The effect of surface energies, strains, and stresses on the size-dependent elastic state of embedded inhomogeneities are investigated. At nanolength scales, due to the increasing surface-to-volume ratio, surface effects become important and induce a size dependency in the otherwise size-independent classical elasticity solutions. In this letter, closed-form expressions are derived for the elastic state of eigenstrained spherical inhomogeneities with surface effects using a variational formulation. Our results indicate that surface elasticity can significantly alter the fundamental nature of stress state at nanometer length scales. Additional applications of our work on nanostructures such as quantum dots, composites, etc. are implied.

The classical formulation of Eshelby (Proc. Royal Society, A241, p. 376, 1957) for embedded inclusions is revisited and modified by incorporating the previously excluded surface/interface stresses, tension and energies. The latter effects come into prominence at inclusion sizes in the nanometer range. Unlike the classical result, our modified formulation renders the elastic state of an embedded inclusion size-dependent making possible the extension of Eshelby’s original formalism to nano-inclusions. We present closed-form expressions of the modified Eshelby’s tensor for spherical and cylindrical inclusions. Eshelby’s original conjecture that only inclusions of the ellipsoid family admit uniform elastic state under uniform stress-free transformation strains must be modified in the context of coupled surface/interface-bulk elasticity. We reach an interesting conclusion in that only inclusions with a constant curvature admit a uniform elastic state, thus restricting this remarkable property only to spherical and cylindrical inclusions. As an immediate consequence of the derivation of modified size-dependent Eshelby tensor for nano-inclusions, we also formulate the overall size-dependent bulk modulus of a composite containing such inclusions. Further applications are illustrated for size-dependent stress concentrations on voids and opto-electronic properties of embedded quantum dots.

Density functional calculations on silicon clusters show that strain effects on the band gap display qualitatively new trends for dots smaller than ϳ2 nm. While the bulk indirect band gap increases linearly with increasing strain, this trend is reversed for small clusters ͑ഛ1 nm͒. In the intermediate 1 -2 nm size range, strain appears to have almost no effect. These results follow from the fact that the bonding/antibonding character of the HOMO and the LUMO change nonmonotonically with size. Since the strain level of the surface atoms dominate this behavior, they strongly stress the role of surface passivation on experimentally measured band gaps.The decrease in size of semiconductor nanoclusters can result in a variety of new properties that are absent in the bulk. For example, while bulk silicon does not emit visible light, photoluminescence is found for clusters smaller than a critical dimension. 1-4 This effect is ascribed to strong quantum confinement. In addition, the band gap in silicon quantum dots has been shown to substantially vary through surface passivation, 5-8 i.e., oxygen and other double bonded atoms have a dramatic effect as compared to hydrogenated clusters. Passivation with single bonded atoms, however, has minimal impact. These findings have led to the possibility of using silicon dots in technological applications ranging from optoelectronic devices to biological labels. 9,10 Several experiments have studied the role of strain and quantum confinement on optical emission. Examples include strained silicon dots ͑with size larger than 2 nm͒ embedded in a silicon dioxide matrix, 11,12 and InAs dots embedded in In x Ga 1−x As confining layers: the strain in this case is controlled by varying the In composition of the confining layers. 13,14 In addition, Buda and Aparisi et al. 15,16 studied the electronic and optical properties of III-V and II-VI semiconductor clusters encapsulated in the zeolites ͑and sodalite͒ cage and observed the shift of the energy gap of the semiconductor clusters due to resultant tensile/compressive strain exerted by the cage on the clusters. While these works find evident strain effects on the band gap, to our knowledge no theoretical study has focused on the coupled effects of size and strain in the case of semiconductor nanodots. This work bridges this gap, especially focusing on the sub-2 nm size range, where we predict that new, interesting phenomena will occur.We have studied the energy gap ͑EG͒ of silicon clusters containing up to Si 123 H 100 ͑ϳ1.7 nm diameter͒ using several computational tools. The starting atomic coordinates for these clusters are obtained from bulk silicon ͑lattice constant of 0.5461 nm͒. The dangling bonds of the surface silicon atoms are terminated by hydrogen at an initial bond length of 0.147 nm. The structure of the cluster is then relaxed using the energy minimization technique. The atomic and electronic structure of these clusters are computed via density functional theory ͑DFT͒ based on the generalized gradient approximation ͑GGA͒ using ...

PACS: 85.35.Be; 68.35.Gy; 62.25.+g; 62.20.Dc Introduction Quantum Dots (QDs) have recently been the focus of several experimental and theoretical researchers due to the promise of improved and new opto-electronic properties [1]. Frequently, embedded QD structures (e.g. InAs/GaAs system), to preserve coherency, must accommodate large lattice mismatch. The ensuing elastic relaxation and the hydrostatic strain state within the QD structure are well known to impact its opto-electronic properties [2,3]. Several works, of varying sophistication (both analytical and numerical), have focused on the "accurate" calculation of the strain state in buried quantum dots [4][5][6][7][8]. Recently, two papers have caught the present authors attention. Pan and Young [4] indicated significant difference in strain calculation when anisotropic elastic behavior is assumed compared to simplified isotropic elasticity. Further, a recent article by Ellaway and Faux [8] indicated (via atomistic simulations) that the elastic properties of QD are staindependent and such a consideration on strain calculation can result in a significant correction (of 16%) to the hydrostatic strain (in their article, for a buried spherical InAs/GaAs QD). In this communication, we show that the so-far unconsidered interfacial elastic properties can also significantly alter the strain calculations; the exact correction sensitively being dependent upon the size of the QD structure and the interfacial elastic constants. We find that the correction resulting from interfacial elasticity is comparable to that due to either anisotropic or strain-dependency effects.Classical elasticity (on which most of the previous works are based) does not admit intrinsic size dependence in the elastic solutions of embedded inhomogeneities. For structures with sizes > 50 nm, typically, the surface-to-volume ratio is negligible and the deformation behavior is governed by classical bulk strain energy. Currently, no formulation exists which combines interface elasticity with bulk elasticity to analyze embedded inclusions. Eshelby's [9] celebrated formalism, often used in QD literature, is based entirely on classical bulk elasticity. In this communication (using a variational approach) we derive a general expression for the correction in hydrostatic strain due to interfacial elasticity (for an embedded spherical quantum dot). Despite the lack of precise data, we are able to show (using InAs/GaAs as an example system) that inclusion of interfacial elasticity effects can result in minimum corrections between 1.8% and 12% in the typical size range of QD structures (2-20 nm).

A closed-form model was proposed to evaluate the elastic properties of nanocrystalline materials as a function of grain size. Grain-boundary sliding, present in nanocrystalline materials even at relatively low temperatures, was included in the formulation. The proposed analytical model agrees reasonably well with the experimental results for nanocrystalline copper and palladium.