) 2003. [6] J. Meneve, E. Dekempeneer, J. Smeets, Diamond Films Technol. 4 (1994) 23. [7] Q. Wei, R.J. Narayan, A.K. Sharma, J. Sankar, J. Narayan, Mater. Res. Soc. Symp. Proc. 498 (1998) 61.
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[8] J. Houska, J. Vlcek, S. Potocky, J. Cizek, V. Perina, J. Zemek, J. Vac. Sci. Technol., A, Vac. Surf. Films, (submitted for publication). [9] J.S. Chen, S.P. Lau, G.Y. Chen, Z. Sun, Y.J. Li, B.K. Tay, J.W. Chai, Diamond Relat. Mater. 10 (2001) 2018. [10] N.A. Marks, Phys. Rev., B 56 (5) (1997) 2441. [11] M.M.M. Bilek, R.N. Tarrant, D.R. McKenzie, S.H.N. Lim, D.G. McCulloch, IEEE Trans. Plasma Sci. 31 (2003) 939. [12] H. Hoffsa¨ss, H. Feldermann, R. Merk, M. Sebastian, C. Ronning, Appl. Phys., A 66 (1998) 153. [13] N.A. Marks, J.M. Bell, G.K. Pearce, D.R. McKenzie, M.M.M. Bilek, Diamond Relat. Mater. 12 (2003). [14] D. Nir, J. Vac. Sci. Technol., A, Vac. Surf. Films 4 (1986) 2954. [15] H. Windischmann, J. Appl. Phys. 62 (1987) 1800. [16] A. Anders (Ed.), Handbook of Plasma Immersion Ion Implantation and Deposition, John Wiley and Sons, New York, 2000. [17] S.H.N. Lim, D.G. McCulloch, M. Bilek, D.R. McKenzie, Surf. Coat. Technol. 174 – 175 (2003) 76. [18] B.K. Gan, M.M.M. Bilek, D.R. McKenzie, Yang Shi, D.A. Tompsett, M.B. Taylor, D.G. McCulloch, J. Phys. D: Condens. Matter 16 (2004) 1751. [19] M.M.M. Bilek, D.R. McKenzie, R.N. Tarrant, T.W.H. Oates, P. Ruch, K. Newton-McGee, Yang Shi, D. Tompsett, H.C. Nguyen, B.K. Gan, D.T. Kwok, Contrib. Plasma Phys. 44 (5 – 6) (2004) 465. [20] S.H.N. Lim, D.G. McCulloch, M. Bilek, D.R. McKenzie, J. Appl. Phys. 93 (7) (2003) 4283. [21] R.N. Tarrant, D.R. McKenzie, M.M.M. Bilek, Diamond Relat. Mater. 13 (2004) 1422. [22] D.R. McKenzie, R.N. Tarrant, M. Bilek, G. Pearce, N.A. Marks, D.G. McCulloch, S.H.M. Lim, NIMB 206 (2003) 741.
[23] M.M.M. Bilek, D.R. McKenzie, W. Moeller, Surf. Coat. Technol. 186 (2004) 21. [24] M.M.M. Bilek, M. Verdon, L. Ryves, T.W.H. Oates, C.T. Ha, D.R. McKenzie, Thin Solid Films (in press). [25] B. Abendroth, R. Gago, A. Kolitsch, W. Moeller, Thin Solid Films 447 – 448 (2004) 131. [26] A. Antilla, L. Lappalainen, V.M. Tiainen, M. Hakovirta, Adv. Mater. 9 (1997) 1161. [27] T.A. Friedmann, J.P. Sullivan, J.A. Knapp, D.R. Tallant, D.M. Follstaedt, D.L. Medlin, P.B. Mirkarimi, Appl. Phys. Lett. 71 (1997) 3820. [28] J.P. Sullivan, T.A. Friedmann, A.G. Baca, J. Electron. Mater. 26 (1997) 1021. [29] J.P. Sullivan, T.A. Friedmann, R.G. Dunn, E.B. Stechel, P.A. Schultz, M.P. Siegal, N. Missert, Mater. Res. Soc. Symp. Proc. 498 (1998) 97. [30] B.K. Tay, X. Shi, E. Liu, S.P. Lau, L.K. Cheah, Z. Sun, J. Shi, Surf. Coat. Technol. 120 – 121 (1999) 448. [31] A.C. Ferrari, B. Kleinsorge, N.A. Morrison, A. Hart, V. Stolojan, J. Robertson, J. Appl. Phys. 85 (1999) 7191. [32] L.G. Jacobsohn, R. Prioli, F.L. Freire Jr., G. Mariotto, M.M. Lacerda, Y.W. Chung, Diamond Relat. Mater. 9 (2000) 680. [33] O.R. Monteiro, J.W. Ager, D.H. Lee, R. Yu Lo, K.C. Walter, M. Nastasi, J. Appl. Phys. 88 (2000) 2395. [34] T.W.H. Oates, L. Ryves, F.A. Burgmann, B. Abendroth, M.M.M. Bilek, D.R. McKenzie, D.G. McCulloch, Diamond Relat. Mater. (in press) (doi:10.1016/j.diamond.2005.02.001). [35] D.G. McCulloch, J.L. Peng, D.R. McKenzie, S.P. Lau, D. Sheeja, B.K. Tay, Phys. Rev., B 70 (2004) 85406. [36] Y. Yin, M.M.M. Bilek, D.R. McKenzie, Surf. Coat. Technol., (in press).
