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Decay of surface modulations in polycrystalline thin films
Acta Materialia 53 (2005) 629–636 www.actamat-journals.com
Decay of surface modulations in polycrystalline thin films E. Rabkin *, L. Klinger Department of Materials Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel Received 1 July 2004; received in revised form 6 October 2004; accepted 10 October 2004 Available online 11 November 2004
Abstract We considered the flattening of the perturbed surface of a thin stress-free polycrystalline film with columnar microstructure deposited on rigid substrate. We show that the mass transport along the film/substrate interface and along the grain boundaries significantly contributes to the overall rate of surface flattening of the film. The diffusion along the film/substrate interface and along the grain boundaries is driven by the capillary stresses in the film. Using the approximation of small surface slopes, we calculated the distribution of capillary stresses in the film, and derived an explicit expression for the temporal behavior of the film topography. The initial distribution of the capillary stresses rapidly relaxes to the steady-state one that does not allow the accumulation of bending strain in the film. For the films with passivated or contaminated surfaces exhibiting reduced surface diffusivity the interfacial and grain boundary diffusion play a leading role in kinetics of surface flattening. The flattening process can be accelerated in this case by several orders of magnitude. The results of our work could be helpful in the design of thin films and coatings with enhanced selfhealing capabilities. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Thin films; Surfaces and interfaces; Interface diffusion; Theory
1. Introduction The flat isotropic surface of a solid tends to regain its flatness after an artificial perturbation is imposed on it. The driving force for this shape restoration is the thrust of the system to decrease its excess surface energy. For the case in which surface self-diffusion is responsible for the mass transfer between the surface sites of convex and concave curvature, Mullins [1] derived an expression that describes the time evolution of a sinusoidal surface perturbation with the wave number x = 2p/k along the X-direction, k being the perturbation wavelength, in the approximation of small surface slopes: H ðX ; tÞ ¼ H ðX ; 0Þ exp½Bs x4 t sinðxX Þ;
*
ð1Þ
Corresponding author. Tel.: +972 482 945 79; fax: +972 482 956
77. E-mail address: [email protected] (E. Rabkin).
where H and t are the height of the perturbation profile and the elapsed time, respectively. Bs is the Mullins coefficient that for isotropic surfaces is defined according to Bs ¼
cV a dDs ; kT
ð2Þ
where c, Va, d and Ds are the isotropic surface energy, atomic volume, the thickness of the surface layer in which the diffusion process proceeds and the surface self-diffusion coefficient, respectively. kT has its usual meaning. Eq. (1) demonstrates that sinusoidal perturbation remains sinusoidal with an amplitude that decays exponentially with time according to the time constant 1/(Bsx4). Mullins also analyzed the decay of sinusoid for transport processes other than surface diffusion, such as volume diffusion in solid or in vapor phase or evaporation/condensation mechanism [1]. It was shown that with the decreasing wavelength of the perturbation the surface diffusion becomes a dominating process. The
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.10.016
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temporal evolution of perturbation profile of arbitrary initial shape can be obtained by integrating Eq. (1) over the initial Fourier components of the profile. For decades, Eq. (1) was successfully used for determining of surface diffusivities [2] and surface energies [3] of solids. In thin polycrystalline films there are two main factors that complicate the outlined above description of surface smoothening given by the classical Mullins model: (1) elastic stresses inherently present in the films; (2) grain boundaries (GBs) and film/substrate interface that represent additional avenues for mass transport in the film. A great deal of literature is dedicated to the shape evolution of the surfaces of stressed films. Under certain circumstances high compressive stresses in the initially flat film can relax by formation of bulges that later evolve into an ordered array of small particles (quantum dots) [4]. However, to our knowledge, no attention has been paid so far to the possible role of self-diffusion along the GBs and film/substrate interface in the surface shape evolution of thin films. The reason for this may be a general belief that surface diffusion is faster than any other type of short-circuit diffusion [5] and thus diffusion along the GBs and film/substrate interface provides only a negligible contribution to the overall mass flux. In our opinion, this is not always the case. Firstly, the difference in diffusivities is not so high: according to the correlation compiled in [5] the surface self-diffusion in face centered cubic (fcc) metals at the temperature T = 0.3Tm (here Tm is the melting temperature; T = 0.3Tm corresponds roughly to the room temperature for Al) is only by about two orders of magnitude faster than GB selfdiffusion. The self-diffusion of metal along the film/substrate interface may be faster than along the GBs and comparable with surface self-diffusion, especially when the film and substrate are made of dissimilar materials (i.e. Cu/TiN). Secondly, in ambient conditions the surfaces of metals are passivated or absorb impurities from the environment. This can drastically decrease surface self-diffusion because the impurities or oxide particles are usually located at the surface kinks/ledges which are most active in surface diffusion. At the same time the GBs and film/substrate interface that were formed during ‘‘clean’’ film deposition process are well protected from the oxygen and other impurities since they are hidden inside the material. An indirect evidence that diffusional drift is faster along GBs and film/substrate interface than along free surface has been obtained during electromigration studies of unpassivated Cu interconnects [6]. In this work we will consider the effect of GB and interfacial diffusion on kinetics of surface corrugation decay in thin stress-free polycrystalline films. This is the first in the series of works in which the effect of
additional short-circuit diffusion paths on surface kinetics is considered.
2. The model and results Let us consider the following model (Fig. 1): the surface of a thin film of a nominal thickness H0 is perturbed sinusoidally with the amplitude DH: H ðX ; tÞ ¼ H 0 þ DH ðtÞ sinðxX Þ:
ð3Þ
The film is supposed to have a one-dimensional columnar grain structure with the grain size d. The corrugation described by Eq. (3) induces surface diffusion fluxes leaving the sites of convex curvature and arriving at the sites with concave curvature (Fig. 1). Detailed consideration of these fluxes was given by Mullins [1] and it leads to the exponential decay of the perturbation described by Eq. (1). In our model, the corrugated surface induces capillary stresses in the film that have a normal component, ryy(X,Y = 0) at the film/substrate interface. Furthermore, we will assume that film/substrate interface is incoherent and can absorb or emit vacancies easily. Under such conditions the normal stress at the interface changes the chemical potential of interfacial atoms by Varyy(X,Y = 0) [7]. The x-variations of this addition to the chemical potential cause the diffusion flux along the interface, in full analogy with the diffusion flux along the free surface. The divergence of this interfacial flux is an additional factor contributing to the changes of the film surface profile. However, in contrast with the surface flux, the divergence of interfacial flux not only changes the film height but also induces continuous bending strain of the film. This bending strain can be compensated for by the GB diffusion fluxes that transport the film material from the film/substrate interface to the free surface and vice versa. In summary, contrary to the Mullins model [2] in which only the surface diffusion was taken into account, in our model the change of film topography is the result of concerted action of three diffusion paths: diffusion along the surface and the film/substrate interface contribute directly to the change of the film topography, while the interfacial diffusion additionally introduces bending strain in the film. This bending strain is compensated for by GB diffusion
Fig. 1. Schematic presentation of the model. The capillary stresses in the film induce self-diffusion fluxes along the film/substrate interface and along the grain boundaries.
