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Capture cross-section of threading dislocations in thin films
Materials Science and Engineering A 551 (2012) 67–72
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Capture cross-section of threading dislocations in thin films Ray S. Fertig III ∗ , Shefford P. Baker Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, United States
a r t i c l e
i n f o
Article history: Received 13 January 2012 Received in revised form 12 April 2012 Accepted 17 April 2012 Available online 2 May 2012 Keywords: Threading dislocations Dislocation interactions Thin films Dislocation simulation
a b s t r a c t The capture cross section for annihilation of two threads with opposite Burgers vectors moving on orthogonal slip planes in a thin film is examined using a numerical model. The initial configurations of threads that lead to annihilation are mapped out for a range of applied film stresses. The area of the region of initial configurations that lead to annihilation at a given stress and thickness is the capture cross-section. The size of the capture cross-section is shown to be highly sensitive to the applied stress relative to the critical stress for dislocation motion imposed by the film thickness. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Thin films are used in a wide variety of applications ranging from microelectronics to biomedical devices to solar cells. In many applications films support stresses in excess of the ultimate stresses of their bulk counterparts by an order of magnitude [1]. The reasons for this are not fully understood. These high stresses contribute to failure mechanisms such as fracture, delamination, and stress voiding. Thus, understanding the mechanical behavior of thin films is crucial to improving thin film performance and reliability. High stresses are only possible when dislocation motion is difficult. In this paper, we consider only large-grained or single crystal films. In such films, threading dislocations (threads), which span the thickness of the film, move through the film to relax the film stress by depositing misfit dislocations (misfits) at film interfaces or creating steps at the film free surfaces [2–4]. Thus, to understand high film stresses we must know what stops threads [4–7]. One such impediment is a critical stress for dislocation motion—the so-called channeling stress, ch [5]—which varies as the reciprocal of the film thickness [2,3] and arises from a balance between the energy increase due to adding a misfit dislocation and the strain energy decrease from the stress relaxed by the dislocation. Above ch , threads move until they are stopped by interacting with other dislocations, both threads and misfits. We used discrete dislocation dynamics (DDD) simulations to study these interactions in detail [4], and found that interac-
∗ Corresponding author. Present address: Department of Mechanical Engineering, Dept. 3295, University of Wyoming, 1000 E. University Ave., Laramie, WY 82071, United States. Tel.: +1 307 766 3647; fax: +1 307 766 2695. E-mail address: [email protected] (R.S. Fertig III). 0921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2012.04.084
tions between two threads stop a large fraction of the mobile threads in a film. Because many thread–thread interactions are very strong, they often permanently immobilize threads, which leads to a reduction in the number of threads available to relax the film stress and, consequently, to strain hardening. Thus, thread–thread interactions are very important to determining film strength. To quantify this relationship, it is necessary to determine the probability that two threads can end up in the same place at the same time to interact. This is the motivation of the present work. The frequency of interaction between mobile threads is controlled by two factors: mobile thread density and the capture cross-sections of the threads. The capture cross-section represents the locus of points around a thread such that another thread with suitable Burgers vector inside this region will form a junction or annihilate with the initial thread. The size of this capture cross-section has been studied previously, but only as it relates to cross-slip, climb, or layer growth [8–16]. Our DDD simulations showed that interactions between threading dislocations on intersecting slip planes played a significant role in stopping threads in (0 0 1) films [4], which makes the capture cross-section of intersecting threads a critical topic of study. The goal of the work reported here is to determine the size and shape of the capture cross-section for a thread annihilating with another thread moving in an orthogonal direction on an intersecting slip plane. In contrast to previous investigations [15,17] that considered cross-slip/climb as the mechanism for annihilation, we focus on dislocation glide in response to an applied stress that leads to annihilation. In particular, we are interested in how the capture cross section varies with applied stress and with film thickness. We find that the capture cross section varies strongly with the applied stress and that ch acts as a limiting stress.
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The force per unit length on the dislocation was then calculated from the Peach–Koehler formula [18],
A
Fi = −εijk j kl bl ,
R
y
where εijk is the permutation tensor. The force on each dislocation was calculated by substituting Eq. (2) into Eq. (3). The glide force on each dislocation was found by taking the dot product of these forces with the slip directions [0 1 0] and [1 0 0], for dislocations A and B, respectively, using
θ B
x
(3)
[0 1 0]
FA,glide = −
b2 y − app b 2R2
(4a)
b2 x − app b. 2R2
(4b)
and
[1 0 0] FB,glide = −
[0 0 1] Fig. 1. Schematic of the model setup. Dislocation A moves in the ±y direction and dislocation B moves in the ±x direction in response to applied stresses and the interaction stresses between the two dislocations. The capture cross section is the locus of initial positions of A with respect to B within which A will move to ultimately annihilate with B.
2. Model In a real film, threads are curved, of mixed character, and may cross-slip or climb [3]. To treat this problem in an efficient and tractable way, we model the threads using straight screw segments that are confined to their slip planes. We consider threads A and B having pure screw character moving on intersecting {1 0 0} slip planes in a cubic (0 0 1) film, as shown in Fig. 1. The stress field of these dislocations was approximated as the stress field of two
dˆy dt˜
dˆx dt˜
⎧
⎪
− yˆ − app − 1 sign − yˆ − app , − yˆ − app ≥ 1 ⎪ ⎨ 2ch Rˆ 2 ch
2ch Rˆ 2 ch 2ch Rˆ 2 ch
=
⎪ ⎪ ⎩ 0, − yˆ − app < 1
2ch Rˆ 2 ch
⎧
⎪
− xˆ − app − 1 sign − xˆ − app , − xˆ − app ≥ 1 ⎪ ⎨ 2ch Rˆ 2 ch
2ch Rˆ 2 2ch Rˆ 2 ch ch
=
⎪ ⎪ ⎩ 0, − xˆ − app < 1
2ch Rˆ 2 ch
infinite screw dislocations with sense vectors A = B = [0 0 1]. The Burgers vectors for the dislocations were bA = [0 0 b] and bB = −bA , such that they would annihilate if they formed a junction. These assumptions allowed the stress field of each dislocation to be written as
dis
b = 2R2
0 0 0 0 y −x
y −x 0
,
(1)
where x and y are coordinates as shown in Fig. 1, R = x 2 + y2 , and is the shear modulus. Uniform shear stresses xz = yz = app were applied to the film so that an equal applied force is applied to both dislocations. The total stress felt by either dislocation due to the applied stress and the stress field of the other dislocation was then
b A = B = 2R2
0 0 0 0 y x
y x 0
+
That is, dislocation A was confined to move in the ±y direction (force due to applied stress acts down in Fig. 1) and dislocation B was confined to move in the ±x direction (force due to applied stress acts to the left in Fig. 1). The interaction stress is, of course, always attractive. For a dislocation to move in a thin film, the glide force must exceed a thickness dependent critical force ch b that is determined by the resolved shear stress at the channeling stress ch [3]. The magnitude of the difference between the applied glide force and the critical force is termed the excess glide force Fge , following [19]. In our model, the dislocation velocity v was related to Fge by a mobility M using a mobility law = MFge . In the case where the magnitude of the glide force is less than the magnitude of the critical force, the dislocation remains stationary. Given this mobility law and that vA and vB represent dy/dt and dx/dt, respectively, the system of equations describing the motion of the dislocations is given by
0 0
0 0
app
app
app app 0
.
