Yield maps for nanoscale metallic multilayers

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1 Scripta Materialia 50 (2004) Yield maps for nanoscale metallic multilayers Adrienne V. Lamm *, Peter M. Anderson Department of Materials Science and Engineering, The Ohio State University, 477 Watts Hall, 2041 College Road, Columbus, OH , USA Accepted 20 November 2003 Abstract This manuscript presents maps for the macroscopic, in-plane, bi-axial tension to macroscopically yield a nanoscale multilayered thin film consisting of alternating phases that can plastically deform. The design parameters are volume fraction, bi-layer thickness, modulus mismatch, and lattice parameter mismatch. Ó 2003 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. Keywords: Design maps; Strength; Multilayers; Theory & modeling 1. Introduction * Corresponding author. Tel.: ; fax: address: lamm.6@osu.edu (A.V. Lamm). Multilayered thin films are a class of engineered layered composite materials with individual layer thicknesses on the order of one to several hundred nanometers. The driving force to reduce interfacial energy can cause large internal stress [1], generate deviations from bulk elastic moduli [2,3], induce phase changes [4], suppress plasticity [5 8], generate significant anelasticity [9,10], and even promote unique rolling textures [11]. Hardness measurements for some A/B metallic systems indicate that a critical bi-layer period exists below which the resistance to plasticity reaches a plateau or even decreases [12 14]. Tensile testing and fractography of Ni/Ni 3 Al multilayers show that multilayers can be macroscopically brittle, but ductile near the fracture surface [15]. The purpose of this manuscript is to determine the role of microstructural parameters such as bi-layer thickness, volume fraction, modulus mismatch, and lattice parameter mismatch on the macroscopic yield strength of metallic multilayer thin films. This will be accomplished by combining mechanical equilibrium, the critical conditions for confined layer slip (CLS) in which dislocations propagate only in alternating layers, and the critical condition for bulk yield whereby extensive co-deformation occurs. Recent embedded atom modeling [3] suggests that for Cu/Ni multilayered thin films with coherent and semi-coherent interfaces, extensive co-deformation across layers occurs when the applied stress is sufficient to eliminate the large compressive stress (2 GPa for the coherent case) in the alternating Cu layers. Thus, dislocations appear to be trapped in individual layers chiefly by the alternating tensile-compressive stress state rather than by the structural resistance provided by the interface. This premise is adopted in the present analysis. The resulting maps predict that decreasing bi-layer thickness (K) is a limited approach to increasing multilayer yield strength, valid only down to a critical K. Below that, decreasing the volume fraction of the compressively stressed phase is most effective. 2. Model development: interface structure and macroscopic yield Multilayered thin films may be modeled as thin sheets of material (1 and 2) that in a detached, stress-free form have in-plane dimensions ðl o 1 ; lo 2 Þ, thicknesses ðho 1 ; ho 2 Þ, and lattice parameters ða o 1 ; ao 2 Þ. The lo k (k ¼ 1 or 2) are chosen so that l o 2 =lo 1 ¼ ao 2 =ao 1, meaning that a coherent interface is produced by first stretching or compressing individual layers elastically so that l 1 ¼ l 2 and then joining the layers. In general, however, elastic (e k )and plastic (e p k ) components of direct in-plane strain may occur e k ¼ e k þ e p k ; e k ¼ r k M k and e p k ¼ ln 1 þ b k s k ð1þ /$ - see front matter Ó 2003 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. doi: /j.scriptamat