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which otherwise does not contribute directly to topography change (see Fig. 1). It is obvious that the bending strain, U, accumulated in the film in the vicinity of film/substrate interface should have the same periodicity as initial surface perturbation described by Eq. (3): U ðX ; tÞ ¼ DU ðtÞ sinðxX Þ:
ð4Þ
Again, the physical reason for the bending strain U is the divergence of the diffusion flux along film/substrate interface. For example, in the case of positive flux divergence at a certain location at the interface, the excess atoms will leave the interfacial layer and join the adjacent bulk layers of the film, thus causing a local lifting of the upper regions of the film and corresponding bending strain in the film. For further analysis it is convenient to define the dimensionless co-ordinates according to: x ¼ xX ; y ¼ xY ; h ¼ xH ; h0 ¼ xH 0 ; g ¼ xDU ; u ¼ xU ; e ¼ xDH :
ð5Þ
In these co-ordinates, Eqs. (3) and (4) are rewritten in the form hðx; tÞ ¼ h0 þ eðtÞ sinðxÞ;
ð6Þ
uðx; tÞ ¼ gðtÞ sinðxÞ:
ð7Þ
It is assumed that e, g 1, which means that topography perturbations and internal bending strain in the film are much smaller than the perturbation wavelength, and that the approximation of small surface slopes [1] can be used. The curvature of the film free surfaces generates the capillary stresses there, r(s), which in the small slope approximation can be written as ðsÞ
r
ryy ðx; y ¼ h0 Þ ¼ xce sinðxÞ:
ð8Þ
These stresses are transmitted to the film/substrate interface where they, together with the film bending, induce the normal stress r(i): rðiÞ ryy ðx; y ¼ 0Þ ¼ xc½ef þ gg sinðxÞ:
ð9Þ
Here f = f(h0) is the stress transmission coefficient describing the fraction of the normal stress applied to the film surface that is transmitted to the film/substrate interface. The second term in parenthesis in the right hand side (RHS) of Eq. (9) is the dimensionless form of an appropriate elastic modulus, G, describing the relationship between the film bending strain in the vicinity of film/substrate interface and the normal stress at the interface caused by this strain. For the films that are not too thin (see below) G g¼ 1: ð10Þ xc Stress transmission coefficient f and elastic modulus G for elastically isotropic film are calculated in Appendix A. Their dependence on dimensionless film thickness, h, is shown in Fig. 2. As expected, f decays rapidly with
Fig. 2. The dependence of stress transmission coefficient, f, and of normalized effective elastic modulus, G, on dimensionless film thickness h. Both f and G are calculated for m = 0.33.
increasing film thickness, since the capillary stresses induced by sinusoidal perturbation at the surface of the film can be approximately described by an array of force dipoles with the distance of 2p/x between the oppositely applied forces. These capillary stresses will not be felt at the film/substrate interface if the film is much thicker than the dipole size. The diffusion fluxes along the film surface, I(s), along the film/substrate interface, I(i), and along the GBs, I(b), can be written down using the Nernst–Einstein relationships for atomic mobilities (all fluxes are given in the units at m1 s1, i.e. per 1 m of the film): I ðsÞ ¼
xdDs orðsÞ ; kT ox
ð11aÞ
I ðiÞ ¼
xdDi orðiÞ ; kT ox
ð11bÞ
xdDb rðsÞ rðiÞ ; kT h0
ð11cÞ
I ðbÞ
where Di and Db are the self-diffusion coefficients along the film/substrate interface and along the GBs, respectively. For the sake of simplicity we assumed that diffusion thicknesses of the film surface, of the film/substrate interface and of GBs are identical. Eq. (11c) is an approximate one in which y-dependent GB flux was replaced by its averaged value. The accuracy of this approximation increases with decreasing film thickness. The kinetics of surface topography and bending strain evolution can be then written in the following form: ðiÞ og I ðbÞ 2 oI sinðxÞ ¼ V a x þ ; ð12aÞ ot ox r ðsÞ oe og oI I ðbÞ sinðxÞ ¼ f sinðxÞ V a x2 ; ot ot ox r
ð12bÞ
where r = xR, R being the grain size in the film. The first term in the RHS of Eq. (12b) describes transmission of the lattice bending strain in the vicinity of film/substrate interface to the film surface. It can be shown (see
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Appendix A) that the fraction of transmitted strain is described by the same ‘‘stress transmission coefficient’’ f that was introduced in Eq. (9): uy ðx; y ¼ hÞ ¼ f gðtÞ sinðxÞ;
ð13Þ
where u is the strain vector. A combination of Eqs. (8), (9), (11) and (12) yields: og ¼ bi ðef þ ggÞ þ bb ðe ef ggÞ; ot oe ¼ bs e bi f ðef þ ggÞ ot þ bb ð1 f Þðe ef ggÞ;
ð14aÞ
ð14bÞ
where bs ¼
cV a dDs 4 x; kT
ð15aÞ
bi ¼
cV a dDi 4 x; kT
ð15bÞ
bb ¼
cV a dDb 2 x: kTH 0 R
ð15cÞ
Eq. (14) is solved in Appendix B. Under the assumption that condition (10) is fulfilled the solution is: 2
e ¼ e0 expðk 2 tÞ
ð1 f Þ b2b f 2 b2i g ð bi þ bb Þ
2
e0 exp ðk 1 tÞ; ð16aÞ
g ¼ e0
ð1 f Þbb fbi ½expðk 2 tÞ exp ðk 1 tÞ; g ð bi þ bb Þ
ð16bÞ
where k 1 ¼ gðbi þ bb Þ; k 2 ¼ bs þ
bi bb : bi þ bb
ð17aÞ ð17bÞ
Under the assumption that condition (10) is valid (Fig. 2 demonstrates that this is not the case for extremely thin films) k1 k2. Therefore, for long annealing times (t 1/k1 ) only the first exponent in Eq. (16) will be significant (steady-state regime): est ¼ e0 expðk 2 tÞ; ð1 f Þbb fbi expðk 2 tÞ: gst ¼ e0 g ð bi þ bb Þ
ð18aÞ ð18bÞ
Eqs. (16)–(18) represent the central result of this work.