(2)
(5a)
(5b)
where t is time, xˆ = x/b, yˆ = y/b, Rˆ = xˆ 2 + yˆ 2 , t˜ = t/t0 , and t0 = 1/(M ch ). Eq. (5a) describes the motion of dislocation A and (5b) describes the motion of dislocation B. Eqs. (5) were solved using a Runge–Kutta technique following [20]. Simulations were conducted by specifying initial positions for the dislocations then iteratively calculating new positions by assuming the velocity obtained from Eqs. (5a) and (5b) was constant over a time step h. The time step was adapted such that it decreased as the rate of approach of the dislocations increased and increased with increasing distance between dislocations. This reduced computation time when the dislocations were initially far from each other and prevented spurious results from occurring when the dislocations were close with high velocities. In the implementation of this solution, h was initially set to h = 0.01 and thereafter was prescribed to be
hn = ˛Rˆ n−1
t˜n−1 − t˜n−2 ˆ Rn−1 − Rˆ n−2
(6)
where t is time, n corresponds to the time step number, and ˛ = 0.005 for the solutions reported here. The value of ˛ was chosen
R.S. Fertig III, S.P. Baker / Materials Science and Engineering A 551 (2012) 67–72
A=
360
360
Ri2 .
(a) 120 100 80
y/b
such that lower values (smaller time steps) did not yield different cross-sectional areas. To determine the capture cross section, simulations were run with dislocation A starting at various distances R from dislocation B along a trajectory established by the angle (Fig. 1). Depending on R and , either or both dislocations could move toward or away from the point of intersection of their slip planes. The threads were allowed to move until either the distance between the dislocations was less than one Burgers vector or the dislocation velocities began to increase in a direction away from interaction, i.e. d2 R/dt2 > 0. For each trajectory there was a distance Ri within which the dislocations eventually came together at the intersection of their slip planes, and beyond which the applied forces won out and the dislocations moved apart. This distance is the boundary of the capture cross section and was found to within ±1%. To determine the shape of the capture cross section, Ri was calculated along a series of trajectories at one degree increments of around thread B. Once the boundary of the cross-section was determined, the cross-sectional area A was calculated as
60
69
Effect of applied stress on the capture cross-section of a thread
τapp = 0 τapp = 0.5τch τapp = 0.85τch
40 20 0 −20 −120 −100 −80 −60 −40 −20
0
20
x/b
(b) 600
Effect of applied stress on the capture cross-section of a thread
500
(7)
i=1
3. Results The paths that the two dislocations followed showed a wide range of different possibilities depending on R, , and app . The applied stress could initially drive both A and B toward the intersection point (x > 0, y > 0), away from the intersection point (x < 0, y < 0), A away from the intersection point and B toward it (x > 0, y < 0), or B away from the intersection point and A toward it (x < 0, y > 0). When the interaction force became stronger than the force due to the applied stress, even dislocations that had been moving apart could reverse and come together. Similarly, even when both A and B were initially moving towards the intersection point, if one got sufficiently ahead of the other, it could continue on its slip plane moving away from the intersection point and the dislocations would never interact. Thus, the R– boundary of the capture cross section could be complicated and depend strongly on app . The size and shape of the capture cross section for a thread centered at (0,0) were examined as a function of applied stress and are shown in Fig. 2, and the variation of the area of the capture cross-section with applied stress is shown in Fig. 3. Both the shape and size of the cross-section were found to exhibit fundamentally different behaviors depending on whether the applied stress was less than or greater than the channeling stress. But in both cases, the area of the capture cross-section varies dramatically with the applied stress. 3.1. Applied stress less than the channeling stress ( app < ch ) The shapes of the representative capture cross-sections for the load cases app = 0, app = 0.5 ch , and app = 0.85 ch are shown in Fig. 2a, where ch = 0.01 for the capture cross sections shown. With zero applied stress (solid curve in Fig. 2a), the threads will come together to form a junction only if the resolved shear stress exerted by one thread on the other exceeds ch . Thus, the capture cross-section is symmetric about the origin. Any applied stress causes asymmetry in the capture cross-section because the applied
400
y/b
Note that for app < ch , the dynamic simulation is not required, as the capture cross-section boundary can be calculated as the point where Fge = 0. The analytical solution to this problem is nontrivial, however, and the simulation provides sufficient accuracy. All results presented here use the simulation described above.
300 200 100
τapp = 0.85τch τapp = 1.01τch τapp = 1.05τch τapp = 2.00τch
0 −600 −500 −400 −300 −200 −100
0
x/b Fig. 2. Boundaries of the capture cross-sections for different applied stresses with dislocation B at the origin. The channeling stress ch is 0.01 m. (a) Applied stress levels below the channeling stress. (b) Applied stress levels above the channeling stress (0.85 ch shown for reference).
stress drives the two dislocations in fixed directions, while the interaction stress drives the dislocations towards each other at all stress levels. For example, at = 135◦ , the applied stress drives both dislocations towards the intersection of their slip planes (Fig. 1) while at = −45◦ the applied stress drives both dislocations away from this point. This asymmetry increases with increasing applied stress and the cross-sectional area becomes increasingly located in the second quadrant. As is evident from Fig. 2a, the capture cross section A increases rapidly as app approaches the channeling stress. The reason for this is intuitively obvious. When the applied stress is below the channeling stress, interactions can occur only when the threads are close enough that the interaction can supply the additional stress needed to exceed the channeling stress. However, for applied stresses near the channeling stress, the additional stress required to move the thread is very low, so the dislocations can be far apart and still draw each other together to annihilate. In fact, in the limit as the applied stress approaches the channeling stress, the threads can be arbitrarily far apart and will still annihilate. A functional form for the area of the capture cross-section below the channeling stress can be approximated by solving the boundary term in Eq. (5b) for yˆ = 0 ⇒ Rˆ = xˆ , which gives the equation for the capture cross-section boundary at the x-axis as
app
1
−
− 2
= 1. ch ˆ ch x
(8)
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R.S. Fertig III, S.P. Baker / Materials Science and Engineering A 551 (2012) 67–72
(a)
the capture cross-sectional area as app approaches the channeling stress. The capture cross section varies roughly as the inverse square of the channeling stress, with the change in area almost entirely due to change of the interaction radius in the second quadrant.
