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Scripta Materialia 65 (2011) 384–387

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Threading dislocation interactions in an inhomogeneous stress field:A statistical model

Ray S. Fertig, III and Shefford P. Baker⇑

Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, United States

Received 31 March 2011; accepted 6 May 2011Available online 14 May 2011

A statistical model is presented to characterize the influence of stress field inhomogeneity on the probability of interactionsbetween threading dislocations in thin films. Any degree of stress field inhomogeneity is shown to increase the likelihood of inter-actions. However, below a critical level, the effect of stress inhomogeneity is small, while above this level a dramatic effect isobserved. The approach taken is applicable to any system of interacting particles with spatially dependent velocities.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Thin films; Dislocation; Plastic deformation; Modelling; Inhomogeneous stresses

Thin metal films have flow stresses as much as an or-der of magnitude higher than those achievable in bulkmetals of similar composition [1] leading to considerableinterest in the contribution of dislocation–dislocationinteractions to thin film strength [2–4]. In large-grainedand single-crystal thin films, dislocation motion can bedescribed in terms of a threading dislocation (“thread”)extending through the thickness of the film that movesthrough the film, laying down or recovering misfit dislo-cations (“misfits”) at interfaces with adjacent layers [5].The requirement that misfits be created/removed leadsto a critical resolved shear stress for dislocation motion,sc [5,6]. At stresses above sc, a thread moves untilstopped by interaction with a misfit crossing its path(TM interaction) or with another thread (TT interac-tion) [7]. TM interactions have been shown to be weak[2], breaking and allowing threads to move on at stresslevels much lower than typical film stresses, while someTT interactions have been shown to be very strong [2],preventing further motion of the participating threadsto stress levels much higher than typical film stresses.

Recently, we conducted dislocation dynamics simula-tions to understand how weak TM and strong TT inter-actions contribute to strength and strain hardening inpassivated thin films on substrates, and found that bothTM and TT interactions are facilitated by inhomoge-neous stresses [7]. For a given applied strain, the stress

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⇑Corresponding author. Tel.: +1 607 255 6679; fax: +1 607 2552365; e-mail: shefford.baker@cornell.edu

is determined by the relaxation due to plastic strain,ep / qMbhxi, where qM is the mobile dislocation density,b the magnitude of the Burgers vector and hxi the aver-age distance that a dislocation travels. Since the appliedstress is relaxed only in the neighborhood of each misfit,the stress field becomes inhomogeneous, even for a reg-ular misfit array [8,9]. In our simulations, an irregularmisfit structure developed with a stress inhomogeneitythat increased in magnitude with increasing misfit den-sity [7]. Stress inhomogeneity in thin films has been ob-served experimentally [10–12] and characterized insimulations [3,7]. We found that inhomogeneous stressesfacilitate both TM and TT interactions, and thus con-tribute to film strength at all stress levels by limiting hxi.

Understanding TM interactions is straightforward.Since TM interactions are weak, they can only stopthreads in regions of low stress. Thus, the distance thata thread can travel before being stopped in a TM inter-action depends on the spatial distribution of regions oflow stress. We found a functional form that accuratelydescribes this “mean free path” in terms of the meanand standard deviation of the stress field, the channelingstress, and the strengths of TM interactions [7].

Interpretation of the results for TT interactions ismore challenging. Since TT interactions are strong, theydo not require regions of low stress. Nonetheless, themajority occurred in low-stress regions [7]. Further-more, TT interactions were more common than TMinteractions at all stress levels, indicating that threadshave efficient means of “finding each other”, limiting

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R. S. Fertig, S. P. Baker / Scripta Materialia 65 (2011) 384–387 385

hxi for TT interactions as well. We have identified twofeatures that contribute to this. The first is the size ofthe interaction radius – the distance within which theattractive force between two threads requires that thethreads interact. The second is the notion that threadsmoving in an inhomogeneous stress field with a velocitythat is related to the local stress must be concentrated inthe regions of low stress because they spend more timethere. We have modeled the interaction radius elsewhere[13]. Here, we develop a statistical model to quantify theeffect of stress inhomogeneity on the likelihood of athread–thread interaction. The results can be used toincorporate the magnitude of the stress inhomogeneityinto models of film strength. We note that, while appliedto dislocations in this case, this model applies equallywell to any system of interacting objects that move witha velocity related by a power law to a spatially inhomo-geneous field.