stage the second exponent in the RHS of Eqs. (16) plays a significant role and the bending strain is intensively accumulated in the film. During the following steadystate stage of the process the first exponent in the RHS of Eqs. (16) is the dominant one and the bending strain follows in its temporal behavior the slowly decaying amplitude e. The corresponding solution given by Eqs. (18) can be obtained directly from Eqs. (12) by assuming og/ot 0. It should be noted, however, that for small values of reduced modulus g, i.e. for very thin films or for very long perturbation wavelengths (see Fig. 2) the initial, transient stage of the process may be rather long. After the steady-state stage of the process is achieved, the temporal evolution of both e and g is determined by exp(k2t) [see Eqs. (18)]. For convenience of further analysis we will rewrite k2 [Eq. (17b)] in the following form: cV a dDs 4 w x 1þ k2 ¼ ; ð19Þ 1 þ nH 0 Rx2 kT where n = Di/Db and w = Di/Ds are dimensionless relative diffusivities. As discussed in Section 1, the upper limit for Di in the case of interface between two dissimilar materials is Ds, the latter being higher than Db by approximately two orders of magnitude at the temperature 0.3Tm. Therefore, n 6 102 at this temperature. For estimating the range of possible w values, we will consider two extreme situations: (i) Clean surface without passivation and free from absorbed impurities. This situation can be realized under ultra high vacuum (UHV) conditions after a series of annealing and sputtering cycles aimed to clean the film from residual impurities with the strong surface segregation tendency. In this case, w 6 1. (ii) ‘‘Dirty’’ or passivated surface. This is the common situation under ambient conditions, and w 1 because of blocking of surface diffusion paths by impurities. Let us now define the critical wavelength of surface perturbation, k*, by k ¼ 2pðnH 0 RÞ
Eqs. (16) demonstrate that there are two stages in temporal evolution of the amplitude of sinusoid perturbation during annealing: during the initial, transient
:
ð20Þ
For the long perturbation wavelengths, k k*, the second term in parenthesis in the RHS of Eq. (19) can be approximated by w, and k2
3. Discussion
1=2
cV a dDs 4 x ð1 þ wÞ: kT
ð21Þ
Eq. (21) demonstrates that the surface evolution of sinusoid perturbation is determined in this case by the classical Mullins-like law (1), albeit with the acceleration factor 1 + w in the time exponent. This acceleration fac-
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tor varies from approximately 2 for ‘‘clean’’ surfaces to approximately w 1 for passivated or ‘‘dirty’’ surfaces. In the latter case the acceleration of film flattening is governed by self-diffusion along film/substrate interface and it can be significant. Let us now consider the surface perturbations with the short wavelengths defined by k k*. For clean surfaces the second term in parenthesis in the RHS of Eq. (19) can be neglected in this case and we arrive at the classical Mullins law (1). No acceleration of film flattening will be observed. For ‘‘dirty’’ surfaces, however, second term in parenthesis in the RHS of Eq. (19) can be of the order of one. In this case the explicit expression for k2 will be cV a d Db 2 Ds x4 þ x : k2 ð22Þ kT H 0R The corresponding kinetic law defined by Eqs. (18) and (22) resembles the Mullins-like law under the parallel action of surface diffusion and evaporation–condensation mechanisms [1]. The kinetics of surface flattening is accelerated if compared with the situation in which surface diffusion alone is active; the acceleration being determined by GB diffusivity and by thickness and microstructure of the film. It is interesting to note that diffusion along film/substrate interface does not play any role in flattening kinetics in this case. It follows from the above discussion that in the case of passivated or ‘‘dirty’’ surfaces the acceleration of surface smoothening can be observed for all perturbation wavelengths, though for the long wavelengths k k* the acceleration is more significant than for short ones. Therefore, a thin film with mechanically damaged or perturbed surface exhibits a profound capability for flattening, or self-healing of initial damage, even in the case of slow surface self-diffusion. Eqs. (18) and (19) describe the kinetics of flattening and can serve as a basis for design of thin films with enhanced self-healing capability. It is instructive to estimate the relative durations of the transient and steady state stages of the flattening and to get a feeling for the characteristic self-healing times at ambient conditions. It should be noted that for relatively thick films or short perturbation wavelength (h P 1) the transient regime of surface topography evolution is extremely short and virtually unobservable. This is related with the high value of Ek/c ratio [see Eqs. (10) and (A1.25)], E being YoungÕs modulus. This ratio determines the smallness of the second exponent responsible for the transient stage in the RHSs of Eqs. (16). Indeed, for the typical metals E 1011 Pa, c 1 J/m2 and Ek/c 1. However, the situation may be different for ultra-thin films in which the effective elastic modulus G can be rather small (see Fig. 2). Let us assume, for the sake of a rough estimate, bi bs, bb (‘‘dirty’’ surface approximation). Using Eqs. (16a), (17) and (A1.25), assuming R H0 (this is not
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uncommon for columnar thin film microstructures), H0 10 nm, m 0.33, n 100 and assuming long perturbation wavelength of k = 50H0 = 500 nm (this corresponds to the periodicity of photolithography pattern easily achievable in microelectronics industry) we arrive at the following expression for the bending strain g accumulated in the film: g
e0 f ½exp ðsÞ exp ð9:3sÞ: g
ð23Þ
where s is the dimensionless time defined according to s = bbt. The dependence g(s) is shown in Fig. 3. It is logical to identify the duration of transient period with the maximal bending of the film, smax 0.27. Assuming Db 1018 m2/s, d 1 nm, T = 300 K [5] and Va 105 m3/mol leads to tmax 11 h, i.e. rather long transient period. The actual flattening of the film can then take up to several days. Finally, it should be noted that in this work we considered only the diffusional path of the relaxation of film surface perturbations. The question arises as to whether the high capillary stresses induced by the curvature of perturbed surface can activate some dislocation activity in the film, thus opening an additional, plasticity path for film flattening. The general answer on this question depends on the details of initial distribution of dislocations and dislocation sources in the film, which is not known a priori. It can be only shown that for thin films that are initially dislocations-free the capillary stresses are too weak to cause the homogeneous nucleation of new dislocations. Indeed, recent nanoindentation experiments show that thin Cu films deposited on Si substrate deform elastically under a sharp diamond indenter until the maximal shear stress under indenter reaches the theoretical shear strength of the film (approximately 12.7 GPa for Cu films) [8]. At this point a spontaneous displacement burst associated with the nucleation of a group of dislocations occurs. Even for a very high surface radius of curvature of, let say, r 10 nm the max-
Fig. 3. The dependence of normalized bending strain accumulated in the film [see Eq. (23)] on dimensionless annealing time for ultrathin film.
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imal amplitude of the capillary stress in the film is about 0.1 GPa, which is by two orders of magnitude lower than the stress needed for dislocations nucleation. Moreover, as it can be seen from Eq. (A1.18) the shear stresses in sufficiently thin (h 1) films scale with he which means that the shear stresses in the film induced by capillary forces are actually much lower than the above estimate of 0.1 GPa. It can be concluded that the capillary-induced shear stresses in the films are by far too low to cause the homogeneous nucleation of dislocations. Therefore, at low homological temperatures at which evaporation of the film material and bulk diffusion in the film are insignificant, the interfacial diffusion considered in the present work provides the only path for surface shape evolution of the dislocation-free films.
4. Conclusions We demonstrated that the temporal evolution of sinusoidal surface perturbation of polycrystalline thin film deposited on rigid substrate is governed by a concerted action of mass transport along the free surface, grain boundaries, and film/substrate interface. The grain boundary and interfacial diffusion are driven by the inhomogeneous distribution of capillary stresses in the film. The explicit expressions describing time dependencies of the perturbation amplitude and of the bending strain in the film were obtained. From their analysis the following conclusions can be drawn: 1. During a very short initial period the bending strain is intensively accumulated in the film. Later, the strain increase stops and a steady-state process of the exponential decay of both surface perturbation and of bending strain begins. 2. The acceleration of surface healing kinetics due to alternative diffusion paths can be very significant in the case of contaminated or passivated surfaces with reduced surface diffusivity. 3. The surface topography evolves differently for short and long perturbation wavelengths, which are defined with respect to critical perturbation wavelength, k*. For long wavelengths, k k*, the surface kinetics follows a classical MullinsÕ model [1], albeit with the acceleration factor in the time exponent which is proportional to the ratio of self-diffusion coefficients along the film/substrate interface and along the free surface. For short wavelength, k k*, the surface kinetics follows a mixed law with the sum of two terms in the time exponent, the first one being proportional to the fourth power of the wave vector and the second one to its second power. In this respect, the obtained kinetic law resembles the Mullins-like law for the case of parallel action of surface diffusion and evaporation-condensation mechanisms
[1]. The critical wavelength k* depends on the ratio of interfacial and grain boundary diffusivities, on the film thickness and on the grain size in the film. 4. Numerical estimates show that for typical metallic films significant flattening can occur due to grain boundary and interfacial diffusion already at room temperature during time period of several days. For ultrathin films, the initial, transient period of surface evolution can take up to several hours.