102 Effect of applied stresses less than the channeling stress on the capture cross-section
A(τch/μb)2
101
3.2. Applied stress greater than the channeling stress ( app > ch )
μ = 100τch μ = 10000τch Curve fit
100
10-1
10-2
A(τch/μbRc)
(b)
0
102
0.2
0.4
0.6
τapp/τch
0.8
1
Effect of applied stresses greater than the channeling stress on the capture cross-section
101
μ = 100τch μ = 10000τch Curve fit
100
10-1 1
2
3
4
τapp/τch
5
Fig. 3. Variation of the capture cross section with applied stress. (a) Applied stress levels below the channeling stress. (b) Applied stress levels above the channeling stress.
Solving for Rˆ gives
⎧ 1 ⎪ ⎨ (1 − ( / )) , xˆ < 0 app ch Rˆ = 2ch ⎪ 1 ⎩ , xˆ > 0
(9)
(1 + (app /ch ))
Guided by observation of the cross-section shown in Fig. 2a, we assume that the area of the cross-section in quadrants I, III, and IV is proportional to the square of the radius along the +x-axis. We assume that the area of the cross-section in quadrant II is proportional to the square of the radius along the −x-axis. Adding the areas of the four quadrants gives the following functional form for the area of the capture cross-section: A
2 ch
b
= 1
0.25 (1 − (app /ch ))
2
+
0.75 (1 + (app /ch ))
2
(10)
Fig. 3a shows quantitatively the relationship between the nondimensional parameter A( ch /b2 ) and app . Two different sets of data from our model are plotted: circles represent data at ch = 0.01 and crossed lines represent data taken at ch = 0.0001. Coincidence of these data gives confidence in the normalization factor used. Eq. (10) was fitted to the data in Fig. 3a and
1 = 7.2374 × 10−2 was found for the system modeled. This fit is shown by the solid curve in Fig. 3a. Agreement with the data suggests that this functional form accurately describes the variation of capture cross section with applied stress for stress levels below ch . The most important result shown here is the divergence of
The shapes of the representative capture cross-sections for the load cases app = 1.01, 1.05, and 2.0 ch are shown in Fig. 2b ( app = 0.85 ch is shown for reference), where ch = 0.01 for the capture cross sections shown. The shapes of these cross sections are also easily understood. When the applied stress exceeds the channeling stress, threads do not require interactions with other threads to move; they are already moving on the trajectories determined by their glide planes. If the threads start out equidistant from their point of intersection along the positive x and y axes, they will interact under any applied biaxial stress. Thus the capture boundary extends to infinity at = 135◦ . If the threads are not equidistant, they can pass through the intersection of their glide planes at different times and not interact. The higher the stress, the higher the velocity and the smaller the region in which they can interact. This gives rise to a capture cross section that is predominantly a band centered at = 135◦ that narrows with increasing applied stress. Consequently, for app > ch , the variation in the capture cross-section area is driven by a change in the shape of the cross-section, which is fundamentally distinct from the variation in the capture cross-section area at applied stresses lower than the channeling stress, which is driven by a change in radius of a similar shape. Because the radius extends to infinity along = 135◦ we cannot use the same definition for the area of the cross-section. Instead we introduce a cutoff radius Rc corresponding to a physically relevant maximum distance apart that two threads may be and still interact; this could be determined by a quantity such as grain size or misfit density. Using this cutoff radius, we then approximate the capture cross-sectional area as simply A = Rc d, where d is the width of the band around = 135◦ calculated as the distance between the intersections of the cross-section boundary with straight lines projected along = 134◦ and = 136◦ . Using the above definition, the area of the capture cross section was plotted as a function of the applied stress, as shown in Fig. 3b. In order to determine an appropriate normalization with respect to the channeling stress, two different sets of data are plotted: circles represent data at ch = 0.01 and squares represent data taken at ch = 0.0001. We chose an area normalization of ( ch /bRc ) since it yields nearly identical curves for the two channeling stresses. A simple derivation of the functional form for this curve was not found, but by trial and error we found that the variation in cross sectional area with applied stress above the channeling stress is well described by an equation of the form A Rc
ch
b
=
2 , ((app /ch ) − 1)
app > ch ,
(11)
where 2 is found to be 0.3556 for the curve fit shown in Fig. 3b. In contrast to the cross-sectional area described in Eq. (10) for films below ch , the cross-sectional area above ch is inversely proportional to the applied stress and the channeling stress, and decreases with increasing stress. Furthermore, changes in cross-section are due almost entirely to changes in shape of the cross-section, rather than changes in radius. 4. Discussion This model helps to clarify the role of thread–thread interactions in determining the strengths of thin films. It has been shown [6] that
R.S. Fertig III, S.P. Baker / Materials Science and Engineering A 551 (2012) 67–72
interactions between threading dislocations and misfit dislocations are weak, with maximum interaction strengths (the applied stress needed to cause the thread to break the interaction and move on) near 1.3 ch , while interactions between threads can be very strong, from several times ch for strong junctions to unbounded for annihilation reactions (threads cannot move at any applied stress if they do not exist). Thus, a naïve view might assume that thread–misfit interactions would dominate at low applied stresses and interactions between threads might dominate at high stresses. According to this view, this is because weaker thread–misfit interactions can stop threads at low stresses and the probability of such interactions must always be high since each thread must cross every misfit that intersects its path. At high stresses, interactions with misfits would not be strong enough to stop threads, so each thread would continue through the film until it was able to interact with another thread. However, DDD simulations showed that this naïve scenario does not occur [4]. Instead, threads are stopped in interactions with both misfits and threads at all stress levels. Surprisingly, at lower stresses, mobile threads were stopped more frequently by other threads than by misfits. We explained the fact that thread–misfit interactions effectively stop dislocations at high stress levels because