A comprehensive model of stress generation and relief processes in thin films deposited with energetic ions M.M.M. Bilek*, D.R. McKenzie School of Physics, A28, University of Sydney, NSW 2006, Australia Received 31 January 2005; accepted in revised form 9 February 2005 Available online 17 June 2005
Abstract Levels of intrinsic stress in thin films deposited using energetic condensation of ions and/or atoms are known to depend strongly on the energy of the depositing species. At low energies (up to a few eV) thin films are often observed to have a large number of voids and show tensile stress. As the energy increases, the stress is observed to rise to a maximum at an intermediate energy—usually at a few tens to a few hundreds of eV. Thereafter as the energy is increased further, the intrinsic stress is found to decrease. Variations on this ‘‘universal’’ behaviour occur when species of two quite different energies are deposited whereupon the higher of the two energies seems to determine the level of intrinsic stress even when the fraction of ions at that energy is comparatively small. This finding has led to the use of the plasma immersion ion implantation (PIII) method during film deposition to produce coatings with low stress. Although 98 – 99% of the ions impacting the substrate have energies in the range which produces very high intrinsic stress, the intrinsic stress in the film is determined predominantly by the 1 – 2% of ions with high energy (kV and above). Recent experiments show that stress relief can be achieved in films deposited with high levels of stress by post-deposition ion implantation with a non-condensing species, such as argon. We carried out in-situ ellipsometric studies during the implantation and relaxation process, and found that the thickness of the film increased. The increase in thickness in carbon was consistent with a transition of material in the treated volume from sp3 rich to sp2 rich chemical bonding. Atomistic simulations also indicate that the film expands by raising the surface when stress is relieved by energetic bombardment. Experiments with a variety of materials show that a phase transition or change in preferred orientation occurs during the stress relief process. This paper will explore the physical processes involved in determining stress levels in thin films using experimental results together with atomistic simulation. Experimental results of intrinsic stress as a function of ion energy when dc bias is applied, so that a high proportion of the depositing species are affected, are compared to results of PIII&D processes where only a small proportion of ions are affected by the applied bias. Mechanisms and simple models of stress generation and relief are proposed and compared with experiment. The carbon system is examined in detail and the effects of thermal annealing (also found to reduce stress but with different effects on microstructure) are compared with that of high energy ion bombardment. The differences are explained in terms of the model presented. D 2005 Published by Elsevier B.V. Keywords: Stress generation; Stress relief; Annealing; Ion implantation; Ion energy; Ion assisted deposition; Model
1. Introduction Early methods of thin film deposition, such as thermal evaporation, involved low energy (less than a few eV) species condensing on substrate surfaces to produce a coating. These methods produced quite porous materials [1] * Corresponding author. Tel.: +61 2 9351 6079; fax: +61 2 9351 7725. E-mail address: [email protected] (M.M.M. Bilek). 0257-8972/$ - see front matter D 2005 Published by Elsevier B.V. doi:10.1016/j.surfcoat.2005.02.161
made up of columnar structures and voids. The use of ions, with substantially higher energies as a component of the condensing flux is found to produce much denser continuous thin films. It was found that only a small component of the flux needs to be energetic to achieve densification of the microstructure, leading to the introduction of ion beam assisted deposition (IBAD). Unfortunately the benefits of energetic ion bombardment during deposition are often accompanied by the appearance of compressive stress in the
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films. Although the densification of structures is desirable, high levels of intrinsic stress are usually not. The stress causes strain energy, which can drive delamination processes, to be stored in the film. The problem is particularly severe in hard materials such as tetrahedrally bonded carbon and cubic boron nitride, which are of interest for applications as protective coatings. Paradoxically, the creation of such desirable hard phases seems to be associated with the formation of high levels of stress. A great deal of research in recent years has been focused on methods of creating hard phases without high levels of intrinsic stress and many reports on the relief of stress both during and after the deposition process have appeared in the literature. In this paper, we review this body of work to identify the factors influencing the development of intrinsic stress in thin film coatings and to determine to what extent stress is linked to the development of microstructure. We explore stress generation and relief processes associated with the growth of thin films, as well as those found to be effective in relieving stress after deposition, in order to draw together a mechanistic model of stress generation and relief processes in thin film coatings deposited with energetic ions. The model is formulated as an equation which is fitted to a number of stress versus energy curves reported in the literature. It is found to produce consistently better fits to the data than the commonly used model of Davis [2].