Appendix A For calculating of stress transmission coefficient f in the small slope approximation we will consider the following elastic problem: to the free surface of the film of thickness h the distributed normal force (capillary stress) of the following amplitude is applied: ryy ðx; hÞ ¼ xce sinðxÞ;
ðA1:1Þ
rxy ðx; hÞ ¼ 0:
ðA1:2Þ
Since the film is deposited on rigid substrate the following conditions should be fulfilled at the film/substrate interface: ux ðx; 0Þ ¼ 0;
ðA1:3Þ
uy ðx; 0Þ ¼ 0:
ðA1:4Þ
The distribution of stresses in the film can be found with the aid of biharmonic stress function U(x,y): Uðx; yÞ ¼ F ðyÞ sinðxÞ
ðA1:5Þ
with F ðyÞ ¼ ðA1 þ B1 y Þ expðyÞ þ ðA2 þ B2 y Þ expðyÞ:
ðA1:6Þ
The constants A1, A2, B1 and B2 can be determined using the boundary conditions of the problem (see below). The components of the stress tensor can be calculated using the stress function U: rxx ¼
o2 U d2 F ¼ 2 sinðxÞ; oy 2 dy
o2 U ¼ F sinðxÞ; ox2 o2 U dF ¼ cosðxÞ; rxy ¼ oxoy dy rzz ¼ m rxx þ ryy ; rxz ¼ ryz ¼ 0; ryy ¼
ðA1:7Þ
where m is PoissonÕs ratio. The components of strain tensor can be found with the aid of HookeÕs law: 1 þ m d2 F uxx ¼ ð1 mÞ þ F m sinðxÞ; ðA1:8Þ E dy 2
E. Rabkin, L. Klinger / Acta Materialia 53 (2005) 629–636
uxy ¼
1 þ m dF cosðxÞ; E dy
ðA1:9Þ
where E is YoungÕs modulus. The components of strain vector u can be determined by the integration of appropriate components of the strain tensor: 1 þ m d2 F ux ¼ ð 1 m Þ þ F m cosðxÞ; ðA1:10Þ E dy 2
1 þ m dF dF ð2 mÞ 3 ð1 mÞ sinðxÞ: uy ¼ E dy dy
The problem of transmission of the bending strain in the vicinity of film/substrate interface to the film surface can be treated in a similar way. The corresponding elastic problem can be formulated in a following way: ux ðx; 0Þ ¼ 0;
ðA1:19Þ
uy ðx; 0Þ ¼ g sinðxÞ;
ðA1:20Þ
ryy ðx; hÞ ¼ 0;
ðA1:21Þ
rxy ðx; hÞ ¼ 0:
ðA1:22Þ
3
ðA1:11Þ
Eqs. (A1.7), (A1.10) and (A1.11) determine the stress and strain fields everywhere in the film, including film surface (y = h) and film/substrate interface (y = 0). For determining the constants A1, A2, B1 and B2 in the stress function, Eqs. (A1.7), (A1.10) and (A1.11) should be substituted in boundary conditions (A1.1)–(A1.6): F ðhÞ ¼ xce; dF ¼ 0; dy y¼h
ðA1:12Þ
The latter two conditions describe constraint-free external surface. It can be shown that y-displacement of the film surface is described in this case by uy ðx; hÞ ¼ gf sinðxÞ;
ðA1:13Þ
d F ð1 mÞ þ F ð0Þm ¼ 0; dy 2 y¼0
ðA1:14Þ
dF d3 F ð2 m Þ 3 dy y¼0 dy
ðA1:15Þ
ð1 mÞ ¼ 0:
Substituting in these equations the definition of F [see Eq. (A1.6)] gives the system of four linear equations for determining of the constants A1, A2, B1 and B2. We will omit here these simple, but rather cumbersome calculations. Once these coefficients are found the stress tensor anywhere in the film can be determined with the aid of Eqs. (A1.7). In particular, at the film/substrate interface (y=0) we get
E ð 1 mÞ sinh ð2hÞ 2h : 2 ð1 þ mÞ 4ð1 mÞ þ ð3 4mÞsinh2 ðhÞ þ h2 ðA1:25Þ
It should be noted that the behavior of G h) is different from that of f h) (see also Fig. 2): G(h ! 0) ! 0 (µh3), and G (h ! 1) ! Gm, where Gm ¼
2Eð1 mÞ : ð1 þ mÞð3 4mÞ
ðA1:26Þ
The other two components of the stress tensor at the film/substrate interface are: m ryy ðx; 0Þ rxx ðx; 0Þ ¼ ðA1:27Þ 1m and rxy ðx; 0Þ ¼
ryy ðx; 0Þ ¼ xcef sinðxÞ;
ðA1:24Þ
where G¼
y¼0
ðA1:23Þ
where the coefficient f is determined by Eq. (A1.16). The component ryy of the stress tensor at the film/substrate interface corresponding to the bending strain (A1.20) is ryy ðx; 0Þ ¼ Gg sinðxÞ;
2
E h2 þ ð1 2mÞsinh2 ðhÞ g cosðxÞ: 1 þ m 4ð1 mÞ2 þ ð3 4mÞsinh2 ðhÞ þ h2 ðA1:28Þ
where f ¼
635
2ð1 mÞ½2ð1 mÞ coshðhÞ þ h sinhðhÞ 4ð1 mÞ2 þ ð3 4mÞsinh2 ðhÞ þ h2
ðA1:16Þ Appendix B
is the stress transmission coefficient we were searching for. As expected, f(h ! 0) ! 1 and f(h ! 1) ! 0. The other two components of the stress tensor at the film/ substrate interface are: rxx ðx; 0Þ ¼ rxy ðx; 0Þ ¼
m ryy ðx; 0Þ; 1m
ðA1:17Þ
2ð1 mÞð1 2mÞ½ð1 2mÞ sinhðhÞ þ h coshðhÞ 2
4ð1 mÞ þ ð3 4mÞsinh2 ðhÞ þ h2 xce cosðxÞ:
ðA1:18Þ
The system of two linear differential equations (14a) and (14b) can be rewritten in the compact form og ¼ a11 g þ a12 e; ot
ðB1:1Þ
oe ¼ a21 g þ a22 e ot
ðB1:2Þ
with a11 ¼ gðbi þ bb Þ;
ðB1:3Þ
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a12 ¼ bi f þ bb ð1 f Þ;
ðB1:4Þ
a21 ¼ g½bi f þ bb ð1 f Þ;
ðB1:5Þ 2
a22 ¼ bs bi f 2 þ bb ð1 f Þ :
ðB1:6Þ
The solution of the system of linear differential equations (B1.1) and (B1.2) with the initial conditions g(0) = 0 and e(0) = e0 is: a12 g ¼ e0 ½exp ðk 2 tÞ exp ðk 1 tÞ; ðB1:7Þ k1 k2 1 k1 k2 ½ðk 2 þ a21 Þ expðk 2 tÞ þ ðk 1 þ a21 Þ exp ðk 1 tÞ;
e ¼ e0
ðB1:8Þ where k1 and k2 are the roots of characteristic equation k 2 þ ða11 þ a22 Þk þ ða11 a22 a12 a21 Þ ¼ 0:
ðB1:9Þ
The expressions (B1.7) and (B1.8), as well as solutions of Eq. (B1.9) can be simplified assuming g 1, which is the case for all films that are not too thin. Eqs. (16) and (17) of this paper represent such a simplification. References [1] Mullins WW. J Appl Phys 1959;30:77. [2] Mehrer H, editor. Diffusion in solid metals and alloys. Landolt– Bo¨rnstein new series, vol. III/26. Berlin: Springer; 1990. [3] Mills B, Leak GM. Acta Metall 1968;16:303. [4] Freund LB, Suresh S. Thin film materials. Cambridge: Cambridge University Press; 2003. [5] Kaur I, Mishin Y, Gust W. Fundamentals of grain and interphase boundary diffusion. Chichester: Wiley; 1995. [6] Koetter TG, Wendrock H, Schuehrer H, Wenzel C, Wetzig K. Microelectron Reliability 2000;40:1295. [7] Herring C. J Appl Phys 1950;21:437. [8] Suresh S, Nieh TG, Choi BW. Scr Mater 1999;41:951.
