Semiconductor Film Grown on a Circular Substrate: Predictive Modeling of Lattice-Misfit Stresses

An effective and physically meaningful analytical predictive model is developed for the evaluation the lattice-misfit stresses (LMS) in a semiconductor film grown on a circular substrate (wafer). The two-dimensional (plane-stress) theory-of-elasticity approximation (TEA) is employed in the analysis. The addressed stresses include the interfacial shearing stress, responsible for the occurrence and growth of dislocations, as well as for possible delaminations and the cohesive strength of a buffering material, if any. Normal radial and circumferential (tangential) stresses acting in the film cross-sections and responsible for its short- and long-term strength (fracture toughness) are also addressed. The analysis is geared to the GaN technology.


Introduction
GaN is a binary III/V direct bandgap semiconductor commonly used in bright light-emitting diodes. GaN based high-electron-mobility-transistor (HEMT) device technology is viewed as a promising one for power amplifier applications [1]. The reliability of GaN devices continues, however, to be a key factor in making promising and viable GaN technology-based devices into reliable products. The elevated lattice-misfit and thermal-mismatch stresses in GaN films are the major limitations for obtaining high-quality GaN systems on technologically important substrates, such as, e.g., Si, SiC, AlN, or diamond (C). A variety of techniques have been suggested to reduce the adverse consequence of the lattice misfit during semiconductor crystal growth (SCG) process. Based on a rather general predictive model for the evaluation of stresses in finite-size bonded joints [2], Luryi and Suhir [3] have shown that the critical thickness of an epitaxial film could be made even infinite, if a properly engineered substrate is used. The idea of employing patterned, porous or otherwise engineered substrates led to many subsequent investigations and to numerous publications exploring the use of nano-sized islands ("towers") as growth nucleation sites. Sagar et al [4] have demonstrated that a reduction in dislocation density from about 10 10 -10 12 cm -2 in a template prepared using molecular beam epitaxy (MBE) could be reduced to about 2.5×10 9 cm -2 , if a porous SiC substrate is employed. The relative level of the latticemisfit and thermal-mismatch stresses in bi-material GaN assemblies was recently addressed, based on analytical (mathematical) predictive modeling, with an objective to evaluate and to compare these two types of stresses [5]. The developed models were based on the strength of material approach (SMA) and treated the GaN assembly as a bi-material elongated rectangular strip. It was determined that even if a reasonably good lattice match takes place (as, e.g., in the case of a GaN film fabricated on a SiC substrate, when the mismatch strain is only about 3%) and, in addition, the temperature change (from the fabrication temperature to the operation temperature) was significant (as high as 1000 C), the thermal stresses were still considerably lower than the lattice-misfit stresses. It was determined also that the interfacial shearing and peeling stresses were as important as the normal stresses acting in the crosssections of the GaN film. While the normal stresses in the GaN film cross-sections are responsible for the fracture toughness of the film material, it is the interfacial stresses that are responsible for the ability of the assembly to resist delaminations (interfacial cracking) and for the performance of the buffering ("bonding") materials, if any. The objective of the analysis that follows is to develop a simple and physically meaningful predictive TEA based LMS model for a GaN film grown on a circular substrate. Our intent is to evaluate, using the developed model, the applicability and accuracy of the SMA (that addresses a bi-material elongated strip as a more or less suitable substitute for an actual circular assembly).

Normal stresses in the assembly mid-portion
The analysis carried out in this section proceeds from the major assumption that neither the circular configuration of the assembly nor its bow affect the normal LMS in the major mid-portion of a large size bi-material assembly. Let the lattice constants for the materials of the components #1 (film) and #2 (substrate) be a1 and a2 ≤ a1, respectively, and, as a result of joining these components into a single bi-material assembly, the final interfacial lattice constant is a. Then the interfacial strains experienced by the component materials are in compression and in tension, respectively. Here E1 and E2 are Young's moduli, and ν1 and ν2 are Poisson's ratios of the materials. The equations (1) reflect the following assumptions: these stresses are the same for all the points in the given cross-section of the given component; the assembly size (in the x-y plane) is significant and the assembly points of interest are sufficiently remote from the assembly edges. The corresponding forces are These formulas indicate that the normal compressive stress in the mid-portion of the component #1 (thin film), as long as it is thin enough, is independent of its thickness, and that the normal tensile stress in the substrate is proportional to the thickness ratio and is very low.

Assumptions
The following major assumptions are used in our analysis:  The assembly components (the film, and the substrate) can be treated as thin circular plates experiencing small deflections, and the engineering theory of bending of thin plates can be used to predict their physical behavior;  The peeling stresses do not affect the interfacial shearing stresses and need not be accounted for when evaluating the shearing stresses;  The interfacial compliances of the assembly in its plane is due to the joint interfacial compliances  The interfacial radial displacements, u1(r), of the component #1 (film) can be evaluated as the sum of the radial displacements, u(r), caused by the lattice-misfit-induced forces, and additional displacements, κ1τ0(r), of the interfacial point at the given radius r, with respect to the displacements u(r) of the inner points of the cross-section: In this formula, τ0(r) is the interfacial shearing stress in the given cross-section, and κ1 is the interfacial compliance of the film layer. The displacements u(r) can be evaluated based on the Hooke's law, and are considered the same for all the points of the given (circumferential) cross-section. The second term in this relationship is, in effect, a correction that considers the deviation of the given cross-section from planarity;  The interfacial radial displacements,. u2(r), of the substrate can be evaluated as  Assembly bow has a small effect on the state of stress in the film and need not be accounted for.  The interfacial shearing stress τ0(r) increases with an increase in the film thickness and with an increase in the shearing stress gradient () xz r z    in the through-thickness direction; in an approximate analysis the interfacial shearing stress τ0(r) can be sought as a product