2. Densification and stress generation by ‘‘constrained insertion’’ In the case of amorphous films where stress cannot be generated by interface lattice parameter mismatch, internal compressive stress is generally correlated with the formation of a high density phase. There has been a great deal of debate about whether the high density phase is formed due to ‘‘subplantation’’ of atoms [3] or due to the presence of the internal stress [4]. In this section, we examine all of the factors experimentally known to affect the generation of stress in thin film coatings as well as some results from simulation and show that this debate is really about semantics rather than fundamental processes. A recent simulation by Marks et al. [5] using the Environmental Dependent Interaction Potential (EDIP) to simulate carbon thin film growth by energetic ion impacts has shown that stress is generated when the condensing atoms have enough energy to enter the surface and insert into the surface layer of atoms. If the notion of ‘‘subplantation’’ is defined to mean to bury below the surface, as the word suggests, then insertion into the surface is not subplantation and dense phases with high compressive stress can be generated in the absence of subplantation. However, in the literature, the word ‘‘subplantation’’ is sometimes taken to include insertion into the surface layer of atoms as well as burial under the first few monolayers. In all of these cases stress is generated along with the insertion of the atom. The
direct result of the atom insertion is densification of the local region. This results in the development of stress and the formation of a high density phase if the material is unable to undergo plastic deformation. In the case of some materials, especially metals, the insertion of an atom into the surface or sub-surface does not result in local densification and high levels of stress because the material can undergo some plastic flow. In most covalently bonded materials, plastic flow is strongly resisted and the local density will exceed the normal density. In cases where there are two or more distinct structural phases, a high-density metastable phase characterised by highcoordination bonding states is formed. In these cases both the stress and the metastable phase can be regarded as the direct result of the atomic insertion due to the energetic ion impact. The level of macroscopic stress generated by this process depends on the yield characteristics of the material and on the energy penalty for deviations from optimum bond angles. Reductions in stress due to the incorporation of elements with more flexible bonding states have been reported. For example, stress in amorphous carbon [6,7] and amorphous carbon boron nitride grown by pulsed magnetron sputtering [8] is reduced by the addition of silicon into the structure. The stress in vacuum arc or pulsed laser deposited amorphous carbon has also been reduced when metals, such as Fe [9], Ti or Cu [7], are incorporated. It should be noted that in the case of iron incorporation in amorphous carbon, the structure relaxes to the sp2 phase under conditions of ion bombardment which would otherwise generate the sp3 metastable phase in the absence of iron (Wei et al. [7]). This may indicate that the presence of intrinsic stress during growth is required to form the highly coordinated sp3 material as suggested by McKenzie et al. [4]. Based on the ideas discussed in this section, we now seek to formulate a phenomenological model of the stress generation process in materials which do not flow readily. The process begins with the insertion into the surface or sub-surface monolayers of an ion or atom, with impact energy sufficient to penetrate into the growing film. The requirement that the atom has sufficient energy to insert into the surface layer defines a threshold energy for stress generation. This is consistent with experimental observations which show a transition from tensile stress in films deposited with very low energies, such as used in thermal evaporation, to the generation of compressive stress at higher energies in ion assisted processes for deposition, such as IBAD. The process of surface or sub-surface insertion has as its immediate effect an increase in local density and local stress at the point of insertion. The energy brought into the structure by the ion or atom is shared with the neighbouring atoms and will result in thermal vibrations and some local bond breaking. The time for the energy to be removed from the affected volume scales with the size of the volume, which in turn scales with the energy of the impacting ion.
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Marks [10] has shown that the quench time of a spherical thermal spike is proportional to the square of the radius; i.e., proportional to E 2/3, where E is the energy brought in by the impacting atom and distributed within the thermal spike. For energies below a few hundred eV, where the ion impacts generate stress, the quench times (for carbon) are lower than 0.3 ps [10,11]. It can be shown, by estimating velocities of the heated atoms, that such short quench times do not allow significant atom movement. This means that the heated atoms do not retain energy long enough to substantially rearrange their bonding and to allow the heated volume to expand upward through the free surface and reduce the local density. Thus the high local density remains with highly coordinated bonding. The small volume of mobile atoms and the necessity to form bonds with non-mobile atoms on the periphery of the affected volume, leads to a limited number of energetically favorable configurations that can be accessed in the available time. Thus the material has limited opportunity to increase its volume. The result is the formation of high coordination bonding which minimizes energy in the high local stress field, i.e. bonding coordinations that minimize the volume of the arrangement. This concept, of total system energy minimization under constraints, drives the generation of stress and is illustrated in Fig. 1. The inserted energetic atom is shown in black. The dark grey region surrounding it contains atoms which have gained enough energy from the impact to rearrange their bonding. Atoms in the light grey region of the figure have gained some thermal energy but not enough to significantly rearrange their positions as they are bonded to many other atoms in the bulk film (dotted region). The energy in the small affected volume will quench very quickly, further limiting the accessible bonding arrangements. None of the atomic arrangements available to the system under these constraints allows a volume expansion of the region. The lowest energy configuration available to the mobile atoms
is that with the lowest strain energy under compressive stress, i.e. a highly coordinated minimum volume structure. This model can thus explain the simultaneous generation of compressive stress and highly coordinated metastable phases as a result of local densification caused by surface or shallow sub-surface insertion (subplantation) of an atom. Translating this phenomenological model into a mathematical expression describing stress generation as a function of ion energy is not straightforward without a deeper understanding of the details of atomic level processes. It is, however, possible to formulate some functional dependencies if we make assumptions about the shape of the thermal spikes. If we assume that the thermal spike is roughly hemispherical as shown in Fig. 1 and that the stress only accumulates in the light grey region, we find that the energy dependence of the size of the stressed volume is proportional to E 2/3. This is because the volume of the heated region (thermal spike) is proportional to E E B (i.e. k(E E B) = 2/3pr 3, where k is a material dependent constant). E E B is the energy of the incoming ion less the energy required to overcome the potential barrier to enter the surface. Once in the surface, the remaining energy will be distributed among neighbouring atoms. The volume of the light grey pinned region can then be expressed as: 2=3 3k 2 Vstressed ¼ 2pr dt ¼ 2p ð E EB Þ2=3 dt ð1Þ 2p where r is the radius of the volume heated by the inserted ion and dt is the thickness of the interfacial region where atom movement is restrained by bonds to unaffected atoms in the bulk. The thickness dt is expected to be of the order of one bond length and will remain constant as the thermal spike grows in size with increasingly energetic ion impacts. The number of stressed sites produced by the impact is proportional to the volume, V stressed we, can write the following parameterized expression for the stress generated: rgenerated ¼ C ð E EB Þ2=3
t ∝ r 2 ∝ (E-EB)2/3
Fig. 1. Conceptual model of a stress generating thermal spike. The inserted energetic atom is shown in black. The dark grey region surrounding it contains atoms which have gained enough energy from the impact to reconfigure their bonding. Atoms in the light grey region of the hemisphere have gained some thermal energy but not enough to rearrange their positions as they are bonded to atoms in the bulk film (dotted region), not affected by the impact induced thermal spike [19].