Decay of surface modulations in polycrystalline thin films E. Rabkin *, L. Klinger Department of Materials Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel Received 1 July 2004; received in revised form 6 October 2004; accepted 10 October 2004 Available online 11 November 2004
Abstract We considered the flattening of the perturbed surface of a thin stress-free polycrystalline film with columnar microstructure deposited on rigid substrate. We show that the mass transport along the film/substrate interface and along the grain boundaries significantly contributes to the overall rate of surface flattening of the film. The diffusion along the film/substrate interface and along the grain boundaries is driven by the capillary stresses in the film. Using the approximation of small surface slopes, we calculated the distribution of capillary stresses in the film, and derived an explicit expression for the temporal behavior of the film topography. The initial distribution of the capillary stresses rapidly relaxes to the steady-state one that does not allow the accumulation of bending strain in the film. For the films with passivated or contaminated surfaces exhibiting reduced surface diffusivity the interfacial and grain boundary diffusion play a leading role in kinetics of surface flattening. The flattening process can be accelerated in this case by several orders of magnitude. The results of our work could be helpful in the design of thin films and coatings with enhanced selfhealing capabilities. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Thin films; Surfaces and interfaces; Interface diffusion; Theory
1. Introduction The flat isotropic surface of a solid tends to regain its flatness after an artificial perturbation is imposed on it. The driving force for this shape restoration is the thrust of the system to decrease its excess surface energy. For the case in which surface self-diffusion is responsible for the mass transfer between the surface sites of convex and concave curvature, Mullins [1] derived an expression that describes the time evolution of a sinusoidal surface perturbation with the wave number x = 2p/k along the X-direction, k being the perturbation wavelength, in the approximation of small surface slopes: H ðX ; tÞ ¼ H ðX ; 0Þ exp½Bs x4 t sinðxX Þ;
*
ð1Þ
Corresponding author. Tel.: +972 482 945 79; fax: +972 482 956
77. E-mail address: [email protected] (E. Rabkin).
where H and t are the height of the perturbation profile and the elapsed time, respectively. Bs is the Mullins coefficient that for isotropic surfaces is defined according to Bs ¼
cV a dDs ; kT
ð2Þ
where c, Va, d and Ds are the isotropic surface energy, atomic volume, the thickness of the surface layer in which the diffusion process proceeds and the surface self-diffusion coefficient, respectively. kT has its usual meaning. Eq. (1) demonstrates that sinusoidal perturbation remains sinusoidal with an amplitude that decays exponentially with time according to the time constant 1/(Bsx4). Mullins also analyzed the decay of sinusoid for transport processes other than surface diffusion, such as volume diffusion in solid or in vapor phase or evaporation/condensation mechanism [1]. It was shown that with the decreasing wavelength of the perturbation the surface diffusion becomes a dominating process. The
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.10.016
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temporal evolution of perturbation profile of arbitrary initial shape can be obtained by integrating Eq. (1) over the initial Fourier components of the profile. For decades, Eq. (1) was successfully used for determining of surface diffusivities [2] and surface energies [3] of solids. In thin polycrystalline films there are two main factors that complicate the outlined above description of surface smoothening given by the classical Mullins model: (1) elastic stresses inherently present in the films; (2) grain boundaries (GBs) and film/substrate interface that represent additional avenues for mass transport in the film. A great deal of literature is dedicated to the shape evolution of the surfaces of stressed films. Under certain circumstances high compressive stresses in the initially flat film can relax by formation of bulges that later evolve into an ordered array of small particles (quantum dots) [4]. However, to our knowledge, no attention has been paid so far to the possible role of self-diffusion along the GBs and film/substrate interface in the surface shape evolution of thin films. The reason for this may be a general belief that surface diffusion is faster than any other type of short-circuit diffusion [5] and thus diffusion along the GBs and film/substrate interface provides only a negligible contribution to the overall mass flux. In our opinion, this is not always the case. Firstly, the difference in diffusivities is not so high: according to the correlation compiled in [5] the surface self-diffusion in face centered cubic (fcc) metals at the temperature T = 0.3Tm (here Tm is the melting temperature; T = 0.3Tm corresponds roughly to the room temperature for Al) is only by about two orders of magnitude faster than GB selfdiffusion. The self-diffusion of metal along the film/substrate interface may be faster than along the GBs and comparable with surface self-diffusion, especially when the film and substrate are made of dissimilar materials (i.e. Cu/TiN). Secondly, in ambient conditions the surfaces of metals are passivated or absorb impurities from the environment. This can drastically decrease surface self-diffusion because the impurities or oxide particles are usually located at the surface kinks/ledges which are most active in surface diffusion. At the same time the GBs and film/substrate interface that were formed during ‘‘clean’’ film deposition process are well protected from the oxygen and other impurities since they are hidden inside the material. An indirect evidence that diffusional drift is faster along GBs and film/substrate interface than along free surface has been obtained during electromigration studies of unpassivated Cu interconnects [6]. In this work we will consider the effect of GB and interfacial diffusion on kinetics of surface corrugation decay in thin stress-free polycrystalline films. This is the first in the series of works in which the effect of
additional short-circuit diffusion paths on surface kinetics is considered.
2. The model and results Let us consider the following model (Fig. 1): the surface of a thin film of a nominal thickness H0 is perturbed sinusoidally with the amplitude DH: H ðX ; tÞ ¼ H 0 þ DH ðtÞ sinðxX Þ:
ð3Þ
The film is supposed to have a one-dimensional columnar grain structure with the grain size d. The corrugation described by Eq. (3) induces surface diffusion fluxes leaving the sites of convex curvature and arriving at the sites with concave curvature (Fig. 1). Detailed consideration of these fluxes was given by Mullins [1] and it leads to the exponential decay of the perturbation described by Eq. (1). In our model, the corrugated surface induces capillary stresses in the film that have a normal component, ryy(X,Y = 0) at the film/substrate interface. Furthermore, we will assume that film/substrate interface is incoherent and can absorb or emit vacancies easily. Under such conditions the normal stress at the interface changes the chemical potential of interfacial atoms by Varyy(X,Y = 0) [7]. The x-variations of this addition to the chemical potential cause the diffusion flux along the interface, in full analogy with the diffusion flux along the free surface. The divergence of this interfacial flux is an additional factor contributing to the changes of the film surface profile. However, in contrast with the surface flux, the divergence of interfacial flux not only changes the film height but also induces continuous bending strain of the film. This bending strain can be compensated for by the GB diffusion fluxes that transport the film material from the film/substrate interface to the free surface and vice versa. In summary, contrary to the Mullins model [2] in which only the surface diffusion was taken into account, in our model the change of film topography is the result of concerted action of three diffusion paths: diffusion along the surface and the film/substrate interface contribute directly to the change of the film topography, while the interfacial diffusion additionally introduces bending strain in the film. This bending strain is compensated for by GB diffusion
Fig. 1. Schematic presentation of the model. The capillary stresses in the film induce self-diffusion fluxes along the film/substrate interface and along the grain boundaries.