2 758 A.V. Lamm, P.M. Anderson / Scripta Materialia 50 (2004) where M k ¼ E k =ð1 m k Þ is the bi-axial elastic modulus for layer k (E k, m k are Young s modulus and Poisson s ratio of layer k). The plastic strain is due to orthogonal arrays of misfit dislocations of Burgers vector magnitude b k and in-plane spacing s k in layer k. Such arrays can be formed during thin film growth or generated by the effective in-plane plastic stretching of Orowan-type dislocation loops that propagate in a CLS mode (Fig. 1) [5,7,16]. The r k are obtained from the compatibility condition that layers deform elastically (and possibly plastically) to make l 1 ¼ l 2, and from the equilibrium condition relating the macroscopic bi-axial tension R to the r k and volume fractions f k, e 1 e 2 ¼ ln lo 2 and R ¼ r l o 1 f 1 þ r 2 f 2 ð2þ 1 j k r R f 2 M 2 ln ao 1 þ ln 1 þ b 1 a ¼ o 1 s 2 1 ln 1 þ b 2 s 2 : ð3þ M 1 f 1 M 1 þ f 2 M 2 Eq. (3) extends the result in [7] to large (logarithmic) strain theory. Dislocations are assumed to propagate in a CLS mode given that a mismatch in a o k generates an alternating bi-axial stress state to confine dislocations within layers. The bi-axial stress required for CLS (excluding image effects) is [5,7,16] r y k ¼ S k lnðh 0 k =b kþ M k 4pð1 þ m k Þh 0 k =b : ð4þ k The Schmid factor S k is the ratio of the resolved shear stress on the active slip system in layer k to r k and h 0 k is the projected layer height (Fig. 1). Numerous sources of dislocation loops from interfaces are envisioned [6,7,17] so that the yield strength of a confined layer is defined by r y k rather than by source-limited plasticity. An analytic analysis of dislocation arrays in multilayered thin films suggests that r y k does not increase with plastic strain provided e p k < b k=10h k and that pile-ups should not occur until e p k b k=2h k [18]. When an alternating stress state exists, the constitutive relation for an embedded layer is elastic perfectly plastic, at least for je p k j < b k=10h k. Unloading from r k ¼ r y k is assumed to be elastic until r k ¼ r y k, due to dislocation dislocation entanglement. However, experimental evidence suggests that CLS may reverse direction when jr k j < r y k, specifically during thermal cycling involving elevated temperature [19]. All possible scenarios for multilayer yield are predicated on the assumption that dislocation loops will permeate throughout the multilayer once R is large enough to eliminate the compressive stress in alternating layers. Without loss of generality, a o 2 =ao 1 > 1 is assumed, so that the initial states ~r 1 > 0 and ~r 2 < 0 occur. Three possible scenarios for the initial state indicated in Fig. 2 are that the interfaces may be (a) coherent (b 1 =s 1 ¼ b 2 =s 2 ¼ 0); or semi-coherent due to arrays of dislocations deposited on the interface by (b) yield of layer 1 ðb 1 =s 1 > 0; b 2 =s 2 ¼ 0Þ or (c) yield of layer 2 ðb 1 =s 1 ¼ 0; b 2 =s 2 < 0Þ. Scenario (a) prevails if the coherent stress state ð~r c 1 ; ~rc 2Þ obtained from Eq. (3) with R ¼ 0 and ðb 1 =s 1 ¼ 0; b 2 =s 2 ¼ 0Þ does not exceed yield (that is, ~r c 1 < ry 1 and j~rc 2 j < ry 2 ). Alternately, scenario (b) occurs with ð~r 1 ; ~r 2 Þ¼ðr y 1 ; ry 1 f 1=f 2 Þ provided that j~r 2 j < r y 2. Finally, scenario (c) occurs with ð~r 1 ; ~r 2 Þ¼ðr y 2 f 2= f 1 ; r y 2 Þ provided that ~r 1 < r y 1. For scenario (b), the nonzero value of b 1 =s 1 is determined by setting r 1 ¼ r y 1 in Eq. (3) and solving for b 1 =s 1 with R ¼ 0andb 2 =s 2 ¼ 0. An upper limit, ðb 1 =s 1 Þ max ¼ða o 2 =ao 1Þ 1, is obtained using Eq. (3) with R ¼ 0andðr 1 ; r 2 Þ¼ð0; 0Þ, so that the interface is fully incoherent. Corresponding values of b 2 =s 2 for scenario (c) are determined in a similar fashion, but with indices 1 and 2 reversed. Fig. 2 shows that a macroscopic strain e ¼ ð0 ~r 2 Þ=M 2 is needed to reduce r 2 to 0. Provided that e < e 1 ðry 1 ~r 1Þ=M 1, macroscopic yield in tension Fig. 1. Schematic of a multilayer with alternating bi-axial stress ðr 1 ; r 2 Þ. Fig. 2. Graphic of bi-axial responses of phase 1 (tensile) and phase 2 (compressive).

