Grain boundary diffusion in thin films under stress fields

Applied Surface Science 175±176 (2001) 312±318

Grain boundary diffusion in thin ®lms under stress ®elds Anatoly S. Ostrovsky*, Boris S. Bokstein Department of Physical Chemistry, Steel and Alloys Institute, P.O. Box 024, Leninsky Pr. 4, 117936 Moscow, Russia Accepted 2 January 2001

Abstract Some experimental data concerning stress effect on grain boundary (GB) diffusion and models describing the GB diffusion (GBD) in thin polycrystalline ®lms for the B- and C-regimes under a stress ®eld are reviewed. Numerical solutions for the conditions of re¯ecting free surface and rapid surface diffusion on the sink surface for the case of a constant source are obtained. The results of the model calculations are compared with the results of the investigation of Cu GBD in Ni thin ®lms and with experimental data from the literature. The comparative in¯uence of the GBD atomic mechanisms on the shape of GBD penetration plots is discussed. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Grain boundary; Diffusion; Thin ®lms; Stresses; Diffusion mechanisms

1. Introduction A mechanical stress ®eld can in¯uence both the diffusion coef®cient (the well-known pressure effect) and the diffusion driving force that is connected with the chemical potential gradient [1]. Also there are many cases where the main effect of a stress is to modify the boundary conditions of the diffusion problem [2]. It relates to the ef®ciency of the sources and sinks of vacancies or interstitials, such as, for example, grain or interface boundaries. Coble creep is a classic example [3]. The interrelation of stress and diffusion is of practical as well as theoretical interest as it appears in the different phenomena, for example, creep phenomenon, Gorsky effect, pore formation during interdiffusion, etc. [4,5]. It is particularly important in the case of thin ®lms and multilayers where diffusion differs from that in * Corresponding author. Tel.: ‡7-095-230-4466/‡7-095-9633734 (Residence); fax: ‡7-095-237-8007.

massive materials [6±8] due to the peculiarities of their structure. First of all, these objects demonstrate high values of mechanical stress after their preparation, exceeding 1 GPa [9,10] and the small thickness of thin ®lms can lead to a great stress gradient which acts as a driving force for GBD. The experimental results concerning stress measurements in thin metallic ®lms (TMF) are covered in [6,10,11]. A suf®ciently full review of stress in polycrystalline thin metal ®lms was published in [10]. Secondly, at low temperatures, where stresses have not yet relaxed, the diffusion processes in thin ®lms are enhanced, as compared with massive materials and due to small enough GB activation energy the pressure effect on GBD is more signi®cant. Then the ®nite size of thin ®lms can modify the boundary conditions of the GBD problem [2]. And last but not least, there is the high density of GBs in polycrystalline thin ®lms. In the ®eld of microelectronic devices, which are based on thin ®lm multilayered structures, short-circuit diffusion along grain and interface boundaries is

0169-4332/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 1 6 4 - 7

A.S. Ostrovsky, B.S. Bokstein / Applied Surface Science 175±176 (2001) 312±318

313

known to be the dominant mode of device failure. Knowledge of the characteristics of GBD is thus essential for optimizing the design parameters of the various microelectronic devices from the viewpoint of reliability.

tensor is sii ˆ sxx ‡ syy ‡ szz ˆ 2s. The stress gradient acts only along the y-axis.

2. Theory: basic relations

Unfortunately, at the moment it is impossible to carry out the full enough comparison of stress and diffusion parameters for the TMFs, as there is only one work with simultaneous measurements of this type using one set of sample [13]. Nonetheless, one can try to compare the following [14] results of different experiments. The values of GBD coef®cients Db in TMFs at 0.35±0.4 of melting temperature are listed in Table 1. The ®lms were deposited on different substrates. All ®lms were stressed; the mean values of stresses vary with each ®lm. The experimental values of stresses have an asterisk. All other values were estimated [14] as thermal stresses which are caused by the difference of linear thermal expansion coef®cients of the ®lm and substrate during the cooling. As was shown [14] it is a good approximation for TMFs with melting temperatures from approximately 1350±1900 K. The values of effective activation enthalpy of GBD Hb in TMFs with different stress values are also listed in Table 1. The analysis of these data from Table 1 leads to the conclusions that greater the mean value of stresses, greater effective the GBD coef®cient and smaller the effective activation enthalpy of GBD.