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ð2Þ
where C and E B are constants. A slightly different expression is arrived at if the volume of the region affected by the thermal spike is assumed to be cylindrical following Hoffsass [12]. This is consistent with volumes found by molecular dynamics simulation for ions with energy of 500 eV [13] and also according to the energy dependence of implantation depths of ions of greater energy calculated using TRIM [11]. Assuming a cylindrical stress generation region of volume pr 2 d = k(E E B) we arrive at 2
Vstressed ¼ 2prd þ pr dt ¼
2k ð E EB Þ 2 þ pr dt r
ð3Þ
Assuming r is constant, as suggested by the simulations referred to above, we can derive the following parameterized
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expression for the stress generated (again assumed proportional to the volume containing stressed sites): rgenerated ¼ C ð E EB Þ
ð4Þ
where C and B are constants. Note that the same result follows for any shape of stress generation region, including hemispherical, if its volume is proportional to (E E B). These expressions, as well as the square root dependence reported in the work of Nir [14], formulated by Windischmann [15], and utilized in Davis’ parameterized stress model [2], will be combined with a stress relief term and used to fit experimental data in subsequent sections of this paper. The Davis equation has been widely used to fit experimental data in the literature.
3. Stress relief due to ion bombardment during deposition by ‘‘unconstrained rearrangement’’ As the energies of the incident species are increased beyond the stress generating range discussed in the previous section they are observed to start relieving or reducing [2] the stress in the growing film. To the authors’ knowledge, this phenomenon has been reported in the case of all materials deposited as thin films using IBAD methods. The downward trend in stress is continued as the impact energy of the energetic flux component is increased further. In a form of IBAD known as plasma immersion ion implantation and deposition (PIII&D) [16], where the high energy flux component (usually only 1 – 2% of the total flux) is accelerated to energies of over a keV and often as high as 20 or 50 keV, the downward trend in stress is seen to saturate at very low levels [17,18]. The effectiveness of the stress relief is correlated with the product of the energy and the flux of the very energetic (>keV) ions [11,19]. In all of the cases the authors are familiar with (AlN, TiN and carbon), the reduction in stress is associated with a relaxation of the material to the phase or crystalline preferred orientation which is thermodynamically stable at low levels of biaxial stress [11,18,20,21]. Table 1 summarises the changes in phase and crystalline preferred orientation which occur in amorphous carbon, TiN and AlN. These observations led to the development of a model based on the concept of thermal spikes which have a lifetime that is long enough to allow the atoms involved to access all available bonding configurations and to expand
Table 1 Microstructural changes associated with transitions from high stress to low stress induced by ions with high bombardment energies Thin film material
High stress microstructure
Low stress microstructure
Amorphous carbon Titanium nitride (fcc) Aluminium nitride (hexagonal)
sp3 coordination (111) orientation c-axis in plane
sp2 coordination (200) orientation c-axis normal
their volume by raising the level of the free surface above them [11] i.e., unconstrained rearrangement. The volume of such spikes is large enough that the fixed atoms on the periphery no longer determine the system’s minimum energy structure by constraining the structures available to the molten core [19]. As in the case of stress generation, the process begins with the insertion of an ion into the sub-surface region (subplantation). In this case, the ion goes deeper than in the stress generating insertions described in the previous section. As before, the ion shares its energy with surrounding atoms in the structure. In this case, the volume affected by the ion’s energy is much larger so that many coordination shells of atoms around the ion damage track have sufficient energy to participate in bonding rearrangements. Because of the large volume affected, the time for the energy to dissipate is also significantly extended (t ” r 2 ” (E E B)2/3) [10]. The system can now minimize its energy by relieving the stress created by the inserted atom through a significant rearrangement of bonding and a net expansion of the affected volume, resulting in a rise of the surface. Effectively, the system can access many more configurations, including ones relatively distant from the starting state. This is in