E. Rabkin, L. Klinger / Acta Materialia 53 (2005) 629–636
631
which otherwise does not contribute directly to topography change (see Fig. 1). It is obvious that the bending strain, U, accumulated in the film in the vicinity of film/substrate interface should have the same periodicity as initial surface perturbation described by Eq. (3): U ðX ; tÞ ¼ DU ðtÞ sinðxX Þ:
ð4Þ
Again, the physical reason for the bending strain U is the divergence of the diffusion flux along film/substrate interface. For example, in the case of positive flux divergence at a certain location at the interface, the excess atoms will leave the interfacial layer and join the adjacent bulk layers of the film, thus causing a local lifting of the upper regions of the film and corresponding bending strain in the film. For further analysis it is convenient to define the dimensionless co-ordinates according to: x ¼ xX ; y ¼ xY ; h ¼ xH ; h0 ¼ xH 0 ; g ¼ xDU ; u ¼ xU ; e ¼ xDH :
ð5Þ
In these co-ordinates, Eqs. (3) and (4) are rewritten in the form hðx; tÞ ¼ h0 þ eðtÞ sinðxÞ;
ð6Þ
uðx; tÞ ¼ gðtÞ sinðxÞ:
ð7Þ
It is assumed that e, g 1, which means that topography perturbations and internal bending strain in the film are much smaller than the perturbation wavelength, and that the approximation of small surface slopes [1] can be used. The curvature of the film free surfaces generates the capillary stresses there, r(s), which in the small slope approximation can be written as ðsÞ
r
ryy ðx; y ¼ h0 Þ ¼ xce sinðxÞ:
ð8Þ
These stresses are transmitted to the film/substrate interface where they, together with the film bending, induce the normal stress r(i): rðiÞ ryy ðx; y ¼ 0Þ ¼ xc½ef þ gg sinðxÞ:
ð9Þ
Here f = f(h0) is the stress transmission coefficient describing the fraction of the normal stress applied to the film surface that is transmitted to the film/substrate interface. The second term in parenthesis in the right hand side (RHS) of Eq. (9) is the dimensionless form of an appropriate elastic modulus, G, describing the relationship between the film bending strain in the vicinity of film/substrate interface and the normal stress at the interface caused by this strain. For the films that are not too thin (see below) G g¼ 1: ð10Þ xc Stress transmission coefficient f and elastic modulus G for elastically isotropic film are calculated in Appendix A. Their dependence on dimensionless film thickness, h, is shown in Fig. 2. As expected, f decays rapidly with
Fig. 2. The dependence of stress transmission coefficient, f, and of normalized effective elastic modulus, G, on dimensionless film thickness h. Both f and G are calculated for m = 0.33.
increasing film thickness, since the capillary stresses induced by sinusoidal perturbation at the surface of the film can be approximately described by an array of force dipoles with the distance of 2p/x between the oppositely applied forces. These capillary stresses will not be felt at the film/substrate interface if the film is much thicker than the dipole size. The diffusion fluxes along the film surface, I(s), along the film/substrate interface, I(i), and along the GBs, I(b), can be written down using the Nernst–Einstein relationships for atomic mobilities (all fluxes are given in the units at m1 s1, i.e. per 1 m of the film): I ðsÞ ¼
xdDs orðsÞ ; kT ox
ð11aÞ
I ðiÞ ¼
xdDi orðiÞ ; kT ox
ð11bÞ
xdDb rðsÞ rðiÞ ; kT h0
ð11cÞ
I ðbÞ
where Di and Db are the self-diffusion coefficients along the film/substrate interface and along the GBs, respectively. For the sake of simplicity we assumed that diffusion thicknesses of the film surface, of the film/substrate interface and of GBs are identical. Eq. (11c) is an approximate one in which y-dependent GB flux was replaced by its averaged value. The accuracy of this approximation increases with decreasing film thickness. The kinetics of surface topography and bending strain evolution can be then written in the following form: ðiÞ og I ðbÞ 2 oI sinðxÞ ¼ V a x þ ; ð12aÞ ot ox r ðsÞ oe og oI I ðbÞ sinðxÞ ¼ f sinðxÞ V a x2 ; ot ot ox r
ð12bÞ
where r = xR, R being the grain size in the film. The first term in the RHS of Eq. (12b) describes transmission of the lattice bending strain in the vicinity of film/substrate interface to the film surface. It can be shown (see
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Appendix A) that the fraction of transmitted strain is described by the same ‘‘stress transmission coefficient’’ f that was introduced in Eq. (9): uy ðx; y ¼ hÞ ¼ f gðtÞ sinðxÞ;
ð13Þ
where u is the strain vector. A combination of Eqs. (8), (9), (11) and (12) yields: og ¼ bi ðef þ ggÞ þ bb ðe ef ggÞ; ot oe ¼ bs e bi f ðef þ ggÞ ot þ bb ð1 f Þðe ef ggÞ;
ð14aÞ
ð14bÞ
where bs ¼
cV a dDs 4 x; kT
ð15aÞ
bi ¼
cV a dDi 4 x; kT
ð15bÞ
bb ¼
cV a dDb 2 x: kTH 0 R
ð15cÞ
Eq. (14) is solved in Appendix B. Under the assumption that condition (10) is fulfilled the solution is: 2
e ¼ e0 expðk 2 tÞ
ð1 f Þ b2b f 2 b2i g ð bi þ bb Þ
2
e0 exp ðk 1 tÞ; ð16aÞ
g ¼ e0
ð1 f Þbb fbi ½expðk 2 tÞ exp ðk 1 tÞ; g ð bi þ bb Þ
ð16bÞ
where k 1 ¼ gðbi þ bb Þ; k 2 ¼ bs þ
bi bb : bi þ bb
ð17aÞ ð17bÞ
Under the assumption that condition (10) is valid (Fig. 2 demonstrates that this is not the case for extremely thin films) k1 k2. Therefore, for long annealing times (t 1/k1 ) only the first exponent in Eq. (16) will be significant (steady-state regime): est ¼ e0 expðk 2 tÞ; ð1 f Þbb fbi expðk 2 tÞ: gst ¼ e0 g ð bi þ bb Þ
ð18aÞ ð18bÞ
Eqs. (16)–(18) represent the central result of this work.