3 A.V. Lamm, P.M. Anderson / Scripta Materialia 50 (2004) will be reached without any prior yield in tension of layer 1. Thus, R y ¼ M full e ; M full ¼ f 1 M 1 þ f 2 M 2 ðvalid for e 6 e 1 Þ ð5þ Alternately, layer 1 may yield in bi-axial tension prior to r 2! 0ife > e 1. Under such circumstances, the multilayer extends with a full bi-axial modulus only until e 1 is reached. For the remaining portion, e e 1, the multilayer extends with a reduced bi-axial modulus, M red ¼ f 2 M 2, valid when layer 1 is at yield. Thus, R y ¼ M full e 1 þ M redðe e 1 Þ; M red ¼ f 2 M 2 ðvalid for e P e 1 Þ ð6þ 3. Results and discussion 3.1. Internal stress maps The thin, shaded lines in Fig. 3a and b are contours of the initial stress ~r k =M k in layers 1 and 2, respectively, as a function of volume fraction f 1 and normalized bi-layer thickness K=b. Here, M 2 =M 1 ¼ 0:91 and a o 2 =ao 1 ¼ 1:01 so that layer 2 has a smaller bi-axial elastic modulus and is in compression initially. For simplicity, the respective Schmid factors, Burgers vectors, and projected heights are assumed to be S 1 ¼ S 2 ¼ 1, b 1 ¼ b 2 ¼ b, andh 0 k ¼ h k. The thick, darker lines define regions for which the initial interfacial structure is coherent, semi-coherent with initial phase 1 yield, and semi-coherent with initial phase 2 yield. The labels I ¼ 0, 25%, and 50% describe ðb k =s k Þ¼Iðb k =s k Þ max ; thus, I ¼ 0 serves as the boundary between coherent and semi-coherent regions. The results imply that the magnitude of ~r k =M k increases with decreasing f k, regardless of the phase or interfacial regime, and it is unaffected by K in the coherent regime. The model is not valid in the shaded region, where at very small individual layer thickness (taken as < 3b) nonlinearity dominates in Eq. (4). The positions and asymmetries of the contour lines in Fig. 3 depend on M 2 =M 1 and a o 2 =ao 1. First, Eqs. (3) and (4) indicate that the contours for various I will shift to the left as a o 2 =ao 1 is increased from the present value of Also, semi-coherent regions for layers 1 and 2 are unequal in size when M 2 =M 1 6¼ 1, since r y k M k according to Eq. (4). More detailed calculations that account for image effects on r y k are expected to reduce the magnitude of the asymmetry [20] Macroyield maps The lightly shaded contours in Fig. 4 display the macroscopic bi-axial tension R y (in GPa) needed to fully yield the multilayer thin film, using M 1 ¼ 110 GPa and M 2 ¼ 100 GPa as well as the parameter values used in Fig. 3. The largest yield strength arises when K and f 2 are decreased to move into the upper left region of the map. A significant result is that decreasing K at constant f 1 causes the yield strength to increase initially, and then reach a plateau at small K, similar to experimental results for several systems [12 14]. The yield process is understood by noting the four distinct regions defined inside the darker contour lines. At larger K, semi-coherent interfaces are expected prior to loading, due to initial yield of either the tensile phase Fig. 3. Internal stress maps of: (a) ~r 1 =M 1 and (b) ~r 2 =M 2 as a function of K=b and f 1, assuming M 2 =M 1 ¼ 0:91, a o 2 =ao 1 ¼ 1:01, S 1 ¼ S 2 ¼ 1, b 1 ¼ b 2 ¼ b and h 0 k ¼ h k.