of the inhomogeneous stress field that arises due to the misfit structure [4]. At all applied strain levels there are regions where the local stress is low enough to allow interactions with misfits to stop threads. To better understand the probability of interactions between threading dislocations, we developed a model in which threads move through a film at a velocity that depends on the local stress [21]. We found that any stress inhomogeneity concentrates threads in regions of low stress leading to an increased probability of their interaction. That model, however, requires that threads move some distance before starting to concentrate, while the DDD simulations showed thread–thread interactions to be highly probable even at the lowest strain levels. Thus, the model presented here was developed to understand the effect that capture cross-section might play on the likelihood of threading dislocation interactions at different stress levels. The results provide an explanation for the predominance of thread–thread interactions at low applied stresses. The dramatic increase in the area of the capture cross-section as the stress in the film approaches the channeling stress from above or below increases the likelihood of thread–thread interactions at low applied stresses. Together with previous work [4,21], this understanding leads to a picture of film strength, stress relaxation, and strain hardening in which threading dislocations move through an inhomogeneous stress field, engaging in interactions with other dislocations that have a range of interaction strengths. No single type of interaction can be assumed to control the strength of the film; the controlling interactions will vary depending on dislocation density and film stress. The mean free path that a threading dislocation may travel before being stopped in an interaction with another dislocation is a controlling feature governing strain hardening [22]. It is controlled by the spatial stress distribution in the film, as set up by the misfit structure [4], and by the likelihood of interacting with another thread, which is, in turn, affected by the capture cross section of the thread and the thread density. The results from this work, namely, a mathematical description for the variation of capture cross-section with applied stress and film channeling stress (or film thickness), may be useful in developing a mechanistic description of strain hardening in films. Because film strength is determined by many features in addition to the capture cross section, it is, of course, not appropriate to think of the present model as quantitatively predicting film strength in real films. Instead its value lies in the functional form
71
for the variation in capture cross section with applied stress, which can be used in more complex conceptual or numerical models. Thus, while it is important that the functional form be more-or-less correct, the quantitative “accuracy” in terms of the actual size of the capture cross section is secondary. Indeed, we made a number of simplifying assumptions—that the interacting dislocations are infinite and straight, that the stress state is equal biaxial, that thickness effects are correctly accounted for by the channeling strain, and that the dislocation velocity is linearly proportional to the excess stress—that may affect the magnitude of the capture cross section, but are expected to have only secondary effects on its variation with applied stress. Of course the functional form is exact only for the modeled case (two threads in infinite film). In a real film, the cross-section will not truly diverge, because the stress field set up by misfit dislocations as well as stresses from nearby threads that repel or are not on intersecting slip planes will overshadow any effect of some thread very far away. Nevertheless, particularly for the case of threads in relatively close proximity, the predicted capture cross-section gives a measure of the likelihood of interaction, which is a primary quantity of interest in developing a mechanistic description of film relaxation. 5. Conclusions We have presented an analytical model to calculate the capturecross section of a thread gliding in response to an applied stress, such that another thread with opposite Burgers vector on an intersecting path will annihilate with the thread when it is inside this capture cross-section. This model is expected to accurately represent the qualitative behavior of real threading dislocations in thin films. For applied stresses below the channeling stress, the size of the capture cross-section increases with the inverse of the square of the applied stress. For applied stresses above the channeling stress, the size of the capture cross section decreases with the inverse of the applied stress. When the applied stress is equal to the channeling stress, the size of the capture cross-section diverges. This behavior of the capture cross-section of a thread in response to applied film stresses is a critical element in understanding the link between local dislocation behavior and macroscopic film relaxation. Acknowledgements This work was funded by the National Science Foundation Contract No. DMR-0311848. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
S.P. Baker, Mater. Sci. Eng. A 319–321 (2001) 16–23. L.B. Freund, J. Appl. Mech. 54 (1987) 553–557. W.D. Nix, Metall. Trans. A 20 (1989) 2217–2245. R.S. Fertig III, S.P. Baker, Acta Mater. 58 (2010) 5206–5218. W.D. Nix, Scripta Mater. 39 (1998) 545–554. P. Pant, K.W. Schwarz, S.P. Baker, Acta Mater. 51 (2003) 3243–3258. R.S. Fertig, S.P. Baker, Progr. Mater. Sci. 54 (2009) 874–908. U. Essmann, H. Mughrabi, Philos. Mag. A: Phys. Condens. Matter Struct. Defect Mech. Prop. 40 (1979) 731–756. A.A. Kusov, V.I. Vladimirov, Phys. Status Solidi B 138 (1986) 135–142. B.W. Dodson, J. Electron. Mater. 19 (1990) 503–508. G.E. Beltz, M. Chang, J.S. Speck, W. Pompe, A.E. Romanov, Philos. Mag. A: Phys. Condens. Matter Struct. Defect Mech. Prop. 76 (1997) 807–835. A.E. Romanov, W. Pompe, G. Beltz, J.S. Speck, Phys. Status Solidi B 199 (1997) 33–49. A.E. Romanov, W. Pompe, G. Beltz, J.S. Speck, Phys. Status Solidi B 198 (1996) 599–613. A.E. Romanov, W. Pompe, G.E. Beltz, J.S. Speck, Appl. Phys. Lett. 69 (1996) 3342–3344. A.E. Romanov, W. Pompe, S. Mathis, G.E. Beltz, J.S. Speck, J. Appl. Phys. 85 (1999) 182–192.
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[16] J.S. Speck, M.A. Brewer, G. Beltz, A.E. Romanov, W. Pompe, J. Appl. Phys. 80 (1996) 3808–3816. [17] G.E. Beltz, M. Chang, M.A. Eardley, W. Pompe, A.E. Romanov, J.S. Speck, Mater. Sci. Eng. A: Struct. 234 (1997) 794–797. [18] J.P. Hirth, J. Lothe, Theory of Dislocations, John Wiley and Sons, Inc., New York, 1982.