3. Experimental results on grain boundary diffusion in thin metallic ®lms under a stress ®eld

The component k diffusion driving force is connected with its chemical potential gradient. Considering hydrostatic pressure p or mechanical stress s the ¯ux for the component k in an in®nitely dilute solution [1]:   cV j ˆ D rc rsii ; (1) 3RT where D is the component k diffusion coef®cient, R the gas constant, T the temperature, c the component k concentration, V its partial molar volume, sii ˆ 3p the ®rst invariant of the stress tensor (Einstein convention is applied for repeated indices: sii ˆ s11 ‡ s22 ‡ s33 ). In Eq. (1) we propose that partial molar volume does not depend on co-ordinates. The dependence of diffusion coef®cient on stress is de®ned as @ ln D…s† ˆ @p

O ; RT

(2)

where O is the diffusion activation volume. Usually, the stress state of the thin ®lm is binary axis [12]. The stress tensor components are sxx ˆ szz ˆ s; syy ˆ sxy ˆ sxz ˆ syz ˆ 0. The ®rst invariant of stress Table 1 Experimental studies of stress effect on GBD in TMF Film Al Cu Au Pt Au Ni

Substrate Sitall Glass Al2O3 Glass Al2O3 Sitall GaAs SiO2/Si Sitall Glass Si Si

Diffusant Cu Cu Ni Ni Cu Cu Au Au Cu Cu Cu Cu

s (GPa) 

0 0 1.6 0.06 0.73 0.04 0.86 1.6 0.04 0.2 1.1 0.06

Db (sm2/s) 13

1:4  10 0:61  10 13 3:8  10 19 1:2  10 22 2:4  10 17 10 19

T (K)

Db0 (sm2/s)

407 407 407 407 300 300

6:0  10 7:1  10

16 15

Hb (kJ mol 1)

Literature

159 93 94.6 87.5 62.4 58.1

[15] [16] [17] [18] [19] [20] [21] [22] [20] [23] [13] [13]

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A.S. Ostrovsky, B.S. Bokstein / Applied Surface Science 175±176 (2001) 312±318

In the case that the stresses are low or absent (aluminum ®lms) diffusion coef®cients are within the experimental error; otherwise Db differ by some orders. However, the only work with simultaneous measurements of stress and GBD parameters (nickel± copper system on Si substrate) leads to the opposite results (see Table 1, Db0 is the pre-exponential factor of the GBD coef®cient). The ®lms were prepared by vapor deposition (stretched ®lms) and magnetron sputtering technique (stress-free ®lms). It was shown that for stretched nickel ®lms GBD coef®cients of copper were less than that for stress-free ®lms. This was explained by the effect of stress distribution through the ®lm [13]. It was shown that although the mean value of stressed nickel ®lms was positive (‡1.1 GPa), the compression from 1.44 to 0.64 GPa had taken place near the Ni±Cu interface, where GBD measurements were performed.

where Db, cb and Ob are the GBD coef®cient, the GB concentration and the activation volume of GBD, respectively. The initial diffusant concentration everywhere is zero. Due to the symmetry of the system the solution of the problem must satisfy the condition jx jxˆ0 ˆ 0. Also, it was proposed that the segregation effects are absent, i. e., cV ‰12 …dS  d†; y; tŠ ˆ cb . The stress distribution function was presumed to be linear y  0:5 ; (5) sii …y† ˆ sii ‡ Dsii h