4 760 A.V. Lamm, P.M. Anderson / Scripta Materialia 50 (2004) Fig. 4. Macroyield map of R y as a function of K=b and f 1, assuming M 2 =M 1 ¼ 0:91 (with M 1 ¼ 110 GPa and M 2 ¼ 100 GPa), a o 2 =ao 1 ¼ 1:01, S 1 ¼ S 2 ¼ 1, b 1 ¼ b 2 ¼ b, and h 0 k ¼ h k. 1 when f 1 is smaller (Region 1) or the compressive phase 2 when f 2 is smaller (Region 2). In Region 1, tensile R causes immediate plastic flow in phase 1 and incrementally reduces the compressive stress in layer 2 to 0 with M multilayer ¼ M red. In Region 2, r 2 is reduced to zero with M multilayer ¼ M full initially, then M multilayer ¼ M red during yield of layer 1. At smaller K, interfaces are predicted to be coherent and M multilayer ¼ M full initially. However, in Region 3, the tensile phase 1 yields and M multilayer ¼ M red before r 2 decreases to 0. In Region 4, interfaces remain coherent and no yield occurs before r 2 reaches 0. Overall, Region 4 is described by Eq. (5) and Regions 1, 2, and 3 use Eq. (6). R y reaches the largest value in the upper left section of the macroyield map, where Regions 3 and 4 dominate and interfaces are coherent initially. The boundary between these regions is given by e ¼ e 1, equivalent to lnða o 2 =ao 1 Þ¼ry 1 =M 1 lnðh 1 Þ=h 1 using Eqs. (2) (4), and equivalent to h 1 f 1 K ¼ constant for a system with a given lattice parameter mismatch a o 2 =ao 1. In Region 3 where e > e 1, R y is increased most effectively by decreasing K and thereby increasing e 1. In Region 4 where e < e 1, R y is increased most effectively by increasing f 1 and thereby increasing e. Movement along the boundary from point A with ðh 1 =b 1 ; h 2 =b 2 Þ¼ð15; 35Þ to point B with ðh 1 =b 1 ; h 2 =b 2 Þ¼ð15; 4Þ changes ð~r 1 =M 1 ; ~r 2 =M 2 Þ from roughly ð0:007; 0:003Þ to ð0:002; 0:008Þ and increases R y from 0.3 to 0.9 GPa. Thus, an optimal R y is achieved by decreasing both K and f 2,so that the multilayer has coherent interfaces with compressive layers that are thin and highly stressed. 4. Conclusions Internal stress maps and macroscopic yield maps for two-phase multilayered thin films are developed assuming that (1) local yield of individual layers is controlled by confined layer slip rather than source availability and (2) macroscopic yield occurs when the applied, in-plane, bi-axial stress reduces the compressive stress state in alternating layers to zero. The resulting maps predict that decreasing bi-layer thickness (K) is a limited approach to increasing the tensile yield strength of multilayers, valid only down to a critical K. Below that, decreasing the volume fraction of the compressively stressed phase is most effective. The principles outlined in this analysis can be extended to account for an attached substrate, applied states other than bi-axial tension, and modifications to assumption (2) that more accurately account for the effect of pile-ups and interfacial structure. Acknowledgements The authors gratefully acknowledge the support of the Air Force Office of Scientific Research Metallic Materials Program (F ) and the National Science Foundation Mechanics & Materials and Metallic Materials Programs ( ). References [1] Sperling EA, Banerjee R, Thompson GB, Fain JP, Anderson PM, Fraser HL. J Mater Res 2003;18(4):979. [2] Huang H, Spaepen F. Acta Mater 2000;48(12):3261. [3] Hoagland RG, Mitchell TE, Hirth JP, Kung H. Philos Mag A 2002;82(4):643. [4] Thompson GB. PhD. Thesis, The Ohio State University, Columbus Ohio, USA; [5] Nix WD. Metall Trans A 1989;20(11):2217. [6] Was GS, Foecke T. Thin Solid Films 1996;286(1 2):1. [7] Anderson PM, Foecke T, Hazzledine P. MRS Bull 1999;24(2): 27. [8] Misra A, Hirth HP, Kung H. Philos Mag A 2002;82(16):2935. [9] Yu D, Spaepen F. Work in progress; June [10] Haque MA, Saif MTA. Scr Mater 2002;47(12):863. [11] Anderson PM, Bingert JF, Misra A, Hirth JP. Acta Mater 2003;51(20):6059. [12] Clemens BM, Hung H, Barnett SA. MRS Bull 1999;24(2):20. [13] McKeown J, Misra A, Kung H, Hoagland RG, Nastasi M. Scr Mater 2002;46(8):593. [14] Tixier S, Boni P, Van Swygenhoven H. Thin Solid Films 1999; 342(1 2):188.

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