Mixed FEM for quantum nanostructured solar cells

Abstract

The gradient theory of piezoelectricity is developed for 3D analyses of QDs with the functionally graded lattice mismatch between the QD and the matrix. Governing equations in the gradient theory contain higher order derivatives than in conventional approaches. Then, it is needed to apply the C¹-elements for approximation of primary fields in the FEM. The mixed FEM with the C⁰ continuous interpolation and collocation approach for kinematic constraints between strains and displacements is developed. The high-density arrays of quantum dots requires to consider various sizes for the representative volume element of the nanostructured solar cell created by the QD (InAs) and matrix (GaAs).

Introduction

Quantum dots (QDs) are tiny nanocrystals made of a semiconducting material, which are buried into a piezoelectric matrix. They have wide range applications in microelectronic, optoelectronic devices and solar cell technology [1], [2]. There are also plenty applications in bioengineering and biomedical studies [3]. The QD solar cells are called as third-generation solar cells. In last years their properties are intensively studied to achieve a high-efficiency solar energy conversion [4]. Recently, Zheng et al. [5] have published a progress towards quantum dot solar cells with enhanced optical absorption in a comprehensive review.

A reliable analysis of elastic and electric fields are crucial to the design and fabrication of such structures [6], [7], [8], [9]. These fields are created in this system due to the lattice mismatch between the QD and matrix [10]. The intrinsic strains have remarkable effects on electronic and optical properties of quantum nanostructure cells. Strains change the interatomic distances and consequently energy levels and bonding of electrons. Then, knowledge of induced strains coupled with electric fields is essential for both understanding and the design of such photovoltaic components [11]. The QDs can work at extreme thermal conditions, therefore, Patil and Melnik [12] applied the classical thermo-piezoelectricity theory for QDs under stationary boundary conditions. Recently, Duc et al. [13] have developed the first analytical approach to investigate the nonlinear dynamic response and vibration of imperfect rectangular nanocomposite multilayer organic solar cell subjected to mechanical loads using the classical plate theory. However, classical continuum mechanics neglects the interaction of material microstructure and the results from it are size-independent. In a nanocomposite structure the size effect should be considered.

In many research works authors deal with uniform initial misfit strains. However, consideration of nonuniform misfit strains would be of great importance since the lattice mismatch occurs only at the interface of the QD and matrix. Inside of the QD it should be considered a functional variation with vanishing misfit at the center. Recently, Bishay et al. [14] have analysed the grading of the material composition as well as the lattice mismatch strain between the QDs and the host matrix. High-density arrays of quantum dots are extensively investigated as components of high performance solar cells. At high densities of QDs the interaction between them should be considered. However, in literature there are missing such analyses. Recently, Rashidinejad and Naderi [15] have studied electro-elastic fields in quantum nanostructure solar cells with consideration of the inter-nanostructure couplings and geometrical effects. One can observe a strong influence of the inter-nanostructure couplings significantly on the induced electro-mechanical fields. Authors [14] applied the classical theory of electro-elasticity for this problem.

The size of quantum dots is only several nanometers. Thus, the material length scale is comparable with the dimensions of the QDs and classical electro-elasticity should not be applied to analyze QD systems. In the classical theory the interaction with the material structure is neglected and results are size-independent. The first attempt to apply the gradient theory to a QD system is given by authors [16]. The gradient theory of thermo-piezoelectricity there is based on the simplified Aifantis theory [17], [18], [19] with one scaling parameter.

This paper presents, for the first time, the application of the gradient theory of piezoelectricity for 3D analyses of QDs with functionally graded lattice mismatch between the QD and matrix. The influence of distances between QDs is investigated too. For this purpose the FEM is developed and applied. In gradient theory it is needed to use C¹-continuous elements to guarantee the continuity of the problem variables and their derivatives at the element boundaries. Since it is not an easy task to develop C¹ continuous elements in 3D problems, the mixed FEM formulation is developed here. The C⁰ continuous interpolation is independently applied for displacement and displacement gradients. The independent interpolation requires to satisfy the kinematic constraints between strains and displacements. A collocation at some cleverly chosen internal points in elements is applied [20]. The present approach reduces the number of DOFs with respect to the Lagrangian mixed FEM formulation [21].

Numerical results for high-density arrays of quantum dots with functionally graded lattice mismatch between the QD (InAs) and matrix (GaAs) are presented, and the influence of the size effect parameter is discussed.