4. Model of grain boundary diffusion under a stress ®eld

4.1. B-regime

ILet us consider the diffusion in an array of uniformly spaced parallel GBs for a specimen of ®nite dimension in the direction of diffusion. The GBs are assumed to be homogenous slabs with constant thickness d and perpendicular to both ®lm surfaces [24]. They are assumed to be centered at …n ‡ 12†dS , where n ˆ 0, 1, 2 and dS is the GB spacing. The ®lm of the thickness h has a strong adherence to the substrate at y ˆ h and it contacts with the diffusion source at y ˆ 0. The thin ®lm is stressed in x and y directions. The stress gradient acts only along the y-axis. The balance equation for bulk diffusion is the following: @cV @ 2 cV @ 2 cV ˆ DV …s† ‡ @t @x2 @y2

2 3RT

@cV @s …V @y @y

2 3RT

@cb @s …V @y @y

Now let us solve this problem numerically for the B-regime after Harrison [25]. It was assumed that the source is constant: c…x; 0; t† ˆ c0 . The boundary condition for the case of rapid surface diffusion on the interface y ˆ h is @cS @ 2 cS 1 ˆ DS …s† 2 ‡ DV …s† dS @t @x   @cV 2V @s cV  ; @y 3RT @y yˆh;x6ˆdS =2

(6)

where DS and cS are the surface diffusion coef®cient and the surface concentration, respectively, dS the

  !!! @ 2 s 2OV @s 2 O V † ‡ cV V ‡ ; @y2 3RT @y

where cV, DV and OV are the bulk concentration, the bulk diffusion coef®cient and the activation volume of bulk diffusion, respectively, and t the annealing time. The balance equation for GBD is @cb @ 2 cb ˆ Db …s† @t @y2

where sii is the mean value and Dsii the difference between the values at the source and sink ®lm surfaces. In this case the non-dimensional parameters S ˆ sii V=…3RT† and G ˆ Dsii V=…3RT† describe the effect of stress on GBD.

(3)

effective thickness of the sink surface. In Eq. (6) it was assumed that there is no diffusion ¯ux from substrate±®lm interface to substrate.

  !!! @ 2 s 2Ob @s 2 2 @cV O b † ‡ cb V ‡ DV …s† ‡ ; @y2 3RT @y d @x xˆ…dS d†=…2†

(4)

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315

At the intersection of the GB and the surface, the boundary condition to be satis®ed is   @cb 2V @s @c cb dDb …s† ˆ 2d D …s† : S S @x xˆdS =2 @y 3RT @y yˆh (7) Introducing the non-dimensional parameters X ˆ x=dS , Y ˆ y=h, TF ˆ DV t=h2 , u ˆ c=c0 , Db ˆ Db =DV , DS ˆ DS =DV and assuming that the thickness of the ®lm h is equal to the GB spacing dS one can obtain a non-dimensional system of equations similar to Eqs. (3)±(7). It was assumed here that the activation volume of GBD is equal to the activation volume of bulk diffusion and to the molar volume of the matrix component [26,27] as it is usually consistent with the vacancy mechanism of GBD. For example, it has been experimentally proven for silver self-diffusion [28] and zinc heterodiffusion in aluminum [29]. The solutions of the system were obtained by a numerical method based on ®nite-difference equations. The concentration pro®les resulting from numerical calculations for different S and G values are shown in Figs. 1 and 2. The parameter values were the following: TF ˆ 10 3 , d=h ˆ 5  10 3 , Db ˆ 103 , DS ˆ 1 (Fig. 1, the plate y ˆ h acts as a diffusion barrier) and

Fig. 1. The concentration pro®les in a thin ®lm (the surface diffusion barrier case).

Fig. 2. The concentration pro®les in a thin ®lm (the rapid surface diffusion case).

