Analysis of quantum-dot systems under thermal loads based on gradient elasticity

Abstract

Since quantum-dot (QD) nanostructures in many applications are subjected to cyclic electrical and thermal loads, it is very important to analyze such nanostructures under transient thermal loads that affect the induced strains which in turn affects the response of the QD system. When the dimensions of the QDs are of the same order of magnitude as the material length scale, gradient elasticity theory should be used to account for the size-dependent behavior of such nano-sized QDs. In this work, a new finite element formulation for gradient-based thermo-electro-mechanical three-dimensional model of strained QDs in piezoelectric matrix under both static and dynamic thermal loads is developed. The proposed formulation requires only C0 shape functions since the original fourth order partial differential equation (PDE) is split into two PDEs of lower order. A unit cell of Indium Arsenide QD in a finite sized Gallium Arsenide (GaAs) substrate is analyzed. Results under dynamic loads demonstrate that the magnitude of the strain component in the thermal loading direction increases significantly with the thermal shock time. On the other hand, at a fixed time, the magnitude of the induced field quantities, such as the strain, decreases remarkably with increasing the size effect parameter. Given that large lattice mismatch induced strain fields are located in or near the QD regions, these newly observed features need to be carefully considered when designing such QD nanostructures.

Size effect in piezoelectric semiconductor nanostructures

“…In semiconductor (SC) industry, piezoelectric semiconductors are particularly appealing when their sizes are reduced to make use of the quantum effect (Bimberg et al, 1999). Various analytical and numerical approaches were proposed, for example, the Green’s function method (Yang and Pan, 2003), boundary integral equation method (Pan et al, 2007), and the FEM (Sladek et al, 2018a). Strain/polarization controlling via strain energy band engineering is the key for achieving stable and reliable operation of wide bandgap devices (Pan et al, 2009; Yang et al, 2019).…”

“…Classical continuum theory was applied to this problem in earlier work (Yang et al, 2019). However, in nanostructures it is needed to apply advanced continuum models where size effect can be considered (Sladek et al, 2018a).…”

Three‐field mixed finite element formulations for gradient elasticity at finite strains

“…Compared to atomistic models they are computationally much cheaper. Examples of the simulation of size-effects with a focus on the general formulation and numerical aspects can be found in Askes et al, [2,4,5] while investigations of gradient elasticity approaches with a focus on specific applications can be found in Aifantis et al [6][7][8][9][10] Another application of gradient elasticity approaches is the instable elastic energy of domains with martensitic phase transformations. [2,3,11] Significant theoretical foundations of these various applications date back to the 1960s gradient enriched elastic continuum theories with notable works of Mindlin [12] and Toupin.…”