Finite Element Modeling Piezoelectric Materials under Thermal Loads with Application for Quantum Dots

Quantum dots (QDs) are very small semiconductor particles, only several nanometres in size [1], so small that their optical and electronic properties differ from those of larger particles. In QDs the free carriers are confined to a small region by potential barriers in all three directions of 3D space. If the size of the region is less than the electron wavelength, the electronic states become quantized at discrete energy levels as it happens in an atom. They are a central theme in nanotechnology. Many types of QDs will emit light of specific frequencies if electricity or light is applied to them, and these frequencies can be precisely tuned by changing the dots’ size, shape and material, giving rise to many applications [2 3]. The exploitation of this kind of nanostructures towards the improvement of the devices performance mainly relies on the ability to control their size and uniformity [4]. To bury QDs into piezoelectric matrix induces not only an elastic field, but also a piezoelectric field. The induction of both fields is given by the lattice mismatch between the QDs and the surrounding piezoelectric matrix. Since both the elastic and piezoelectric fields are equally important in the understanding of the photonic and electronic features in semiconductors [5] a reliable analysis on these fields is crucial to the design of such structures. QDs can exist in a wide variety of shapes, including cuboidal, pyramidal [6], truncated pyramidal, and lens shaped. The stress and the strain distributions in and around the QDs have recently been investigated by many researchers [7]. The finite element method (FEM) has become an essential solution technique in many areas of engineering and physics [8]. The FEM is an efficient computational tool in the development of efficient micro and nanoscale systems [9 11]. In this paper a coupled 3D ANSYS Multiphysics FE model is used to simulate a cubic matrix with an embedded cubic QD under static thermal loadings. Our particular focus is on the influence of thermo-electromechanical effects on properties of QDs, where InAs (Indium arsenide) is embedded into GaAs (Gallium arsenide) matrix. The paper is organized as follows: section 2 and section 3 briefly present the governing equations for thermo-piezoelectricity and the finite element equations. Section 4 describes used simulation model in the ANSYS Multiphysics section 5 presents the numerical results. Conclusions are summarized in section 6.


Introduction
Quantum dots (QDs) are very small semiconductor particles, only several nanometres in size [1], so small that their optical and electronic properties differ from those of larger particles. In QDs the free carriers are confined to a small region by potential barriers in all three directions of 3D space. If the size of the region is less than the electron wavelength, the electronic states become quantized at discrete energy levels as it happens in an atom. They are a central theme in nanotechnology. Many types of QDs will emit light of specific frequencies if electricity or light is applied to them, and these frequencies can be precisely tuned by changing the dots' size, shape and material, giving rise to many applications [2 -3]. The exploitation of this kind of nanostructures towards the improvement of the devices performance mainly relies on the ability to control their size and uniformity [4].
To bury QDs into piezoelectric matrix induces not only an elastic field, but also a piezoelectric field. The induction of both fields is given by the lattice mismatch between the QDs and the surrounding piezoelectric matrix. Since both the elastic and piezoelectric fields are equally important in the understanding of the photonic and electronic features in semiconductors [5] a reliable analysis on these fields is crucial to the design of such structures. QDs can exist in a wide variety of shapes, including cuboidal, pyramidal [6], truncated pyramidal, and lens shaped.
The stress and the strain distributions in and around the QDs have recently been investigated by many researchers [7].
The finite element method (FEM) has become an essential solution technique in many areas of engineering and physics [8]. The FEM is an efficient computational tool in the development of efficient micro and nanoscale systems [9 -11].
In this paper a coupled 3D ANSYS Multiphysics FE model is used to simulate a cubic matrix with an embedded cubic QD under static thermal loadings. Our particular focus is on the influence of thermo-electromechanical effects on properties of QDs, where InAs (Indium arsenide) is embedded into GaAs (Gallium arsenide) matrix. The paper is organized as follows: section 2 and section 3 briefly present the governing equations for thermo-piezoelectricity and the finite element equations. Section 4 describes used simulation model in the ANSYS Multiphysics section 5 presents the numerical results. Conclusions are summarized in section 6.

Governing equations for thermo-electro-elastic fields
The base requirement of QDs is to have initial strain, which induces electric field in a piezoelectric material. This initial strain is given by the lattice mismatch. Usually the QD has almost vanishing thermal conduction and thermal expansion , ; where ij l is the thermal conductivity tensor. Both materials have transversely isotropic properties [13].
The following essential and natural boundary conditions are assumed for the mechanical field , , for the electric field and for the thermal field , , , , , where u C is the part of the global boundary with prescribed displacements, and on ti C , p C , Q C , Ci , and q C the traction vector, the electric potential, the surface charge density, the temperature and the heat flux are prescribed, respectively.

Finite element equations
Substituting (9), (10) and (11) into (6) gives the coupled finite matrix element equation for an element Mq Rq Kq f f where ; ; ## ### (15) where N(S) is the matrix of the shape functions evaluated on the boundary S. In a strongly coupled thermo-piezoelectric analysis, coefficients. In our analysis we consider a periodic distribution of QDs in the matrix and for a numerical simulation we can select a representative volume element (RVE) illustrated in Fig. 1. This piezoelectric composite structure under a thermal load can be described by the theory of thermo-piezoelectricity.