DS ˆ 103 (Fig. 2). The average layer concentration u as de®ned by numerical layer-by-layer integration: Z 2 dS =2 uˆ u…X; Y; TF † dX: (8) dS 0 One can see (Figs. 1 and 2) that the homogenous …G ˆ 0† ®lm tension (S > 0, curve 1 in Fig. 1) accelerates the resulting diffusion ¯ux, but the homogenous ®lm compression (S < 0, curve 5 in Fig. 1) decreases compared to the free stress case (curve 3 in Fig. 1). In the case of non-homogenous stress …G 6ˆ 0† one can see that if the stress and concentration gradients have the same sign …G < 0† then this increases the diffusion at the ®rst part of a ®lm (closed to the source surface) and decreases the diffusion at the second part of a ®lm (closed to the sink surface, see curve 4 in Fig. 1) compared to the homogenous stress (curve 3 in Fig. 1) and vice versa (see curves 2 and 3 in Fig. 1). These conclusions from the model are in accordance with the conclusions from the experimental investigation of the stress effect on GBD of copper in nickel thin ®lms [13] (see Table 1). In the case of rapid surface diffusion (see Fig. 2), the above-mentioned conclusions are also true. For the stress free case our calculation (curve 3 in Fig. 1 and curve 5 in Fig. 2) follows the results obtained by Gilmer and Farrell [30,31].

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A.S. Ostrovsky, B.S. Bokstein / Applied Surface Science 175±176 (2001) 312±318

Note that if we assume that the plate y ˆ h acts as a diffusion barrier (Fig. 1, the ¯ux beyond y ˆ h is equal zero): @c 2V @s ˆ c ; (9) @y yˆh 3RT @y yˆh the penetration curves are determined by the stress gradient sign. The examination of the plots near y ˆ h give us an indication of the stress distribution through the ®lm: the negative slope of the penetration curve (see curve 4 in Fig. 1) con®rms the negative stress gradient and the positive slope of the penetration curve (see curve 2 in Fig. 1) con®rms the positive stress gradient. Note also that in the case of the rapid surface diffusion (see Fig. 2) the sink surface concentration does not depend on the stress distribution through ®lm but depends on the sink surface stress value (see curves 1 and 2, 3 and 4 in Fig. 2). 4.2. C-regime At low temperatures the volume diffusion coef®cient is negligibly small compared to the GBD coef®cient. This condition allows us to simplify substantially Eqs. (3) and (4). Now, we must solve only Eq. (4) without the bulk diffusion-containing term. This problem is very current, since the lower the temperature of diffusion experiments, the greater the stress effect on GBD. Moreover, stress relaxation is much slower during diffusion annealing at low temperatures. Models describing the GBD in thin polycrystalline ®lms in the C-regime under a stress ®eld were proposed in [32,33] for different source conditions. The analytical quasi-steady solution for the conditions of linear stress distribution (Eq. (5)), rapid surface diffusion (Eq. (6) without the volume diffusioncontaining term) and constant source was obtained as  cS ˆL 1 c0

 exp

2tb ML1=2 ln L L2 1

 ;

(10)

where cS is the average sink surface concentration, L ˆ exp…G†, M ˆ exp…S†, and tb ˆ dDb 0 t=…dS hdS † the non-dimensional time, where Db 0 the GB coef®cient for stress-free diffusion. It was assumed here

and below that the surface diffusion coef®cient is much larger than the GBD coef®cient, as it is usually for the C-regime [34]. Following [35], it was proposed that in the case of the quasi-steady GBD regime the divergence of the GBD ¯ux is equal to zero. This corresponds to the situation in which every time a suf®cient amount of material is transported to the accumulation surface causing a signi®cant change inconcentration, the GB concentration changes rapidly to maintain a quasi-stationary pro®le throughout diffusion. It was shown that the stress in a non-homogenous …G < 0† ®lm decreases the saturation concentration at the sink surface compared to the homogenous ®lm, and that the greater the stress gradient, lesser the saturation concentration. The saturation concentration cS sat ˆ c0 exp…G†. The homogenous ®lm tension …S > 0† leads to an acceleration of GBD and to a decrease in saturation time. The homogenous ®lm compression …S < 0† leads to a decrease in the GBD rate and to an increase in the saturation time. The effective coef®cient of GBD Db ef may be expressed in this case as Db 0 exp…S†. As was shown [32] using the experimental surfacesaturation plots for chromium GBD in platinum thin ®lms [36,37] the difference in the value of the GBD activation energy Eb which can be determined from this model and the stress-free model [35] is about 12 kJ mol 1. It is important to note that in the case of an in®nite diffusion source the quasi-stationary model is only true if G  0, because in the opposite case the sink surface capacity will be exhausted before the stationary state is achieved and we must solve the non-quasistationary problem. In the case of a ®nite diffusion source one can write the boundary condition at the source interface …y ˆ 0† as @cSO ˆ @t

djy : dSO dS

(11)