contrast to the case of the lower energy, small volume and fast cooling thermal spikes, discussed in the previous section, which generate stress. The atomic bonding most favoured by the system when volume expansion is allowed and a larger range of bonding arrangements can be sampled is that which produces the lowest total energy in a low stress field. Due to the availability of free surface expansion this does not have to be the configuration of lowest volume and may now be the configuration with lowest bonding and strain energies. The assumption that the volume affected is proportional to the energy of the implanting ion is supported by atomistic simulation. Experimental observations [22] of the Raman spectrum of carbon films, subjected to PIII during growth, showed an increase in the size of graphitically bonded clusters with increasing implantation energy. As the energy of implanting ions increases, so too does the size of the thermal spike. Since there are now many atoms involved in each thermal spike and all of their bonds undergo significant stress relaxation as a result of the energy dissipated by a single high energy ion, it is possible to relieve large levels of stress with a small fraction of high energy ions. In fact, with ion impacts of several keV, around 1% of the total depositing flux is sufficient to saturate the stress relief and no further relief is gained by increasing the energy or flux. According to our model such saturation would be expected when the volumes relieved by the energetic ions begin to overlap [11]. The model described above leads naturally to an exponential analytical expression describing the relationship between stress and the product of energy and flux of the high energy ions [23]. Assume that the majority (97.5 –
stress
M.M.M. Bilek, D.R. McKenzie / Surface & Coatings Technology 200 (2006) 4345 – 4354 subplantation increases density and compressive stress while quench time is too short to allow relaxation of over dense increasing quench time allows structures more bonding configurations to be accessed and increasingly surface expansion can occur to relieve stress surface insertion some stress remains from residual first generates damage and strained bonds compressive stress
energy
underdense condensation on surface with shadowing creates voids and tensile stress
Fig. 2. Schematic diagram showing the regions of the typical stress – energy curve with explanations for the main features.
99.5%) of the ions incident to the growth surface arrive with moderate energies in the range 10 – 200 eV and thus generate stress, while the remainder arrive with an energy sufficient for stress relief applied by high voltage pulsing of the substrate as in the PIII process. Consider a slab of depth, d, equal to the depth of each thermal spike created by the high energy, stress relieving ion fraction. Suppose that the volume of the slab is Vo. As the film grows, many such slabs are deposited and all are subject to a number of high energy impacts (N) before they are covered by another slab of thickness d and no further impacts affect them. We assume that the moderate energy deposition creates a film with a particular level of stress, r o, by atomic insertion and densification processes. In addition, the regions treated by the high energy, stress relieving thermal spikes are left with a lower residual value, r residual, of stress. The volume in the slab as yet untreated by thermal spikes is V. The rate at which the untreated volume decreases with each new spike is proportional to both the
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volume of each spike (V spike) and the proportion of the total volume left untreated. Thus dV V Vo ¼ Vspike ½lnV VVo ¼ ½nNo dn Vo Vspike
ð5Þ
Vspike N V ¼ exp Vo Vo
ð6Þ
now V spike = E/qE A and N = qVoD = qVowf where the duty factor, D = w/T = wf with high voltage pulse width w and repetition frequency f. Also q is the number density of atoms in the growing film and E A is the average energy distributed to each of the film atoms involved in the thermal spike. Substituting using these relations gives: V Ef w ¼ exp ð7Þ Vo EA Assuming that the stress varies with the volume fraction treated, from the deposited value, r o, to the relaxed value, r residual, it can be described by: V Ef w r ¼ ro þ rresidual ¼ ro exp ð8Þ þ rresidual Vo EA This expression was fitted to experimental data from a number of materials deposited as thin films using the PIII&D method. In all cases the stress generating ion flux was kept constant (i.e. constant r o) while the flux and energy of the high voltage stress relieving population was varied. Eq. (8) produced a very good fit to the data [23,24]. A schematic diagram combining the models for generation and relief into one picture is shown in Fig. 2.