We consider a film with M threads moving through itin response to a stress that is inhomogeneously distrib-uted in the plane of the film. For simplicity, the inclina-tion of a thread on its slip plane is ignored so that eachthread can be represented by a single point in the x–yplane, as shown in Figure 1. To model the stress distri-bution, we divide the film into N bins of equal size(Fig. 1) and assign a discrete stress to each bin. If the re-solved shear stress s on the thread exceeds sc, we definethe excess stress as sexc = s � sc [14]. We obtain arealistic stress distribution by assigning a normalizedexcess stress si ¼ sexc;i

scto each bin i such that

PNi¼1si ¼

NR1�1 sQðsÞds, where Q(s) is the probability density

function that describes the distribution of film stresses.We assume that any thread in bin i moves with velocitymi ¼ bsn

i [15], where b is the thread mobility and n is thestress exponent.

As threads move through the film, they spend timeti ¼ ðr=bÞs�n

i in each bin, where r is the bin width. Theprobability pi that a dislocation will be in a particularbin i at a given time is then just the fraction of time spentin bin i relative to the time required to move through allthe bins,

1 2 3 4

N = 16

r

Bin #

Threadingdislocation

Figure 1. A plan-view schematic of a film divided into N equally sizedbins. Each bin has a particular shear stress associated with it consistentwith a realistic probability distribution. M threading dislocations,indicated by filled circles, are initially randomly distributed throughoutthe film and move at a rate that depends on the local stress.

pi ¼tiPNj¼1tj

¼ s�niPN

j¼1s�nj

¼ 1

Nsni hs�ni ; ð1Þ

where hs�ni indicates the average value of s�n. Becauses�1 diverges as s approaches zero, we introduce a cut-off parameter c such that s P c in every bin. Physically,c sets a lower limit on thread velocity. The average valuein Eq. (1) can be obtained from Q(s),

hs�ni ¼Z 1

cQðsÞs�nds: ð2Þ

Eq. (1) describes a Bernoulli distribution for each bin,thus the probability Pk,i that k out of M threads are inbin i is

P k;i ¼M !

k!ðM � kÞ! pki ð1� piÞ

M�k: ð3Þ

We require that the width r of each bin be set equal tothe TT interaction radius [13], so that we can assumethat any two threads in the same bin will interact. Theprobability of a TT interaction occurring in any bin iis then

P int;i ¼XM

k¼2

P k;i: ð4Þ

Eqs. (3) and (4) can be simplified by considering only thecase where pi� l, such that the likelihood of twothreads being in a single bin is low and the likelihoodthat three or more threads will be in a bin can be ne-glected. This requires that the number of threads relativeto the number of bins be small, M/N� 1. SinceqTD = M/A and r ¼

ffiffiffiffiffiffiffiffiffiffiA=N

p, where qTD is the thread

density and A is the film area, we have

M jN ¼ qTDr2: ð5Þ

Thus, an equivalent requirement is qTDr2� 1. For high-er dislocation densities, the occurrence of more than twothreads in the same bin cannot be neglected. However,the qualitative conclusions presented here would stillhold, since the factors that facilitate interactions be-tween two threads will also facilitate interactions be-tween multiple threads.

Substituting Eq. (3) into Eq. (4) and ignoring occur-rences of more than two dislocations per bin yields

P int;i � P 2;i ¼MðM � 1Þ

2p2

i ð1� piÞM�2

: ð6Þ

For the case pi� 1, the exponential term in Eq. (6) canbe replaced with the first term of a Taylor series,

P int;i �MðM � 1Þ

2p2

i ð1� ðM � 2ÞpiÞ: ð7Þ

Substituting Eq. (1) into Eq. (7) and assuming thatM� 1 gives

P int;i �M2

2N 2

s�2ni

hs�ni2�M

Ns�3n

i

hs�ni3

!: ð8Þ

The number of interactions / is then determined by tak-ing the average of Pint,i over i bins and multiplying by N,

f/ f h

2

4

6

8

10

12

14

n = 0.5n = 1n = 1.5

(a)

μ = 1c = 0.03

Interaction probability for different stress exponents

386 R. S. Fertig, S. P. Baker / Scripta Materialia 65 (2011) 384–387

/ ¼ NhP inti ¼M2

2Nhs�2nihs�ni2

�MNhs�3nihs�ni3

!: ð9Þ

Multiplying / by two threads per interaction and divid-ing by M gives the fraction of threads f involved in TTinteractions, which can be written, using Eq. (5), as

f ¼ qTDr2 hs�2nihs�ni2

� qTDr2 hs�3nihs�ni3

!: ð10Þ

The fraction of interacting threads in a homogeneousfield fh, where hs�2ni = hs�ni2, is then given by fn =qTDr2(1 � qTDr2).