stage the second exponent in the RHS of Eqs. (16) plays a significant role and the bending strain is intensively accumulated in the film. During the following steadystate stage of the process the first exponent in the RHS of Eqs. (16) is the dominant one and the bending strain follows in its temporal behavior the slowly decaying amplitude e. The corresponding solution given by Eqs. (18) can be obtained directly from Eqs. (12) by assuming og/ot 0. It should be noted, however, that for small values of reduced modulus g, i.e. for very thin films or for very long perturbation wavelengths (see Fig. 2) the initial, transient stage of the process may be rather long. After the steady-state stage of the process is achieved, the temporal evolution of both e and g is determined by exp(k2t) [see Eqs. (18)]. For convenience of further analysis we will rewrite k2 [Eq. (17b)] in the following form: cV a dDs 4 w x 1þ k2 ¼ ; ð19Þ 1 þ nH 0 Rx2 kT where n = Di/Db and w = Di/Ds are dimensionless relative diffusivities. As discussed in Section 1, the upper limit for Di in the case of interface between two dissimilar materials is Ds, the latter being higher than Db by approximately two orders of magnitude at the temperature 0.3Tm. Therefore, n 6 102 at this temperature. For estimating the range of possible w values, we will consider two extreme situations: (i) Clean surface without passivation and free from absorbed impurities. This situation can be realized under ultra high vacuum (UHV) conditions after a series of annealing and sputtering cycles aimed to clean the film from residual impurities with the strong surface segregation tendency. In this case, w 6 1. (ii) ‘‘Dirty’’ or passivated surface. This is the common situation under ambient conditions, and w 1 because of blocking of surface diffusion paths by impurities. Let us now define the critical wavelength of surface perturbation, k*, by k ¼ 2pðnH 0 RÞ
Eqs. (16) demonstrate that there are two stages in temporal evolution of the amplitude of sinusoid perturbation during annealing: during the initial, transient
:
ð20Þ
For the long perturbation wavelengths, k k*, the second term in parenthesis in the RHS of Eq. (19) can be approximated by w, and k2
3. Discussion
1=2
cV a dDs 4 x ð1 þ wÞ: kT
ð21Þ
Eq. (21) demonstrates that the surface evolution of sinusoid perturbation is determined in this case by the classical Mullins-like law (1), albeit with the acceleration factor 1 + w in the time exponent. This acceleration fac-
E. Rabkin, L. Klinger / Acta Materialia 53 (2005) 629–636
tor varies from approximately 2 for ‘‘clean’’ surfaces to approximately w 1 for passivated or ‘‘dirty’’ surfaces. In the latter case the acceleration of film flattening is governed by self-diffusion along film/substrate interface and it can be significant. Let us now consider the surface perturbations with the short wavelengths defined by k k*. For clean surfaces the second term in parenthesis in the RHS of Eq. (19) can be neglected in this case and we arrive at the classical Mullins law (1). No acceleration of film flattening will be observed. For ‘‘dirty’’ surfaces, however, second term in parenthesis in the RHS of Eq. (19) can be of the order of one. In this case the explicit expression for k2 will be cV a d Db 2 Ds x4 þ x : k2 ð22Þ kT H 0R The corresponding kinetic law defined by Eqs. (18) and (22) resembles the Mullins-like law under the parallel action of surface diffusion and evaporation–condensation mechanisms [1]. The kinetics of surface flattening is accelerated if compared with the situation in which surface diffusion alone is active; the acceleration being determined by GB diffusivity and by thickness and microstructure of the film. It is interesting to note that diffusion along film/substrate interface does not play any role in flattening kinetics in this case. It follows from the above discussion that in the case of passivated or ‘‘dirty’’ surfaces the acceleration of surface smoothening can be observed for all perturbation wavelengths, though for the long wavelengths k k* the acceleration is more significant than for short ones. Therefore, a thin film with mechanically damaged or perturbed surface exhibits a profound capability for flattening, or self-healing of initial damage, even in the case of slow surface self-diffusion. Eqs. (18) and (19) describe the kinetics of flattening and can serve as a basis for design of thin films with enhanced self-healing capability. It is instructive to estimate the relative durations of the transient and steady state stages of the flattening and to get a feeling for the characteristic self-healing times at ambient conditions. It should be noted that for relatively thick films or short perturbation wavelength (h P 1) the transient regime of surface topography evolution is extremely short and virtually unobservable. This is related with the high value of Ek/c ratio [see Eqs. (10) and (A1.25)], E being YoungÕs modulus. This ratio determines the smallness of the second exponent responsible for the transient stage in the RHSs of Eqs. (16). Indeed, for the typical metals E 1011 Pa, c 1 J/m2 and Ek/c 1. However, the situation may be different for ultra-thin films in which the effective elastic modulus G can be rather small (see Fig. 2). Let us assume, for the sake of a rough estimate, bi bs, bb (‘‘dirty’’ surface approximation). Using Eqs. (16a), (17) and (A1.25), assuming R H0 (this is not
633
uncommon for columnar thin film microstructures), H0 10 nm, m 0.33, n 100 and assuming long perturbation wavelength of k = 50H0 = 500 nm (this corresponds to the periodicity of photolithography pattern easily achievable in microelectronics industry) we arrive at the following expression for the bending strain g accumulated in the film: g
e0 f ½exp ðsÞ exp ð9:3sÞ: g
ð23Þ
where s is the dimensionless time defined according to s = bbt. The dependence g(s) is shown in Fig. 3. It is logical to identify the duration of transient period with the maximal bending of the film, smax 0.27. Assuming Db 1018 m2/s, d 1 nm, T = 300 K [5] and Va 105 m3/mol leads to tmax 11 h, i.e. rather long transient period. The actual flattening of the film can then take up to several days. Finally, it should be noted that in this work we considered only the diffusional path of the relaxation of film surface perturbations. The question arises as to whether the high capillary stresses induced by the curvature of perturbed surface can activate some dislocation activity in the film, thus opening an additional, plasticity path for film flattening. The general answer on this question depends on the details of initial distribution of dislocations and dislocation sources in the film, which is not known a priori. It can be only shown that for thin films that are initially dislocations-free the capillary stresses are too weak to cause the homogeneous nucleation of new dislocations. Indeed, recent nanoindentation experiments show that thin Cu films deposited on Si substrate deform elastically under a sharp diamond indenter until the maximal shear stress under indenter reaches the theoretical shear strength of the film (approximately 12.7 GPa for Cu films) [8]. At this point a spontaneous displacement burst associated with the nucleation of a group of dislocations occurs. Even for a very high surface radius of curvature of, let say, r 10 nm the max-
Fig. 3. The dependence of normalized bending strain accumulated in the film [see Eq. (23)] on dimensionless annealing time for ultrathin film.
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E. Rabkin, L. Klinger / Acta Materialia 53 (2005) 629–636
imal amplitude of the capillary stress in the film is about 0.1 GPa, which is by two orders of magnitude lower than the stress needed for dislocations nucleation. Moreover, as it can be seen from Eq. (A1.18) the shear stresses in sufficiently thin (h 1) films scale with he which means that the shear stresses in the film induced by capillary forces are actually much lower than the above estimate of 0.1 GPa. It can be concluded that the capillary-induced shear stresses in the films are by far too low to cause the homogeneous nucleation of dislocations. Therefore, at low homological temperatures at which evaporation of the film material and bulk diffusion in the film are insignificant, the interfacial diffusion considered in the present work provides the only path for surface shape evolution of the dislocation-free films.
4. Conclusions We demonstrated that the temporal evolution of sinusoidal surface perturbation of polycrystalline thin film deposited on rigid substrate is governed by a concerted action of mass transport along the free surface, grain boundaries, and film/substrate interface. The grain boundary and interfacial diffusion are driven by the inhomogeneous distribution of capillary stresses in the film. The explicit expressions describing time dependencies of the perturbation amplitude and of the bending strain in the film were obtained. From their analysis the following conclusions can be drawn: 1. During a very short initial period the bending strain is intensively accumulated in the film. Later, the strain increase stops and a steady-state process of the exponential decay of both surface perturbation and of bending strain begins. 2. The acceleration of surface healing kinetics due to alternative diffusion paths can be very significant in the case of contaminated or passivated surfaces with reduced surface diffusivity. 3. The surface topography evolves differently for short and long perturbation wavelengths, which are defined with respect to critical perturbation wavelength, k*. For long wavelengths, k k*, the surface kinetics follows a classical MullinsÕ model [1], albeit with the acceleration factor in the time exponent which is proportional to the ratio of self-diffusion coefficients along the film/substrate interface and along the free surface. For short wavelength, k k*, the surface kinetics follows a mixed law with the sum of two terms in the time exponent, the first one being proportional to the fourth power of the wave vector and the second one to its second power. In this respect, the obtained kinetic law resembles the Mullins-like law for the case of parallel action of surface diffusion and evaporation-condensation mechanisms
[1]. The critical wavelength k* depends on the ratio of interfacial and grain boundary diffusivities, on the film thickness and on the grain size in the film. 4. Numerical estimates show that for typical metallic films significant flattening can occur due to grain boundary and interfacial diffusion already at room temperature during time period of several days. For ultrathin films, the initial, transient period of surface evolution can take up to several hours.