Here cSO is the average source surface concentration and dSO the effective thickness of the source surface. The initial condition at the source surface is cSO …x; 0; 0† ˆ c0 . If the GB capacity is much smaller than the surface capacity, under these conditions a quasi-steady state may be attained in the GB [35].

A.S. Ostrovsky, B.S. Bokstein / Applied Surface Science 175±176 (2001) 312±318

The solutions of a diffusion problem now are    cS L 2tb …L ‡ 1†ML1=2 ln L 1 exp ˆ ; L2 1 c0 L ‡ 1 (12) cSO 1 ˆ L‡1 c0    2tb …ds =dSO †…L‡1†ML1=2 ln L  1‡L exp : L2 1 (13) In the absence of stresses Eqs. (10), (13) and (14) transform to the Hwang±Balluf® equations [35]. If the stress gradient is in the ®lm and the stress and concentration gradients have the same sign …G < 0†, then it decreases the saturation concentration at the sink surface and increases at the source surface. If the stress and concentration gradient have different signs …G > 0†, then it decreases the saturation concentration at the source surface and increases at the sink surface. One can see from Eqs. (12) and (13) that the sink and surface saturation concentrations correlate as cS sat =cSO sat ˆ exp…G† and in the sum give us the initial surface concentration. 5. A note about the in¯uence of different GBD mechanisms on the shape of GBD penetration plots All above-mentioned calculations were performed for a vacancy mechanism, where the value of diffusion activation volume is closed to the molar volume of the matrix. However there are other atomic mechanisms of diffusion, for example, interstitial. For the interstitial mechanism of diffusion the diffusion activation volume is only a small fraction of the molar volume, for example, the activation volume of cobalt GBD in a-Zr is …0:15  0:14†V-Zr [38]. The in¯uence of the atomic mechanism on the penetration plot displays through Eq. (2). One can see that in the case of homogenous ®lm tension the diffusion on an interstitial mechanism is slower than with the vacancy mechanism, and in the case of a homogenous compression the rate of diffusion on an interstitial mechanism increases compared with a vacancy mechanism.

317

In the case of non-homogenous stress state in thin ®lms if the stress …G < 0† and concentration gradients have the same sign then the effect of the diffusion with a low activation volume on penetration plots is to shift the overall pro®le by a speed of approximately GDV/h for type B kinetics and of GDb/h for type C kinetics, and to increase the pro®le penetration of the force as a result of enhanced grain boundary mobility for type B diffusion kinetics [39]. In the situation of the reversing driving force, the force acting on diffusion at the GB would retard the resultant pro®le. 6. Conclusions 1. Numerical solutions for the conditions of re¯ecting free surface and rapid surface diffusion on the sink surface for the case of a constant source are obtained. 2. The results of the model calculations are analyzed and agree with the experimental data in the literature. 3. An analysis of the comparative in¯uence of interstitial and vacancy diffusion mechanisms under the stress ®eld on the rate of GBD is carried out. Acknowledgements This work was ®nancially supported by the Russian Foundation for Basic Researches (RFBR) through grant No. 98-03-32252a. One of the authors (AO) is grateful to the RFBR for a grant No. 00-03-42666 and the Organizing Committee of the International Conference on Solid Films and Surfaces (Princeton University) or the ICSFS-10 Travel Award. References [1] J. Philibert, Diffusion and Mass Transport in Solids, Edition du Physique, Paris, 1991. [2] J. Philibert, Defect Diff. Forum 129±130 (1996) 3. [3] R. Coble, J. Appl. Phys. 34 (1963) 1679. [4] D. Beke, Defect Diff. Forum 129±130 (1996) 9. [5] F. Larche, J. Cahn, Acta Met. 33 (1985) 331. [6] J.M. Poate, K.N. Tu, J.W. Mayer (Eds.), Thin Films Ð Interdiffusion and Reactions, Wiley/Interscience, New York, 1978.