In order to examine the effect of stress inhomogeneityon f in Eq. (10), a distribution Q(s) must be constructed.Our discrete dislocation dynamics simulations [7]showed that a normal distribution characterizes thestresses arising from misfits well. Thus, the stresses hereare assumed to be normally distributed, following

F ðsÞ ¼ 1

rffiffiffiffiffiffi2pp exp � 1

2

s� lr

� �2� �

; ð11Þ

where l and r are the mean and standard deviation of s,respectively. Because only values of s between c andinfinity are allowed, the distribution in Eq. (11) is trun-cated below c and normalized such that the integral overthe new probability density function Q(s) is unity,

QðsÞ ¼R1

c F ðsÞds� ��1

F ðsÞ; s P c

0; s < c

(: ð12Þ

Finally, f is determined from Eq. (10), using Q(s) as inEq. (2) to solve for the necessary averages. We examinedthe effect of varying r, l, n, and c on the fraction ofinteracting threads f normalized by fh. Figure 2 showsthe effect of varying l and r on f/fh, where c and n areheld constant at 0.03 and 1, respectively. The data areplotted as a function of the coefficient of variation, r/

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

σ/μ

f/ f h

μ = 0.25μ = 1μ = 3

I II III

Probability of threading dislocation interactions

n = 1c = 0.03

Figure 2. The probability of interactions between threading disloca-tions, f, normalized by the probability in a homogeneous stress field, fh,vs. the coefficient of variation of the stress distribution. Anyinhomogeneity increases the likelihood of thread–thread interactions(f/fh > 1), but the effect becomes particularly dramatic for r/l P 0.25.

l, which represents the degree of inhomogeneity in thestress field. f/fh is greater than unity for all values of rand l, indicating that stress inhomogeneity always in-creases the likelihood of TT interactions from that ofa homogeneous stress field. Furthermore, any increasein inhomogeneity increases the likelihood of interaction.The curves can be divided up into three regions rela-tively independent of l: from r/l = 0 to r/l � 0.25, in-creases in r/l increase the likelihood of interaction onlyslightly; from r/l � 0.25 to r/l � 0.5, rapid increases ininteraction likelihood are observed for relatively smallincreases in r/l; and for r/l > 0.5, the interaction likeli-hood continues to increase, but at a decreasing rate sincef/fh is bounded by 1/fh. The threshold value of r/l � 0.25 is unexpected and represents a critical pointin the influence of stress inhomogeneity. The curves dis-played in Figure 2 were found to be essentially un-changed for different values of qTDr2� 1 (not shown).

The effect of the stress exponent n is illustrated in Fig-ure 3a, where l and c are held constant at 1.0 and 0.03,respectively. The primary effect of changing n is a drasticchange in the effect of inhomogeneous stresses forr/l > 0.25, where increases in n result in substantial

0

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

σ/μ

f / f h

c = 0.1c = 0.05c = 0.03

(b) Interaction probability for different cutoff parameters

μ = 1n = 1

Figure 3. The probability of interactions between threading disloca-tions normalized by the probability in a homogeneous stress field vs.the coefficient of variation of the stress distribution for different (a)stress exponents n and (b) cutoff parameters c. Neither n nor c

qualitatively changes the results shown in Fig. 2.

R. S. Fertig, S. P. Baker / Scripta Materialia 65 (2011) 384–387 387

increases in interaction likelihood. The qualitativetrends for n = 1/2 and 3/2 are the same as for n = 1:inhomogeneous stress facilitates TT interactions, espe-cially for r/l > 0.25. Below this threshold value, the ef-fect of changing n is much less pronounced.

Figure 3b shows the effect of changing the cutoffparameter c, for l = 1 and n = 1. The results are quali-tatively similar to those observed in Figure 2. Decreas-ing c has the effect of increasing the likelihood ofinteraction due to the fact that lower dislocation veloc-ities are allowed with lower values of c. Again, the effectsof changing c are confined predominantly to the regionr/l > 0.25.

Three comments regarding the applicability of thismodel to real films are warranted. First, for Eq. (1) toaccurately describe the probability of a thread being inbin i, the thread must sample every bin. If a film withsome inhomogeneous stress field is initially seeded ran-domly with threads, the thread locations would beuncorrelated to the stresses and the corresponding frac-tion of threads in TT interactions would be character-ized by fh. In this case, Eq. (1) becomes more preciseas the threads move through an increasing number ofbins. Thus, for a particular instant in time, the likeli-hood of interactions in a real film will have lower andupper bounds fh and f, respectively. Cellular automatasimulations of thread motion have shown that threadsneed only sample a small number of bins before themodel presented here describes their interaction likeli-hood well [16]. Second, in this model, threads are as-sumed to be able to move into any bin, but in realfilms threads are confined to move on their slip planes.This does not invalidate the results presented here pro-vided that the dislocations sample Q(s). Finally, thestresses are continuous and correlated in real films, butin the proposed model they are discrete and random. Be-cause the stresses in the model were only used to get adistribution of velocities, we do not believe this assump-tion introduces significant qualitative differences be-tween real films and model films.

