Misfit stresses in a core-shell nanowire with core in the form of long parallelepiped

We present an analytical solution to the boundary-value problem in the classical theory of elasticity for a core-shell nanowire with a core in the form of a long parallelepiped with a square cross section. The core is placed symmetrically with respect to the cylindrical shell surface. The misfit stresses are found in a concise and transparent form of trigonometric series, which is convenient for practical use in theoretical modeling of misfit relaxation mechanisms. This work was supported by Peter the Great St. Petersburg Polytechnic University and ITMO University.


Introduction
Radially inhomogeneous nanowires (NWs) demonstrate excellent electronic and optical properties, which encourage their use in various devices of optoelectronics, nanoscale field effect transistors, storage and data transmission devices, logic devices, sensors, etc. [1][2][3][4][5][6][7][8][9][10]. Physical properties of NWs depend on the shape, size, chemical composition, and types of crystalline lattices of NW components as well as on presence of various defects in their structure. In particular, the shape effects can be tightly related with peculiarities in misfit stress distribution over NWs and with the mechanisms of misfit stress relaxation in them. However, in theoretical description of these mechanisms, they commonly use the model of cylindrically symmetric core-shell NWs [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] as it is much simpler for analytical modeling. The problem is that this simple approach strictly limits the variety of possible relaxation mechanisms available for theoretical examination. For example, it excludes the glide of straight misfit dislocations along the flat areas of the core-shell interface that is sometimes observed in experiments [27,28]. It is worth noting that flat interface regions often form in radially inhomogeneous NWs [29][30][31][32].
It seems that the simplest case of a core with flat faces is the core in the form of a long parallelepiped with a square cross section. To our best knowledge, today there is only one analytical solution which describes the elastically strained state in a core-shell NW with such a core placed symmetrically with respect to the shell surface [33]. The solution was found in the model case of plane strain through the complex potentials method and illustrated by stress maps in Cartesian coordinates. The disadvantage of the work [33] is that the authors did not demonstrate the evidence of the boundary condition fulfillment. Moreover, the case of plane strain is obviously quite far from the case of threedimensional mismatch of crystalline lattices in real core-shell NWs.
The present work is aimed at the analytical calculation and numerical analysis of the misfit stresses in a core-shell NW with a symmetrically placed core which is characterized by a three-dimensional dilatation eigenstrain (3D misfit strain) with respect to the shell material and has the shape of a long parallelepiped with a square cross section. We show the analytical formulas for the misfit stress components applicable for practical use in theoretical modeling of misfit stress relaxation processes and demonstrate their distribution in the NW cross section, from which the fulfillment of boundary conditions on the shell free surface is evident.

Model
Consider a long core-shell NW of radius R, which consists of elastically homogeneous and isotropic shell and a core having the shape of a long parallelepiped (figure 1). The shear modulus G and the Poisson ratio ν are the same for the shell and the core. The core cross section is a square with a side 2а. The core is centered symmetrically with respect to the shell surface and subjected to the 3D homogeneous dilatation eigenstrain ε * .
The desired stress field ij σ caused by the eigenstrain ε * can be represented by the sum of a similar stress field ∞ ij σ created by the core in an infinite medium, and an extra stress field * ij σ which is needed to satisfy the boundary conditions on the shell free surface: .
The boundary conditions are: The non-vanishing stress components ∞ ij σ are given as follows [34]: To satisfy the boundary conditions (2), we should represent Eqs. (3a)-(3c) in the cylindrical coordinate system on the shell surface R r = . First, let us consider the following derivative: Integrating series (5) over θ, we find the trigonometric series for the desired function: Applying the same procedure to every function figuring in Eqs. (3a)-(3c), after some algebra we find the following compact formulas for the stress fields ∞  Subtracting Eq. (10b) from Eq. (10a), we obtain the following formula which is convenient for the fulfillment of the boundary conditions: Taking into account that the stress must be finite at r → 0, we will search the functions ) (ξ

Results
To illustrate our solution, we show in figures 2(a) and 2(b) the distribution of stress components

Conclusions
We can conclude that our analytical solution (1) satisfies the boundary conditions of the problem (2), gives an opportunity to analyze the misfit stress distribution in detail, and (3) is represented in a concise and transparent form which is applicable for theoretical modeling of misfit stress relaxation processes.