Appendix A For calculating of stress transmission coefficient f in the small slope approximation we will consider the following elastic problem: to the free surface of the film of thickness h the distributed normal force (capillary stress) of the following amplitude is applied: ryy ðx; hÞ ¼ xce sinðxÞ;
ðA1:1Þ
rxy ðx; hÞ ¼ 0:
ðA1:2Þ
Since the film is deposited on rigid substrate the following conditions should be fulfilled at the film/substrate interface: ux ðx; 0Þ ¼ 0;
ðA1:3Þ
uy ðx; 0Þ ¼ 0:
ðA1:4Þ
The distribution of stresses in the film can be found with the aid of biharmonic stress function U(x,y): Uðx; yÞ ¼ F ðyÞ sinðxÞ
ðA1:5Þ
with F ðyÞ ¼ ðA1 þ B1 y Þ expðyÞ þ ðA2 þ B2 y Þ expðyÞ:
ðA1:6Þ
The constants A1, A2, B1 and B2 can be determined using the boundary conditions of the problem (see below). The components of the stress tensor can be calculated using the stress function U: rxx ¼
o2 U d2 F ¼ 2 sinðxÞ; oy 2 dy
o2 U ¼ F sinðxÞ; ox2 o2 U dF ¼ cosðxÞ; rxy ¼ oxoy dy rzz ¼ m rxx þ ryy ; rxz ¼ ryz ¼ 0; ryy ¼
ðA1:7Þ
where m is PoissonÕs ratio. The components of strain tensor can be found with the aid of HookeÕs law: 1 þ m d2 F uxx ¼ ð1 mÞ þ F m sinðxÞ; ðA1:8Þ E dy 2
E. Rabkin, L. Klinger / Acta Materialia 53 (2005) 629–636
uxy ¼
1 þ m dF cosðxÞ; E dy
ðA1:9Þ
where E is YoungÕs modulus. The components of strain vector u can be determined by the integration of appropriate components of the strain tensor: 1 þ m d2 F ux ¼ ð 1 m Þ þ F m cosðxÞ; ðA1:10Þ E dy 2
1 þ m dF dF ð2 mÞ 3 ð1 mÞ sinðxÞ: uy ¼ E dy dy
The problem of transmission of the bending strain in the vicinity of film/substrate interface to the film surface can be treated in a similar way. The corresponding elastic problem can be formulated in a following way: ux ðx; 0Þ ¼ 0;
ðA1:19Þ
uy ðx; 0Þ ¼ g sinðxÞ;
ðA1:20Þ
ryy ðx; hÞ ¼ 0;
ðA1:21Þ
rxy ðx; hÞ ¼ 0:
ðA1:22Þ
3
ðA1:11Þ
Eqs. (A1.7), (A1.10) and (A1.11) determine the stress and strain fields everywhere in the film, including film surface (y = h) and film/substrate interface (y = 0). For determining the constants A1, A2, B1 and B2 in the stress function, Eqs. (A1.7), (A1.10) and (A1.11) should be substituted in boundary conditions (A1.1)–(A1.6): F ðhÞ ¼ xce; dF ¼ 0; dy y¼h
ðA1:12Þ
The latter two conditions describe constraint-free external surface. It can be shown that y-displacement of the film surface is described in this case by uy ðx; hÞ ¼ gf sinðxÞ;
ðA1:13Þ
d F ð1 mÞ þ F ð0Þm ¼ 0; dy 2 y¼0
ðA1:14Þ
dF d3 F ð2 m Þ 3 dy y¼0 dy
ðA1:15Þ
ð1 mÞ ¼ 0:
Substituting in these equations the definition of F [see Eq. (A1.6)] gives the system of four linear equations for determining of the constants A1, A2, B1 and B2. We will omit here these simple, but rather cumbersome calculations. Once these coefficients are found the stress tensor anywhere in the film can be determined with the aid of Eqs. (A1.7). In particular, at the film/substrate interface (y=0) we get
E ð 1 mÞ sinh ð2hÞ 2h : 2 ð1 þ mÞ 4ð1 mÞ þ ð3 4mÞsinh2 ðhÞ þ h2 ðA1:25Þ
It should be noted that the behavior of G h) is different from that of f h) (see also Fig. 2): G(h ! 0) ! 0 (µh3), and G (h ! 1) ! Gm, where Gm ¼
2Eð1 mÞ : ð1 þ mÞð3 4mÞ
ðA1:26Þ
The other two components of the stress tensor at the film/substrate interface are: m ryy ðx; 0Þ rxx ðx; 0Þ ¼ ðA1:27Þ 1m and rxy ðx; 0Þ ¼
ryy ðx; 0Þ ¼ xcef sinðxÞ;
ðA1:24Þ
where G¼
y¼0
ðA1:23Þ
where the coefficient f is determined by Eq. (A1.16). The component ryy of the stress tensor at the film/substrate interface corresponding to the bending strain (A1.20) is ryy ðx; 0Þ ¼ Gg sinðxÞ;
2
E h2 þ ð1 2mÞsinh2 ðhÞ g cosðxÞ: 1 þ m 4ð1 mÞ2 þ ð3 4mÞsinh2 ðhÞ þ h2 ðA1:28Þ
where f ¼
635
2ð1 mÞ½2ð1 mÞ coshðhÞ þ h sinhðhÞ 4ð1 mÞ2 þ ð3 4mÞsinh2 ðhÞ þ h2
ðA1:16Þ Appendix B
is the stress transmission coefficient we were searching for. As expected, f(h ! 0) ! 1 and f(h ! 1) ! 0. The other two components of the stress tensor at the film/ substrate interface are: rxx ðx; 0Þ ¼ rxy ðx; 0Þ ¼
m ryy ðx; 0Þ; 1m
ðA1:17Þ
2ð1 mÞð1 2mÞ½ð1 2mÞ sinhðhÞ þ h coshðhÞ 2
4ð1 mÞ þ ð3 4mÞsinh2 ðhÞ þ h2 xce cosðxÞ:
ðA1:18Þ
The system of two linear differential equations (14a) and (14b) can be rewritten in the compact form og ¼ a11 g þ a12 e; ot
ðB1:1Þ
oe ¼ a21 g þ a22 e ot
ðB1:2Þ
with a11 ¼ gðbi þ bb Þ;
ðB1:3Þ
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E. Rabkin, L. Klinger / Acta Materialia 53 (2005) 629–636
a12 ¼ bi f þ bb ð1 f Þ;
ðB1:4Þ
a21 ¼ g½bi f þ bb ð1 f Þ;
ðB1:5Þ 2
a22 ¼ bs bi f 2 þ bb ð1 f Þ :
ðB1:6Þ
The solution of the system of linear differential equations (B1.1) and (B1.2) with the initial conditions g(0) = 0 and e(0) = e0 is: a12 g ¼ e0 ½exp ðk 2 tÞ exp ðk 1 tÞ; ðB1:7Þ k1 k2 1 k1 k2 ½ðk 2 þ a21 Þ expðk 2 tÞ þ ðk 1 þ a21 Þ exp ðk 1 tÞ;
e ¼ e0
ðB1:8Þ where k1 and k2 are the roots of characteristic equation k 2 þ ða11 þ a22 Þk þ ða11 a22 a12 a21 Þ ¼ 0:
ðB1:9Þ
The expressions (B1.7) and (B1.8), as well as solutions of Eq. (B1.9) can be simplified assuming g 1, which is the case for all films that are not too thin. Eqs. (16) and (17) of this paper represent such a simplification. References [1] Mullins WW. J Appl Phys 1959;30:77. [2] Mehrer H, editor. Diffusion in solid metals and alloys. Landolt– Bo¨rnstein new series, vol. III/26. Berlin: Springer; 1990. [3] Mills B, Leak GM. Acta Metall 1968;16:303. [4] Freund LB, Suresh S. Thin film materials. Cambridge: Cambridge University Press; 2003. [5] Kaur I, Mishin Y, Gust W. Fundamentals of grain and interphase boundary diffusion. Chichester: Wiley; 1995. [6] Koetter TG, Wendrock H, Schuehrer H, Wenzel C, Wetzig K. Microelectron Reliability 2000;40:1295. [7] Herring C. J Appl Phys 1950;21:437. [8] Suresh S, Nieh TG, Choi BW. Scr Mater 1999;41:951.