Table 2 Details of best fit equations for films deposited with a single energetic flux Data and Model
C
EA/D
r residual
EB
R2
v2
Rank
a-C [2] (A) a-C [2] (B) a-C [2] (C) a-C [2] (Davis) AlN [2] (A) AlN [2] (B) AlN [2] (C) AlN [2] (Davis) SiBCN [8] (A) SiBCN [8] (B) SiBCN [8] (C) SiBCN [8] (Davis) a-SiH [36] (A) a-SiH [36] (B) a-SiH [36] (C) a-SiH [36] (Davis) a-C [24] (A) a-C [24] (B) a-C [24] (C) a-C [24] (Davis)
1.4 T 0.1 1.098 T 0.013 2.2 T 7.4 2 T 1149 1.9 T 3.1 0.44 T 0.05 2.4 T 0.1 0.37 T 288 0.15 T 0.06 0.021 T 0.005 0.23 T 0.07 0.25 T 50.38 0.08 T 0.02 0.016 T 0.003 0.17 T 0.03 0.0009 T 0.1424 0.40 T 0.02 0.071 T 0.007 0.90 T 0.04 0.7 T 22
68.1 T 2.4 25.36 T 0.23 66 T 184 1 T 489 38.1 T 29.6 58.1 T 2.2 75.5 T 2.4 0.51 T 398 75.1 T16.6 88.6 T 18.4 92.3 T 23.5 1.6 T 323.2 155 T 50 135 T 32 192 T 68 0.01 T1.65 295 T 17 229.5 T 15.2 365.4 T 20.1 0.8 T 27
1.38 T 0.05 0.48 T 0.05 0.4 T 2.1 0T1 0.7 T 1 1.39 T 0.06 1.37 T 0.04 0.00015 T 0.1 1.20 T 0.06 1.16 T 0.10 1.19 T 0.08 0.0002 T 0.0297 0.09 T 0.26 0.09 T 0.22 0.01 T 0.33 1e6 T 0.00017 0.7 T 0.3 1.04 T 0.30 0.23 T 0.34 0.00007 T 0.002
61 T1 6.38 T 0.09 6.4 T 97 0T9 6.4 T 33.8 52.7 T 1.8 65.4 T 0.7 0.07 T 3.32 68.4 T 5.3 43.2 T 8.5 70.3 T 4.7 0.21 T 0.61 42 T 20 18.4 T 19.0 46.3 T 18.1 0T0 6.6 T 6.6 16 T 11 12.6 T 4.5 3e14 T 0.7
0.99959 0.99995 0.88759 0.80181 0.97735 0.99943 0.9997 0.85657 0.91549 0.9041 0.89961 0.81147 0.85884 0.90403 0.85488 0.64036 0.93195 0.9215 0.93263 0.89937
0.00448 0.00292 4.59068 15.87066 1.06973 0.00621 0.00327 4.18454 0.00368 0.00675 0.00437 0.01327 0.00672 0.00587 0.00691 0.01047 0.35555 0.41013 0.352 0.52577
2 1 3 4 3 2 1 4 1 2 2 4 2 1 3 4 2 3 1 4
Best fit curves are plotted in Figs. 3 – 7.
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4. Comparison of models with experimental data Combining the models outlined above for the stress generation and relief processes to describe the variation of intrinsic stress with ion energy over a wide range of energies, gives ER D Model A : r ¼ C ðEG EB Þ2=3 exp þ rresidual EA ð9Þ Model B :
ER D r ¼ C ðEG EB Þexp þ rresidual EA ð10Þ
Model C :
r ¼ C ðEG EB Þ1=2 exp
ER D EA
þ rresidual
Fig. 4. Aluminium nitride data taken from Davis [2]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
where r o is a constant. Table 2 shows parameters for fits to experimental data from the literature, deposited with only one energetic flux component (i.e. using the models A – C with E = E G = E R as well as the model of Davis [2] to fit the data). The model of
Davis was extended with an additive constant to allow a saturation value of the intrinsic stress and to ensure that the number of parameters was the same across the models. The fitted curves are shown in Figs. 3– 7 and are presented in the same order as the data in Table 2. The last column shows the ranking (1 to 4) of the ‘‘goodness of fit’’ for all four models with each data set. Models A – C fit all of the data well and always rank between 1 and 3. The model of Davis [2] provides an inferior fit in all cases (i.e. always ranks 4). Table 3 shows the parameters for fits of model D to data obtained with PIII&D. The first three rows give parameters of fits published elsewhere [23] and the final row shows parameters from a fit to boron nitride data of Abendroth [25]. The fitted curve is shown with the data in Fig. 8. The fit is very good despite the fact that the data is plotted against damage per atom (dpa) instead of the energy– frequency – pulse length product (Efw) or energy-duty cycle product. We believe that this is because the damage per atom in the energy range used in these experiments is proportional to Efw. Damage per atom is a measure of the
Fig. 3. Amorphous carbon data taken from Davis [2]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
Fig. 5. SiBCN data from Vlcek [8]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
ð11Þ where E G is the energy of the stress generating flux component, E R is the energy of the stress relieving flux component and D is its duty cycle (i.e. the flux of high energy stress relieving impacts as a fraction of the total flux). In cases where there is only one energetic flux component which varies in energy, the energy E is equal to E G = E R. In cases such as PIII with an earthed or floating substrate, where the stress generating ions do not vary in energy E G is constant and all of the above expressions reduce to: ER D ð12Þ Model D : r ¼ ro exp þ rresidual EA
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Table 3 Details of best fit for BN produced by PIII&D [25] using model D Data set/parameters r residual (GPa) r o (GPa) TiN [23] AlN [23] Carbon [23] BN [25]
1.11 T 0.35 0.41 T 0.06 0.53 T 0.08 1.57 T 0.04
E A/D (V/s)
10.84 T 1.73 938 T 190 6.82 T 0.50 184 T 17 7.81 T 0.35 325 T 21 41.24 T 8.28 0.129 T 0.010a
R2 0.939 0.989 0.941 0.999
a The data for BN was plotted [25] in terms of damage per atom (dpa) instead of bias multiplied by frequency (Vf ) so the units for parameter c will be dpa instead of V/s.