Basic equation and its solution
The taken assumption and the condition u1(r) = u2(r) of the displacement compatibility result in the following formula for the radial interfacial displacements of the film: (4) where κ = κ1 + κ2 is the total interfacial compliance of the assembly. The formula (4) Introducing the obtained formulas into the equilibrium equation 0 , the following basic equation of Bessel type for the shearing stress function, τ0(r) can be obtained: is the parameter of the interfacial shearing stress, and λ1 is the radial compliance of the film. Note that when the SMA is used and the film is significantly thinner than the the substrate, the longitudinal axial compliance λ of the assembly is due primarily to the compliance λ1 of the film, and the parameter of the shearing stress is .The difference should be attributed to the circumferential loading in circular assemblies. The equation (13) has the following solution: (7) where C1 is the constant of integration, k is the parameter of the interfacial shearing stress, and I1(kr) is the modified Bessel function of the first kind of the first order [8]. The Bessel function in (7) obeys the following rules of differentiation:  [2]. Introducing the solution (7) into the formulas (5), we have: By integration, we find There are no external loads acting on the assembly edges, and therefore the boundary condition σr(r0) = 0 should be fulfilled. This condition and the expression (12) yield: This formula indicates particularly that while the radial normal stress is proportional to the 1 1 C kh ratio and is, hence, film thickness independent, the interfacial shearing stress is proportional to the C1 value and increases linearly with an increase in the film thickness.
Using the solution (7) and the formula (16) for the constant of integration 1 C , the following formula for the interfacial shearing stress can be obtained: considers the effect of the product kr0 of the parameter k of the interfacial shearing stress and the assembly size (radius) r0 on the maximum interfacial shearing stress. Thus, the maximum interfacial shearing stress at the assembly edge increases with an increase in the effective Young's modulus of the material of the film, with an increase in the parameter k of the interfacial shearing stress and with the increase in the thickness h1of the film. The stress increases, of course, with an increase in the lattice that plays the role of the "external loading". The maximum shearing stress is inversely proportional to the longitudinal (axial) compliance  Table 1 for ν1 = 0.25. As evident from the calculated data, the maximum shearing stress increases with an increase in the parameter kr0 when this parameter changes from zero to about kr0 ≈ 10.0, and then remains constant, i.e., assembly size independent.

Theory-of-elasticity (TEA) vs. strength-of-materials (SMA) solutions
Let us compare the TEA solution with the SMA solution for the interfacial shearing stress. One can write the SMA solution as [2] International (20) Table 1. Tabulated function χ(kr0) that considers the effect of the product kr0 on the maximum interfacial shearing stress and the function χ1(kr0) that considers the effect of this product on the ratio of the maximum interfacial stresses computed based on the theory-of-elasticity approach to the maximum interfacial stress calculated using the strength-of-materials approach where the function is the maximum interfacial stress ratio.
The half-assembly-length l is replaced here with the radius r0 value, and the notation k = kTEA is used. The function χ(kr0) changes from zero to one, when the product kr0 changes from zero to infinity. For small kr0 values this function can be computed as the TEA and in the SMA based solutions. This means that TEA predicts somewhat higher stress concentration at the assembly ends than the SMA (for the same maximum stress at the assembly end). The predicted relative ordinates of the interfacial shearing stress are rather close though.
For large arguments z ( 10) z , the modified Bessel function of the order n can be evaluated by the approximate formula [8]:  I  kr  I  kr  I  kr  I  kr  kr kr we obtain: where the maximum value of the shearing stress may be derived from the formulas (20) and (21). As evident from the formula (23), the interfacial shearing stress, τ0(r), concentrates along a narrow peripheral ring, and is next to zero for the inner radii of the assembly 0 rr .

Normal stresses
Introducing the expression (16) for the constant C1 into the formula (15) for the normal stress σr(r) in the film, we obtain: where σ1 is the normal stress in the mid-portion of the assembly. The expression in the brackets is, in effect, a "correction" that considers the role of the finite radius r0 of the assembly. By differentiation we find: Then the circumferential stress σθ can be found from the equilibrium equation as follows: These formulas indicate that the normal stresses, σr and σθ, in the film are uniformly distributed over the inner portion of the assembly ( 0 rr ). At the assembly end the radial stress σr is zero, and the circumferential stress is i.e., by the factor of 2 -ν1 higher than the normal stresses, σ1, in the mid-portion of the film.

Calculated plots
The calculated stresses for the example of 2µm GaN film on a SiC substrate are presented in figures 1 and 2. Figure 1 displays plots of interfacial shearing stress calculated by using TEA and SMA, respectively and the plots of radial and tangential stresses based on TEA are shown in figure 2. Experimental studies show that thermally induced stresses in epitaxial grown GaN layers may relax by cracking or occurrence of high dislocation densities during the cooling down from deposition temperatures. Still biaxial stresses with values exceeding 1 GPa have been observed as a consequence of lattice-misfit and thermal-mismatch strains on GaN epitaxial films deposited on SiC substrates [4,9].