[19] L.B. Freund, R. Hull, J. Appl. Phys. 71 (1992) 2054. [20] W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 6 ed., John Wiley & Sons, Inc., New York, 1997. [21] R.S. Fertig III, S.P. Baker, Scripta Mater. 65 (2011) 384–387. [22] B. Devincre, T. Hoc, L. Kubin, Science 320 (2008) 1745–1748.
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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Capture cross-section of threading dislocations in thin films Ray S. Fertig III ∗ , Shefford P. Baker Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, United States
a r t i c l e
i n f o
Article history: Received 13 January 2012 Received in revised form 12 April 2012 Accepted 17 April 2012 Available online 2 May 2012 Keywords: Threading dislocations Dislocation interactions Thin films Dislocation simulation
a b s t r a c t The capture cross section for annihilation of two threads with opposite Burgers vectors moving on orthogonal slip planes in a thin film is examined using a numerical model. The initial configurations of threads that lead to annihilation are mapped out for a range of applied film stresses. The area of the region of initial configurations that lead to annihilation at a given stress and thickness is the capture cross-section. The size of the capture cross-section is shown to be highly sensitive to the applied stress relative to the critical stress for dislocation motion imposed by the film thickness. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Thin films are used in a wide variety of applications ranging from microelectronics to biomedical devices to solar cells. In many applications films support stresses in excess of the ultimate stresses of their bulk counterparts by an order of magnitude [1]. The reasons for this are not fully understood. These high stresses contribute to failure mechanisms such as fracture, delamination, and stress voiding. Thus, understanding the mechanical behavior of thin films is crucial to improving thin film performance and reliability. High stresses are only possible when dislocation motion is difficult. In this paper, we consider only large-grained or single crystal films. In such films, threading dislocations (threads), which span the thickness of the film, move through the film to relax the film stress by depositing misfit dislocations (misfits) at film interfaces or creating steps at the film free surfaces [2–4]. Thus, to understand high film stresses we must know what stops threads [4–7]. One such impediment is a critical stress for dislocation motion—the so-called channeling stress, ch [5]—which varies as the reciprocal of the film thickness [2,3] and arises from a balance between the energy increase due to adding a misfit dislocation and the strain energy decrease from the stress relaxed by the dislocation. Above ch , threads move until they are stopped by interacting with other dislocations, both threads and misfits. We used discrete dislocation dynamics (DDD) simulations to study these interactions in detail [4], and found that interac-
∗ Corresponding author. Present address: Department of Mechanical Engineering, Dept. 3295, University of Wyoming, 1000 E. University Ave., Laramie, WY 82071, United States. Tel.: +1 307 766 3647; fax: +1 307 766 2695. E-mail address: [email protected] (R.S. Fertig III). 0921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2012.04.084
tions between two threads stop a large fraction of the mobile threads in a film. Because many thread–thread interactions are very strong, they often permanently immobilize threads, which leads to a reduction in the number of threads available to relax the film stress and, consequently, to strain hardening. Thus, thread–thread interactions are very important to determining film strength. To quantify this relationship, it is necessary to determine the probability that two threads can end up in the same place at the same time to interact. This is the motivation of the present work. The frequency of interaction between mobile threads is controlled by two factors: mobile thread density and the capture cross-sections of the threads. The capture cross-section represents the locus of points around a thread such that another thread with suitable Burgers vector inside this region will form a junction or annihilate with the initial thread. The size of this capture cross-section has been studied previously, but only as it relates to cross-slip, climb, or layer growth [8–16]. Our DDD simulations showed that interactions between threading dislocations on intersecting slip planes played a significant role in stopping threads in (0 0 1) films [4], which makes the capture cross-section of intersecting threads a critical topic of study. The goal of the work reported here is to determine the size and shape of the capture cross-section for a thread annihilating with another thread moving in an orthogonal direction on an intersecting slip plane. In contrast to previous investigations [15,17] that considered cross-slip/climb as the mechanism for annihilation, we focus on dislocation glide in response to an applied stress that leads to annihilation. In particular, we are interested in how the capture cross section varies with applied stress and with film thickness. We find that the capture cross section varies strongly with the applied stress and that ch acts as a limiting stress.
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The force per unit length on the dislocation was then calculated from the Peach–Koehler formula [18],
A
Fi = −εijk j kl bl ,
R
y
where εijk is the permutation tensor. The force on each dislocation was calculated by substituting Eq. (2) into Eq. (3). The glide force on each dislocation was found by taking the dot product of these forces with the slip directions [0 1 0] and [1 0 0], for dislocations A and B, respectively, using
θ B
x
(3)
[0 1 0]
FA,glide = −
b2 y − app b 2R2
(4a)
b2 x − app b. 2R2
(4b)
and
[1 0 0] FB,glide = −
[0 0 1] Fig. 1. Schematic of the model setup. Dislocation A moves in the ±y direction and dislocation B moves in the ±x direction in response to applied stresses and the interaction stresses between the two dislocations. The capture cross section is the locus of initial positions of A with respect to B within which A will move to ultimately annihilate with B.
2. Model In a real film, threads are curved, of mixed character, and may cross-slip or climb [3]. To treat this problem in an efficient and tractable way, we model the threads using straight screw segments that are confined to their slip planes. We consider threads A and B having pure screw character moving on intersecting {1 0 0} slip planes in a cubic (0 0 1) film, as shown in Fig. 1. The stress field of these dislocations was approximated as the stress field of two
dˆy dt˜
dˆx dt˜
⎧
⎪
− yˆ − app − 1 sign − yˆ − app , − yˆ − app ≥ 1 ⎪ ⎨ 2ch Rˆ 2 ch
2ch Rˆ 2 ch 2ch Rˆ 2 ch
=
⎪ ⎪ ⎩ 0, − yˆ − app < 1
2ch Rˆ 2 ch
⎧
⎪
− xˆ − app − 1 sign − xˆ − app , − xˆ − app ≥ 1 ⎪ ⎨ 2ch Rˆ 2 ch
2ch Rˆ 2 2ch Rˆ 2 ch ch
=
⎪ ⎪ ⎩ 0, − xˆ − app < 1
2ch Rˆ 2 ch
infinite screw dislocations with sense vectors A = B = [0 0 1]. The Burgers vectors for the dislocations were bA = [0 0 b] and bB = −bA , such that they would annihilate if they formed a junction. These assumptions allowed the stress field of each dislocation to be written as
dis
b = 2R2
0 0 0 0 y −x
y −x 0
,
(1)
where x and y are coordinates as shown in Fig. 1, R = x 2 + y2 , and is the shear modulus. Uniform shear stresses xz = yz = app were applied to the film so that an equal applied force is applied to both dislocations. The total stress felt by either dislocation due to the applied stress and the stress field of the other dislocation was then
b A = B = 2R2
0 0 0 0 y x
y x 0
+
That is, dislocation A was confined to move in the ±y direction (force due to applied stress acts down in Fig. 1) and dislocation B was confined to move in the ±x direction (force due to applied stress acts to the left in Fig. 1). The interaction stress is, of course, always attractive. For a dislocation to move in a thin film, the glide force must exceed a thickness dependent critical force ch b that is determined by the resolved shear stress at the channeling stress ch [3]. The magnitude of the difference between the applied glide force and the critical force is termed the excess glide force Fge , following [19]. In our model, the dislocation velocity v was related to Fge by a mobility M using a mobility law = MFge . In the case where the magnitude of the glide force is less than the magnitude of the critical force, the dislocation remains stationary. Given this mobility law and that vA and vB represent dy/dt and dx/dt, respectively, the system of equations describing the motion of the dislocations is given by
0 0
0 0
app
app
app app 0
.