rearrangement, the total energy available for rearrangement is that of the incoming ion minus any energy barrier to enter through the surface. Fig. 6. a-SiH data from Yin [36]. The curves represent best fits obtained with models (A) dot-dashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
fraction of atoms which are displaced from their lattice positions in a crystal as a result of the collision cascade, i.e. the fraction of atoms that have broken bonds with neighbours. For the purposes of our model this damage represents only one part of the energy available for atomic movements and rearrangements. The other two components are the energy transferred to electrons via electronic stopping and the vibrational energy gained by atoms in the affected region. Electronic stopping increases strongly with energy and can be neglected for impacts less than 500 eV. For higher impact energies, as used in PIII&D, it can be viewed as a contribution to the stress relieving thermal spike because the lifetime to cool is greater than the lifetime of the excited electronic states (i.e. the time it takes for them to decay to phonons and therefore heat the volume under consideration). Atoms that gain vibrational energy from the cascade also contribute to the energy available in the thermal spike. Because all of these dissipation mechanisms contribute energy to the region of material undergoing
Fig. 7. Amorphous carbon data with high energy ion bombardment taken from [24]. The curves represent best fits obtained with models (A) dotdashed line, (B) solid line, (C) dashed line and the Davis model with additive constant—dotted line.
5. Post-deposition stress relief In this section, we review three methods which have successfully been employed to relieve stress in films after deposition. The first of these methods involves the use of high energy non-condensing ions, such as argon, to bombard the surface of the film. This method is based on the mechanisms described in the previous section but because the ions are applied post-deposition, stress relief can only be achieved down to the depth of penetration of the energetic ions. Since this depth varies approximately linearly with ion energy, thicker layers of film can be relieved with higher ion energies. This is in contrast to the case when the method is applied during deposition and the entire film can be treated at any ion energy. The second method, which we will describe in this section, relies on the use of flexible substrates (or in the extreme case removal of the film from the substrate) to relieve stress [26]. The third post-deposition method which we will describe is thermal annealing. The three methods differ substantially in their effects on the microstructure of the relieved material. The ion implantation route results in a change in chemical bonding associated with a relaxation to the thermodynamically favoured phase at low stress (similar to the effect
Fig. 8. Data for boron nitride produced with PIII&D [25]. The solid curve shows the model D fit to the data.
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described for stress relief with ion implantation during deposition in the previous section). In contrast, when the stress is relieved by delamination or use of a flexible substrate, the phase of the as-deposited high stress material is retained. For annealing, in some cases there appears to be relaxation without significant phase change for low temperatures while at higher annealing temperatures a phase change to the thermodynamically favoured phase at low stress (as in the ion implantation route) is observed. In the remainder of this section we will explain the results of these three very different types of stress relief in terms of the model developed in the previous two sections. 5.1. High energy ion bombardment Experiments in which highly stressed carbon thin films deposited by the filtered cathodic vacuum arc technique were ion implanted using the plasma immersion ion implantation (PIII) process after deposition have recently been reported [24,34]. The implanting argon ion energy was 10 keV and the stress, thickness and optical properties were monitored. The stress was found to decrease with implanted argon ion dose and the film thickness was observed to increase [24]. Examination of the film using cross-sectional transmission electron microscopy (X-TEM) showed that the stress relieved implanted layer had been converted to the sp2 phase by the argon ion implantation [34]. This is consistent with the model of the stress relieving thermal spike where the lifetime of the spike is long enough to allow the system to relax to the configuration of lowest energy. This entails the removal of strain energy and relaxation to lowest energy phase at low stress. The volume expansion required to accommodate these changes is facilitated by a net upward expansion of the stress relieved layer. 5.2. Delamination and use of flexible substrates Delamination (removal of the film from the substrate) and deposition onto flexible substrates are the only methods reported to date which allow relief of stress in highly stressed, predominantly sp3 bonded carbon films without producing any changes in the bonding or sp3 fraction [26]. In both cases, the film expands in the lateral direction resulting in an increased surface area. As a result, ripples [26] appear on the surface if the film is still attached to the substrate (as in the case of stress reduction on a flexible substrate). The force required to maintain stress in a thin film is ultimately provided by the substrate which deforms somewhat due to the stress in the film. This elastic deflection (often used to measure stress in the film) provides the force which maintains the stress in the film. If the substrate is flexible like rubber or is capable of plastic flow, it will deform and will not provide a force to maintain the stress built up in the film during deposition. The key reason that films grown on such substrates contain the highly coordi-
nated phases when stress generating energies are used during growth is that the timescales involved in the stress generation mechanism of the short-lived thermal spike are extremely short compared to the time it takes for the substrate to deform and relieve the stress. The low volume phase, stable at high stress, is locked in before the stress relieving deformation occurs. There is no energy available in the system at this stage to facilitate a rearrangement of bonding but the stress is relieved by the reduction of the force applied by the substrate. Because there is no energy available to break bonds, the system does not relax to the lowest energy bonding configuration but maintains the bonding generated under high local stress conditions associated with surface and sub-surface atomic insertion occurring during deposition. 5.3. Post-deposition thermal annealing There are many reports on the use of post-deposition thermal annealing to relieve stress in highly stressed, high sp3 amorphous carbon thin films [27 – 33]. The results show that at temperatures of around 600 -C, annealing does relieve stress in high (¨80%) sp3 carbon films with only very minor microstructural change. Observations show that there is simultaneously an increase in the electrical conductivity [31]. This can be explained by an increase in the conductive pathways caused either by an increase in the fraction of conductive graphite like sp2 material or by an improvement in the quality of conductive pathways in the structure. In the absence of increasing sp2 content, the quality of conductive pathways could be increased by clustering of sp2 sites in sheets or filaments. As the temperature is raised, the structure changes towards the sp2 rich phase and at temperature over 1000 -C there is a rapid transition to sp2 bonding. Recent studies of post-deposition thermal annealing of carbon films containing an intermediate level (<50%) of sp3 sites show that the sp2 fraction increases with annealing at 600 -C. This microstructural change is accompanied by a dramatic increase in stress [35], which can be explained by a conversion of sp3 material to sp2 material of lower density. The increase in volume produces an increase in stress and the graphite like sheets orient normal to the plane of the film to minimize the strain energy. The temperature and the time allowed for annealing are important parameters. The temperature allows for atom motion by increasing the amplitude of thermal vibrations in the network. There is a small increase in the energy available to each atom allowing atoms in higher energy configurations associated with the most highly strained bonds to overcome energy barriers to breaking and reforming bonds and reach lower energy bonding configurations. Many such small probability events can occur given sufficient annealing time. In some materials, particularly those with high Young’s modulus, where small reductions in local strain can translate into large reductions
M.M.M. Bilek, D.R. McKenzie / Surface & Coatings Technology 200 (2006) 4345 – 4354
in stress, this may be enough to significantly relieve the macroscopic stress.