Regardless of the precise description of the stressfield, two features of Eq. (10) are notable. First, the frac-tion of threads that interact is expected to vary approx-imately linearly with the thread density. Second, thenumber of TT interactions is expected to be propor-tional to the square of the interaction radius, r. Fors > sc, r has been shown to be inversely proportionalto sexc [13], so a higher l would decrease r and resultin a lower f. However, as long as qTDr2� 1, changesin r have only small effects on the normalized f/fh shownin Figure 2.

The prediction that inhomogeneous stresses increasethe likelihood of thread–thread interactions is consistentwith our recent dislocation dynamics simulations [7]. Be-cause the stress field was such that r/l was always great-er than the threshold value of r/l � 0.25, the modelexplains both why the majority of threads were immobi-lized by TT rather than TM interactions, and why TTinteractions were observed to occur nearly exclusivelyin regions of low film stress in the simulations.

In summary, a model has been presented to quantifythe effect of stress inhomogeneity on the likelihood ofthread–thread interactions in a thin film. It shows thatthe probability of thread–thread interactions is alwaysgreater in an inhomogeneous field than in a homoge-neous field, and that the fraction of interacting threadsshould be proportional to the thread density and tothe square of the interaction radius, which has been re-lated elsewhere [13] to the mean stress. A threshold valueof the coefficient of variation of the resolved shear stressr/l � 0.25 was found such that, below this value, theinfluence of the stress inhomogeneity and stress expo-nent on the fraction of interacting threads is low. Abovethis value, the effects of the stress inhomogeneity arestriking, with the probability of thread–thread interac-tions becoming significantly greater than in a homoge-neous field. Finally, a strong sensitivity to the stressexponent was shown, which demonstrates that preciseunderstanding of stress–velocity relationships in filmsis critical for determining the frequency with which TTinteractions occur and, consequently, for developingaccurate strain hardening models. We reiterate that theapproach presented here is not limited to dislocationsin films, but is applicable to any system with interactingobjects that move with a velocity related by a power lawto a spatially inhomogeneous field.

This work was funded by the National ScienceFoundation Contract No. DMR-0311848. R.F. thanksS. Hicks for useful discussions on the statistical aspectsof this problem.

[1] S.P. Baker, Mater. Sci. Eng. A 319-321 (2001) 16–23.[2] P. Pant, K.W. Schwarz, S.P. Baker, Acta Mater. 51 (2003)

3243–3258.[3] K.W. Schwarz, Phys. Rev. Lett. 91 (2003) 145501–145503.[4] R.S. Fertig, S.P. Baker, Prog. Mater. Sci. 54 (2009) 874–

908.[5] W.D. Nix, Metall. Trans. A 20 (1989) 2217–2245.[6] L.B. Freund, J. Appl. Mech. 54 (1987) 553–557.[7] R.S. Fertig, S.P. Baker, Acta Mater. 58 (2010) 5206–5218.[8] V. Weihnacht, W. Bruckner, Acta Mater. 49 (2001) 2365–

2372.[9] J.R. Willis, S.C. Jain, R. Bullough, Phil. Mag. A 62 (1990)

115–129.[10] S. Kramer, J. Mayer, C. Witt, A. Weickenmeier, M.

Ruhle, Ultramicroscopy 81 (2000) 245–262.[11] R. Spolenak, W.L. Brown, N. Tamura, A.A. MacDowell,

R.S. Celestre, H.A. Padmore, B. Valek, J.C. Bravman, T.Marieb, H. Fujimoto, B.W. Batterman, J.R. Patel, Phys.Rev. Lett. 90 (2003) 096102.

[12] T.J. Balk, G. Dehm, E. Arzt, Acta Mater. 51 (2003) 4471–4485.

[13] R.S. Fertig, PhD Dissertation, Cornell University, 2010.[14] L.B. Freund, R. Hull, J. Appl. Phys. 71 (1992) 2054.[15] L.B. Freund, Scr. Metall. Mater. 27 (1992) 669–674.[16] R.S. Fertig, S.P. Baker, in: D. LaVan, M. Spearing, S.

Vengallatore, M.d. Silva (Eds.), MicroelectromechanicalSystems – Materials and Devices, Materials ResearchSociety, Boston, MA, 2007, p. DD07-06.