(2)
(5a)
(5b)
where t is time, xˆ = x/b, yˆ = y/b, Rˆ = xˆ 2 + yˆ 2 , t˜ = t/t0 , and t0 = 1/(M ch ). Eq. (5a) describes the motion of dislocation A and (5b) describes the motion of dislocation B. Eqs. (5) were solved using a Runge–Kutta technique following [20]. Simulations were conducted by specifying initial positions for the dislocations then iteratively calculating new positions by assuming the velocity obtained from Eqs. (5a) and (5b) was constant over a time step h. The time step was adapted such that it decreased as the rate of approach of the dislocations increased and increased with increasing distance between dislocations. This reduced computation time when the dislocations were initially far from each other and prevented spurious results from occurring when the dislocations were close with high velocities. In the implementation of this solution, h was initially set to h = 0.01 and thereafter was prescribed to be
hn = ˛Rˆ n−1
t˜n−1 − t˜n−2 ˆ Rn−1 − Rˆ n−2
(6)
where t is time, n corresponds to the time step number, and ˛ = 0.005 for the solutions reported here. The value of ˛ was chosen
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A=
360
360
Ri2 .
(a) 120 100 80
y/b
such that lower values (smaller time steps) did not yield different cross-sectional areas. To determine the capture cross section, simulations were run with dislocation A starting at various distances R from dislocation B along a trajectory established by the angle (Fig. 1). Depending on R and , either or both dislocations could move toward or away from the point of intersection of their slip planes. The threads were allowed to move until either the distance between the dislocations was less than one Burgers vector or the dislocation velocities began to increase in a direction away from interaction, i.e. d2 R/dt2 > 0. For each trajectory there was a distance Ri within which the dislocations eventually came together at the intersection of their slip planes, and beyond which the applied forces won out and the dislocations moved apart. This distance is the boundary of the capture cross section and was found to within ±1%. To determine the shape of the capture cross section, Ri was calculated along a series of trajectories at one degree increments of around thread B. Once the boundary of the cross-section was determined, the cross-sectional area A was calculated as
60
69
Effect of applied stress on the capture cross-section of a thread
τapp = 0 τapp = 0.5τch τapp = 0.85τch
40 20 0 −20 −120 −100 −80 −60 −40 −20
0
20
x/b
(b) 600
Effect of applied stress on the capture cross-section of a thread
500
(7)
i=1
3. Results The paths that the two dislocations followed showed a wide range of different possibilities depending on R, , and app . The applied stress could initially drive both A and B toward the intersection point (x > 0, y > 0), away from the intersection point (x < 0, y < 0), A away from the intersection point and B toward it (x > 0, y < 0), or B away from the intersection point and A toward it (x < 0, y > 0). When the interaction force became stronger than the force due to the applied stress, even dislocations that had been moving apart could reverse and come together. Similarly, even when both A and B were initially moving towards the intersection point, if one got sufficiently ahead of the other, it could continue on its slip plane moving away from the intersection point and the dislocations would never interact. Thus, the R– boundary of the capture cross section could be complicated and depend strongly on app . The size and shape of the capture cross section for a thread centered at (0,0) were examined as a function of applied stress and are shown in Fig. 2, and the variation of the area of the capture cross-section with applied stress is shown in Fig. 3. Both the shape and size of the cross-section were found to exhibit fundamentally different behaviors depending on whether the applied stress was less than or greater than the channeling stress. But in both cases, the area of the capture cross-section varies dramatically with the applied stress. 3.1. Applied stress less than the channeling stress ( app < ch ) The shapes of the representative capture cross-sections for the load cases app = 0, app = 0.5 ch , and app = 0.85 ch are shown in Fig. 2a, where ch = 0.01 for the capture cross sections shown. With zero applied stress (solid curve in Fig. 2a), the threads will come together to form a junction only if the resolved shear stress exerted by one thread on the other exceeds ch . Thus, the capture cross-section is symmetric about the origin. Any applied stress causes asymmetry in the capture cross-section because the applied
400
y/b
Note that for app < ch , the dynamic simulation is not required, as the capture cross-section boundary can be calculated as the point where Fge = 0. The analytical solution to this problem is nontrivial, however, and the simulation provides sufficient accuracy. All results presented here use the simulation described above.
300 200 100
τapp = 0.85τch τapp = 1.01τch τapp = 1.05τch τapp = 2.00τch
0 −600 −500 −400 −300 −200 −100
0
x/b Fig. 2. Boundaries of the capture cross-sections for different applied stresses with dislocation B at the origin. The channeling stress ch is 0.01 m. (a) Applied stress levels below the channeling stress. (b) Applied stress levels above the channeling stress (0.85 ch shown for reference).
stress drives the two dislocations in fixed directions, while the interaction stress drives the dislocations towards each other at all stress levels. For example, at = 135◦ , the applied stress drives both dislocations towards the intersection of their slip planes (Fig. 1) while at = −45◦ the applied stress drives both dislocations away from this point. This asymmetry increases with increasing applied stress and the cross-sectional area becomes increasingly located in the second quadrant. As is evident from Fig. 2a, the capture cross section A increases rapidly as app approaches the channeling stress. The reason for this is intuitively obvious. When the applied stress is below the channeling stress, interactions can occur only when the threads are close enough that the interaction can supply the additional stress needed to exceed the channeling stress. However, for applied stresses near the channeling stress, the additional stress required to move the thread is very low, so the dislocations can be far apart and still draw each other together to annihilate. In fact, in the limit as the applied stress approaches the channeling stress, the threads can be arbitrarily far apart and will still annihilate. A functional form for the area of the capture cross-section below the channeling stress can be approximated by solving the boundary term in Eq. (5b) for yˆ = 0 ⇒ Rˆ = xˆ , which gives the equation for the capture cross-section boundary at the x-axis as
app
1
−
− 2
= 1. ch ˆ ch x
(8)
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(a)
the capture cross-sectional area as app approaches the channeling stress. The capture cross section varies roughly as the inverse square of the channeling stress, with the change in area almost entirely due to change of the interaction radius in the second quadrant.