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[7] R.W. Balluf®, J.M. Blakely, Thin Solid Films 25 (1975) 441. [8] A. Greer, Defect Diff. Forum 129±130 (1996) 163. [9] K.L. Chopra, Thin Film Phenomena, McGraw-Hill, New York, 1969. [10] R. Koch, J. Phys.: Condens. Mat. 6 (1994) 9519. [11] R.W. Hoffman, Thin Solid Films 171 (1989) 5. [12] K.N. Tu, J. Mayer, L. Feildman, Material Science: Thin Film Technology, McGraw-Hill, New York, 1992. [13] N. Balandina, B. Bokstein, A. Ostrovsky, Defect Diff. Forum 156 (1998) 181. [14] N. Balandina, B. Bokstein, A. Peteline, A. Ostrovsky, Defect Diff. Forum 129±130 (1996) 151. [15] M. Chamberlain, S. Lehoczky, Thin Solid Films 45 (1977) 189. [16] M. Shearer, C.L. Bauer, A.G. Jordan, Thin Solid Films 61 (1979) 273. [17] B.C. Jonson, C.L. Bauer, A.G. Jordan, J. Appl. Phys. 59 (1986) 1147. [18] H. Lefakis, J. Cain, P. Ho, Thin Solid Films 102 (1983) 207. [19] J.A. Borders, Thin Solid Films 19 (1973) 359. [20] A.N. Aleshin, B.S. Bokstein, V.K. Egorov, P.V. Kurkin, Thin Solid Films 275 (1996) 144. [21] G. McGuire, W. Wisseman, P. Holloway, J. Vac. Sci. Technol. 15 (1978) 1701. [22] C.C. Chang, K. Quintana, Thin Solid Films 31 (1976) 265. [23] P.M. Hall, J.M. Morabito, Thin Solid Films 41 (1977) 341.

[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

J.C. Fisher, J. Appl. Phys. 22 (1951) 74. L.G. Harrison, Trans. Faraday Soc. 57 (1961) 1191. W. Lojkowski, Defect Diff. Forum 129±130 (1996) 269. G. Martin, D.A. Blackburn, Y. Adda, Phys. Status Sol. 23 (1967) 223. G. Erdely, W. Loikowski, D.L. Beke, I. Godeny, F. Kedves, Phil. Mag. A 56 (1987) 637. H. Mehrer, Defect Diff. Forum 129±130 (1996) 57. G.H. Gilmer, H.H. Farrell, J. Appl. Phys. 47 (1976) 3792. G.H. Gilmer, H.H. Farrell, J. Appl. Phys. 47 (1976) 4373. A. Ostrovsky, Defect Diff. Forum 156 (1998) 249. A. Ostrovsky, N. Balandina, B. Bokstein, Mat. Sci. Forum 294±296 (1999) 553. N.A. Gjostein, in: H.I. Aaronson (Ed.), Diffusion, American Society for Metals, Metals Park, OH, 1973. J.C.M. Hwang, R.W. Balluf®, J. Appl. Phys. 50 (1979) 1339. S. Danyluk, G.E. McGuire, K.M. Kollwad, M.G. Yang, Thin Solid Films 25 (1974) 483. P.H. Holloway, G.E. McGuire, J. Electrochem. Soc. 125 (1978) 2070. K. Vieregge, Ch. Herzig, W. Lojkowski, Scripta Met. Mat. 25 (1991) 1707. D. Gupta, D. Kambell, P. Ho, in: J.M. Poate, K.N. Tu, J.W. Mayer (Eds.), Thin Films Ð Interdiffusion and Reactions, Wiley/Interscience, New York, 1978.