Fig. 1 Quantum dot electronic structure with InAs cubic dot in a finite sized GaAs substrate
The governing equations for thermo-piezoelectricity in a homogeneous medium are given by the balance of momentum, the first Maxwell's equation for the electric displacement vector, and heat conduction equation [12] , , ,  On the interface between InAs and GaAs the continuity of displacements, electric potential and temperature has to be satisfied In the first example stationary conditions can be considered and we consider prescribed temperature at a wide interval from 0 to 500 o C to test behavior of the electronic microstructure. the electric potential and temperature degrees of freedom are coupled [14].

Simulation model
Using the above definitions and governing equations described in the previous section, a Finite Element Model was developed in ANSYS Multiphysics to resemble the work done in ANSYS and validate the assumptions made in the model.
The Piezoelectric Devices (PzD) user interface found under the Structural Mechanics branch in ANSYS Multiphysics, combines Solid Mechanics and Electrostatics for modeling of piezoelectric devices, for which all or some of the domains contain a piezoelectric material. The interface has the equations and features for modeling piezoelectric devices, solving for displacements and electric potential.
Conversion of material properties of piezoelectric materials (such QDs) has caused many users confusion because of the difference between manufacturer-supplied data and the format required by ANSYS. The direct method for performing a coupledfield analysis involves a single analysis using a coupled field elements SOLID226 and SOLID227. Coupled-field elements contain all the necessary degrees of freedom. They handle the field coupling by calculating the appropriate element matrices (strong, or matrix coupling) or element load vectors (weak, or load vector coupling). In linear problems with strong coupling, coupled-field interaction is calculated in one iteration. Weak coupling requires at least two iterations to achieve a coupled response. For detailed descriptions of the elements and their characteristics (DOFs, KEYOPT options, inputs and outputs, etc.), see the Element Reference [15].

Mesh definition
In [16], the authors performed all numerical experiments under the condition for the relative errors between successive refinements to be less than 10 −6 . It was achieved with around 10 5 triangular elements. In our model, we define a slightly finer mesh than the default settings. The mesh is refined around the QDs to get high resolution for the calculated fields. A 3D ANSYS model was created by constructing the geometry shown in Fig. 2 and meshing it using 195,696 SOLID elements with 339,117 nodes.
An important point to note is that the best way to check the numerical results is to use to check the results are symmetric to X and Y. For cubic materials nine planes of elastic symmetry

Material properties
For a periodic distribution of quantum dots in a piezoelectric matrix we consider a representative volume element (RVE) with prescribed temperature on the upper side. The RVE is assumed to have a cubic geometry with side length of 40 nm and a cubic quantum dot with side length of 4nm is embedded in the center. Material properties of the dot correspond to InAs [17 -19]: . GPa, . GPa, . GPa, .
, 22 33 We note that the pyroelectric coefficient p 1 is not considered in the numerical analyses as a quantity with a small influence on the results.

Results and discussion
In this example stationary conditions are considered and we consider prescribed temperature at the value 250 o C to test behavior of the electronic microstructure. All calculations were performed for the GaAs/InAS QDs obtained from finite element analysis carried out in ANSYS Multiphysics program. 22 33 For band-gap calculations, the mismatch strain is subtracted from the actual compatible elastic strain. As a result while in the solid mechanics community the compatible elastic strain is normally expressed and plotted, the QDs research community often illustrates the subtracted strain. This can potentially cause confusion and care must be exercised in interpreting results from the solid mechanics literature.  are prescribed and are derived from tetragonal symmetry [20]. Three of nine planes perpendicular to coordinate axes, are interchangeable. Six planes have normals that contain an angle π/4 with the coordinate axes. Only three independent constants remain. The remaining 12 coefficients are all equal to zero. The potential difference across the top and bottom of the QD decreases with an increase in temperature. The electric field E x in plane XY decreases with an increase in temperature.
However, the values of electromechanical quantities show a slight shift. This shift is significant as the values for quantities like electric potential, X, Y and Z-component of electric field vector and X-component of strain tensor, xx f at the boundaries of QDs system structure.
The electric potential in planes Y = 0 and Z = 0 are similar to that in the plane X = 0 presented in Fig. 3 due to cubic symmetry of the boundary conditions and material properties. We just need to mention that using the symmetry conditions, other E i on the three planes can be easily shown, with similar patterns. The strain components are presented in Figs. 5a-d.
We investigated also influence of temperature prescribed on the upper surface of the quantum dot electronic structure EFGH in Fig. 1 Fig. 6 The variation of strain components the material properties of the piezoelectric QD and matrix. The best way to check the numerical results is to use the results in figures and check if the results are symmetric with respect to x and y. We are dealing with cubic materials and the results should be cubic symmetric. Once we have checked this issue, the second check is to check if the results are reasonably accurate enough by comparing these from two different mesh sizes. In this paper, we did not do this check, because the FE mesh is sufficiently fine.