102 Effect of applied stresses less than the channeling stress on the capture cross-section
A(τch/μb)2
101
3.2. Applied stress greater than the channeling stress ( app > ch )
μ = 100τch μ = 10000τch Curve fit
100
10-1
10-2
A(τch/μbRc)
(b)
0
102
0.2
0.4
0.6
τapp/τch
0.8
1
Effect of applied stresses greater than the channeling stress on the capture cross-section
101
μ = 100τch μ = 10000τch Curve fit
100
10-1 1
2
3
4
τapp/τch
5
Fig. 3. Variation of the capture cross section with applied stress. (a) Applied stress levels below the channeling stress. (b) Applied stress levels above the channeling stress.
Solving for Rˆ gives
⎧ 1 ⎪ ⎨ (1 − ( / )) , xˆ < 0 app ch Rˆ = 2ch ⎪ 1 ⎩ , xˆ > 0
(9)
(1 + (app /ch ))
Guided by observation of the cross-section shown in Fig. 2a, we assume that the area of the cross-section in quadrants I, III, and IV is proportional to the square of the radius along the +x-axis. We assume that the area of the cross-section in quadrant II is proportional to the square of the radius along the −x-axis. Adding the areas of the four quadrants gives the following functional form for the area of the capture cross-section: A
2 ch
b
= 1
0.25 (1 − (app /ch ))
2
+
0.75 (1 + (app /ch ))
2
(10)
Fig. 3a shows quantitatively the relationship between the nondimensional parameter A( ch /b2 ) and app . Two different sets of data from our model are plotted: circles represent data at ch = 0.01 and crossed lines represent data taken at ch = 0.0001. Coincidence of these data gives confidence in the normalization factor used. Eq. (10) was fitted to the data in Fig. 3a and
1 = 7.2374 × 10−2 was found for the system modeled. This fit is shown by the solid curve in Fig. 3a. Agreement with the data suggests that this functional form accurately describes the variation of capture cross section with applied stress for stress levels below ch . The most important result shown here is the divergence of
The shapes of the representative capture cross-sections for the load cases app = 1.01, 1.05, and 2.0 ch are shown in Fig. 2b ( app = 0.85 ch is shown for reference), where ch = 0.01 for the capture cross sections shown. The shapes of these cross sections are also easily understood. When the applied stress exceeds the channeling stress, threads do not require interactions with other threads to move; they are already moving on the trajectories determined by their glide planes. If the threads start out equidistant from their point of intersection along the positive x and y axes, they will interact under any applied biaxial stress. Thus the capture boundary extends to infinity at = 135◦ . If the threads are not equidistant, they can pass through the intersection of their glide planes at different times and not interact. The higher the stress, the higher the velocity and the smaller the region in which they can interact. This gives rise to a capture cross section that is predominantly a band centered at = 135◦ that narrows with increasing applied stress. Consequently, for app > ch , the variation in the capture cross-section area is driven by a change in the shape of the cross-section, which is fundamentally distinct from the variation in the capture cross-section area at applied stresses lower than the channeling stress, which is driven by a change in radius of a similar shape. Because the radius extends to infinity along = 135◦ we cannot use the same definition for the area of the cross-section. Instead we introduce a cutoff radius Rc corresponding to a physically relevant maximum distance apart that two threads may be and still interact; this could be determined by a quantity such as grain size or misfit density. Using this cutoff radius, we then approximate the capture cross-sectional area as simply A = Rc d, where d is the width of the band around = 135◦ calculated as the distance between the intersections of the cross-section boundary with straight lines projected along = 134◦ and = 136◦ . Using the above definition, the area of the capture cross section was plotted as a function of the applied stress, as shown in Fig. 3b. In order to determine an appropriate normalization with respect to the channeling stress, two different sets of data are plotted: circles represent data at ch = 0.01 and squares represent data taken at ch = 0.0001. We chose an area normalization of ( ch /bRc ) since it yields nearly identical curves for the two channeling stresses. A simple derivation of the functional form for this curve was not found, but by trial and error we found that the variation in cross sectional area with applied stress above the channeling stress is well described by an equation of the form A Rc
ch
b
=
2 , ((app /ch ) − 1)
app > ch ,
(11)
where 2 is found to be 0.3556 for the curve fit shown in Fig. 3b. In contrast to the cross-sectional area described in Eq. (10) for films below ch , the cross-sectional area above ch is inversely proportional to the applied stress and the channeling stress, and decreases with increasing stress. Furthermore, changes in cross-section are due almost entirely to changes in shape of the cross-section, rather than changes in radius. 4. Discussion This model helps to clarify the role of thread–thread interactions in determining the strengths of thin films. It has been shown [6] that
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interactions between threading dislocations and misfit dislocations are weak, with maximum interaction strengths (the applied stress needed to cause the thread to break the interaction and move on) near 1.3 ch , while interactions between threads can be very strong, from several times ch for strong junctions to unbounded for annihilation reactions (threads cannot move at any applied stress if they do not exist). Thus, a naïve view might assume that thread–misfit interactions would dominate at low applied stresses and interactions between threads might dominate at high stresses. According to this view, this is because weaker thread–misfit interactions can stop threads at low stresses and the probability of such interactions must always be high since each thread must cross every misfit that intersects its path. At high stresses, interactions with misfits would not be strong enough to stop threads, so each thread would continue through the film until it was able to interact with another thread. However, DDD simulations showed that this naïve scenario does not occur [4]. Instead, threads are stopped in interactions with both misfits and threads at all stress levels. Surprisingly, at lower stresses, mobile threads were stopped more frequently by other threads than by misfits. We explained the fact that thread–misfit interactions effectively stop dislocations at high stress levels because of the inhomogeneous stress field that arises due to the misfit structure [4]. At all applied strain levels there are regions where the local stress is low enough