6. General model for post-deposition stress relief Once a film has been deposited, its total energy can be divided into three contributions as follows: Etotal ¼ Ebonds þ Etopol þ Estrain where E bonds is the energy of all of the existing chemical bonds in their unstrained configuration, E topol is the additional energy resulting from straining the bonds to achieve the topology of the structure in the absence of externally applied stresses. E strain is the energy contribution from externally applied stresses, such as that applied by a substrate. For any post-deposition treatment, the system will move in a direction so as to minimize the total energy E total within constraints. For example, treatments in which the film is detached from the substrate will automatically ensure that E strain vanishes regardless of changes in E bonds or E topol, however when a film is attached to a substrate changes in E topol and E bonds may affect E strain. In the case of high energy ion bombardment as a method of stress relief, the system can minimise all three terms independently in the volume of the thermal spike. The energy available during the spike is sufficient to allow major restructuring of the material to achieve a global energy minimum. The material behaves as if molten for the duration of the spike, and minimizes E bonds and E topol as far as possible during the time allowed. The volume of the final structure can adjust to accommodate the minimum energy bonding arrangement because there is sufficient time available for upward expansion of the surface to occur. For all bonding configurations, the relaxation time allows E strain to be reduced by a net upward expansion. Therefore, we would expect this method to reduce E strain as well as to move the bonding and topological arrangements towards the lowest energy structures that apply in the absence of applied stress. Note that E strain cannot be reduced exactly to zero in the film as a whole because some stress will be generated at the boundaries of the thermal spike volume and by residual defects. In the case of delamination, the externally applied stresses and hence E strain are removed post-deposition by expansion of the material at ordinary temperatures. This does not produce changes in E bonds or E topol because at room temperature the number of bonds and topology are fixed. In the case of flexible substrates, the stress is reduced by a plastic deformation of the substrate without changes to the bonding and network topology. The result in both cases is that the E strain term becomes zero, minimizing the system’s energy. In contrast to the case of the thermal spike there is insufficient vibrational energy to cause any reconfigurations of bonding and therefore E bond and E topol remain unaffected while the film expands in the lateral direction. In the case of a
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flexible substrate, the film’s expansion is evidenced by the development of ripples [26] on the surface caused by expansion of the film’s lateral area. In the case of annealing a film on the substrate after deposition, some vibrational energy is given to all of the structure simultaneously, allowing small changes in E topol and E bond to occur. There are constraints as to the accessible states of the system imposed by the presence of the substrate and by the energy barriers. Allowable changes in structure are those with energy barriers lower than the energy of the thermal vibrations at the annealing temperature. Changes in E bond may cause changes in E topol. Also due to the lateral area constraint of attachment to the substrate, changes in E bond and/or E topol may cause changes in E strain. The system may now trade off increases in any of the three terms for larger energy reductions associated with one or more of the others. Therefore annealing may actually result in stress increases in some cases.
7. Conclusions A new model for stress generation and relief processes occurring during growth with energetic ions has been developed and compared with experimental data from a wide variety of materials. This model has been found to consistently fit the data better than the widely used model of Davis. Post-deposition stress relief by energetic ion impacts, by deposition of material onto flexible substrates and by annealing have been reviewed and the results interpreted in terms of a minimization of the total system energy under constraints. One of the outcomes of energy minimization is that annealing, can create stress whilst minimizing the total energy through structural change. In the case of stress relief with highly energetic ion impacts, the model predicts that the phases and preferred orientations observed will be those which minimise the bonding and topological energy since strain can be removed by upward flow of the material. Thus high energy ions relieve stress and produce phases and orientations favoured in low stress conditions, while flexible substrates relieve stress but retain as-deposited phases.
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