Misfit strain

R. Hull, Misfit Strain Accommodation in SiGe Heterostructures... 

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

The waves then coarsen into a fewer ones, and their valleys or troughs become crack-tips , which advance to the film-substrate interface. The number of the advancing cracktips depends on the misfit strain. In the infinite-toque condition, the tips end at the substrate, but become wider at the triple junction of film-substrate-vapor. A typical island features an acorn shape. In the zero-torque condition, the tips advance much deeper with the pucker-... 

If the misfit strain is less than a critical value, the undulations cannot mount cracktips, as demonstrated in Fig. 4, where a periodic length is equal to 100 a and film thickness is 30 ML. With the same physical parameters employed for Fig. 3, no islands are created if the misfit strain is less than 0.006. When the misfit strain is less than but close to the critical value, a permanent wave structure sets in the film as in the case ofs = 0.005. If the misfit strain is further reduced, coherency-induced undulations are swept away by thermal fluctuations. 

Figure 4. Thin films with small misfit strains under the zero-torque condition. As the misfit strains are less than the critical value, 0.006, no islands are created out of the undulations. The periodic length is equal to 100 a and the film thickness is 30 ML.

The instability of the two lamellar structures may be understood in terms ofEshelby s inclusion theory [6,7]. According to the theory, a hard coherent precipitate with a dilata-tional misfit strain is elastically stable when it takes on a spherical shape in an infinite matrix. A soft coherent precipitate, on the other hand, takes on a plate-like shape as the minimum strain energy shape. Thus, the soft-hard-soft layered structure of Fig. 7 is simply a... 

Sridhar and coworkers studied the kinetics of a compressed film on a viscous substrate [30], They performed linear-stability analysis to determine the onset and maximally unstable mode of this mechanical instability as a function of misfit strain, viscous layer thickness, and viscosity. 

Sc Characteristic length do Misfit strain at ambient temper-... 

Misfit strain, Cl Bilayer distortion Permanent strain, ep Residual crack opening 0-2 x 10 ... 

The development of damage is sensitive to the residual stress caused by the misfit strain, ft. Measurement of these stresses thus becomes an important aspect of the analysis and prediction of damage. These stresses arise at interlaminate and intralaminate levels. Within a laminate, the axial stress in the matrix is,58... 

Once the ambient misfit strain, fto, has been inferred for the above measurements, the temperature dependence can be assessed from the thermal expansion mismatch, using,... 

Most experimental results are consistent with the misfit strain predicted from the thermal expansion difference, am — atf, and the cooling range from the processing temperature. Examples for SiC/CAS and SiC/SiC are given in Table 1.3. 

Analyses of the plastic strains caused by matrix cracks, combined with calculations of the compliance change, provide a constitutive law for the material. The important parameters are the permanent strain, e0 and the unloading modulus, E. These quantities, in turn, depend on several constituent properties the sliding stress, r, the debond energy, T, and the misfit strain, il. The most important results are summarized below. 

Ashby and Brown (1963a) have shown that the image width tv (defined as the width corresponding to a change of transmitted intensity of x percent from background intensity) depends principally on the dimensionless parameter = egro/tg, where c is the misfit strain between the inclusion and the matrix, and Tq the radius of the constrained precipitate particle. They constructed curves of w/tg as a function of logio Ps for values of x of 2, 10, 20, and 50 percent, from which it is possible to determine e by... 

From the standpoint of the quantitative model used to describe this system, we assume that both the matrix and precipitate are elastically isotropic and characterized by the same elastic moduli and that the misfit strain associated with the precipitate is purely dilatational and is given by Further, it is assumed that the interfacial energy y,n (n) is isotropic (i.e. yint(n) = yo). Recall from... 

A regular network of edge-type dislocations on an atomic flat interface, as depicted in Figure 8.3, can in principle relax all the misfit strain, allowing the GaN overgrown layer to be totally free of threading dislocations. In real growths, however, the interface is rarely flat, and the dislocation... 

Gao, M., Boodey, J. B., and Wei, R. P., Misfit Strains and Mechanism for the Precipitation of Hydrides in Thermally Charged Alpha-2 Titanium Aluminides, in Environmental Effects on Advanced Materials, R. H. Jones and R. E. Ricker, eds., The Minerals, Metals and Materials Society, Warrendale, PA (1991), 47-55. 

In its most primitive form the idea behind the coincidence lattice assumes that the forces within a monolayer are sufficiently strong to ignore the potential of the substrate, except for small adjustments of the lattice in the plane of the overlayer that may be required to achieve perfectly coincident registry. In a real situation it is likely that there will be small individual displacements of the overlayer atoms, not only parallel to the plane of the surface but also normal to it. Such relaxations will be expected to offset misfit strains, and should occur in regular fashion within the coincident mesh. Such displacements have been theoretically considered in papers by Tucker (329), Palmberg and Rhodin (165), Fedak and Gjostein (173) and Ducros (330). 

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