to allow interactions with misfits to stop threads. To better understand the probability of interactions between threading dislocations, we developed a model in which threads move through a film at a velocity that depends on the local stress [21]. We found that any stress inhomogeneity concentrates threads in regions of low stress leading to an increased probability of their interaction. That model, however, requires that threads move some distance before starting to concentrate, while the DDD simulations showed thread–thread interactions to be highly probable even at the lowest strain levels. Thus, the model presented here was developed to understand the effect that capture cross-section might play on the likelihood of threading dislocation interactions at different stress levels. The results provide an explanation for the predominance of thread–thread interactions at low applied stresses. The dramatic increase in the area of the capture cross-section as the stress in the film approaches the channeling stress from above or below increases the likelihood of thread–thread interactions at low applied stresses. Together with previous work [4,21], this understanding leads to a picture of film strength, stress relaxation, and strain hardening in which threading dislocations move through an inhomogeneous stress field, engaging in interactions with other dislocations that have a range of interaction strengths. No single type of interaction can be assumed to control the strength of the film; the controlling interactions will vary depending on dislocation density and film stress. The mean free path that a threading dislocation may travel before being stopped in an interaction with another dislocation is a controlling feature governing strain hardening [22]. It is controlled by the spatial stress distribution in the film, as set up by the misfit structure [4], and by the likelihood of interacting with another thread, which is, in turn, affected by the capture cross section of the thread and the thread density. The results from this work, namely, a mathematical description for the variation of capture cross-section with applied stress and film channeling stress (or film thickness), may be useful in developing a mechanistic description of strain hardening in films. Because film strength is determined by many features in addition to the capture cross section, it is, of course, not appropriate to think of the present model as quantitatively predicting film strength in real films. Instead its value lies in the functional form
71
for the variation in capture cross section with applied stress, which can be used in more complex conceptual or numerical models. Thus, while it is important that the functional form be more-or-less correct, the quantitative “accuracy” in terms of the actual size of the capture cross section is secondary. Indeed, we made a number of simplifying assumptions—that the interacting dislocations are infinite and straight, that the stress state is equal biaxial, that thickness effects are correctly accounted for by the channeling strain, and that the dislocation velocity is linearly proportional to the excess stress—that may affect the magnitude of the capture cross section, but are expected to have only secondary effects on its variation with applied stress. Of course the functional form is exact only for the modeled case (two threads in infinite film). In a real film, the cross-section will not truly diverge, because the stress field set up by misfit dislocations as well as stresses from nearby threads that repel or are not on intersecting slip planes will overshadow any effect of some thread very far away. Nevertheless, particularly for the case of threads in relatively close proximity, the predicted capture cross-section gives a measure of the likelihood of interaction, which is a primary quantity of interest in developing a mechanistic description of film relaxation. 5. Conclusions We have presented an analytical model to calculate the capturecross section of a thread gliding in response to an applied stress, such that another thread with opposite Burgers vector on an intersecting path will annihilate with the thread when it is inside this capture cross-section. This model is expected to accurately represent the qualitative behavior of real threading dislocations in thin films. For applied stresses below the channeling stress, the size of the capture cross-section increases with the inverse of the square of the applied stress. For applied stresses above the channeling stress, the size of the capture cross section decreases with the inverse of the applied stress. When the applied stress is equal to the channeling stress, the size of the capture cross-section diverges. This behavior of the capture cross-section of a thread in response to applied film stresses is a critical element in understanding the link between local dislocation behavior and macroscopic film relaxation. Acknowledgements This work was funded by the National Science Foundation Contract No. DMR-0311848. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
S.P. Baker, Mater. Sci. Eng. A 319–321 (2001) 16–23. L.B. Freund, J. Appl. Mech. 54 (1987) 553–557. W.D. Nix, Metall. Trans. A 20 (1989) 2217–2245. R.S. Fertig III, S.P. Baker, Acta Mater. 58 (2010) 5206–5218. W.D. Nix, Scripta Mater. 39 (1998) 545–554. P. Pant, K.W. Schwarz, S.P. Baker, Acta Mater. 51 (2003) 3243–3258. R.S. Fertig, S.P. Baker, Progr. Mater. Sci. 54 (2009) 874–908. U. Essmann, H. Mughrabi, Philos. Mag. A: Phys. Condens. Matter Struct. Defect Mech. Prop. 40 (1979) 731–756. A.A. Kusov, V.I. Vladimirov, Phys. Status Solidi B 138 (1986) 135–142. B.W. Dodson, J. Electron. Mater. 19 (1990) 503–508. G.E. Beltz, M. Chang, J.S. Speck, W. Pompe, A.E. Romanov, Philos. Mag. A: Phys. Condens. Matter Struct. Defect Mech. Prop. 76 (1997) 807–835. A.E. Romanov, W. Pompe, G. Beltz, J.S. Speck, Phys. Status Solidi B 199 (1997) 33–49. A.E. Romanov, W. Pompe, G. Beltz, J.S. Speck, Phys. Status Solidi B 198 (1996) 599–613. A.E. Romanov, W. Pompe, G.E. Beltz, J.S. Speck, Appl. Phys. Lett. 69 (1996) 3342–3344. A.E. Romanov, W. Pompe, S. Mathis, G.E. Beltz, J.S. Speck, J. Appl. Phys. 85 (1999) 182–192.
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[16] J.S. Speck, M.A. Brewer, G. Beltz, A.E. Romanov, W. Pompe, J. Appl. Phys. 80 (1996) 3808–3816. [17] G.E. Beltz, M. Chang, M.A. Eardley, W. Pompe, A.E. Romanov, J.S. Speck, Mater. Sci. Eng. A: Struct. 234 (1997) 794–797. [18] J.P. Hirth, J. Lothe, Theory of Dislocations, John Wiley and Sons, Inc., New York, 1982.
[19] L.B. Freund, R. Hull, J. Appl. Phys. 71 (1992) 2054. [20] W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 6 ed., John Wiley & Sons, Inc., New York, 1997. [21] R.S. Fertig III, S.P. Baker, Scripta Mater. 65 (2011) 384–387. [22] B. Devincre, T. Hoc, L. Kubin, Science 320 (2008) 1745–1748.