Mixed FEM for quantum nanostructured solar cells

Mixed FEM for quantum nanostructured solar cells

Composite Structures 229 (2019) 111460 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/comp...

1MB Sizes 0 Downloads 87 Views

Composite Structures 229 (2019) 111460

Contents lists available at ScienceDirect

Composite Structures journal homepage: www.elsevier.com/locate/compstruct

Mixed FEM for quantum nanostructured solar cells Jan Sladek a b

a,b

a

a

, Vladimir Sladek , Miroslav Repka , Siegfried Schmauder

T b

Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia Institute for Materials Testing, Materials Sciences and Strength of Materials, University of Stuttgart, 70569 Stuttgart, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: High-density arrays of quantum dots Functionally graded lattice mismatch Mixed FEM Gradient theory

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).

1. 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 [6–9]. 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

E-mail address: [email protected] (J. Sladek). https://doi.org/10.1016/j.compstruct.2019.111460 Received 21 June 2019; Accepted 17 September 2019 Available online 19 September 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

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 [17–19] 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

Composite Structures 229 (2019) 111460

J. Sladek, et al.

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.

fα (x α ) = ⎡2 x α − ⎣

Ej = −ϕj ,

ηijk = εij, k =

am − aQD 1, in QD δij, χ = ⎧ ⎨ aQD ⎩ 0, in matrix

2

n

,

+ 4 / hQD ⎤ ⎦

(3)

1 (ui, jk + uj, ik ), 2

(4)

and the elastic strain gradients follow from Eq. (2) e ηijk

= εije, k = (εij − εij∗ )k .

(5)

The density of the electro-elastic enthalpy in gradient theory for piezoelectricity is given by [23,24]

U (ε, ∇ε, E) 1 1 1 = c ijkl εij εkl + g ijklmn ηijk ηlmn − hijEi Ej − eijk εij Ek − c ijkl εij εkl∗ , 2 2 2

(6)

where cijkl , eijk , hij and gijklmn are the material elastic, piezoelectric, dielectric and high-order elastic tensor in a piezoelectric medium, respectively. The Cauchy stress σij , the higher-order stress τijk and the electric displacement Di are obtained from the deformation energy density (6) as

σij (x) =

τijk (x) =

∂U = cijkl (εkl (x) − εkl∗ (x)) − ekij Ek (x), ∂εij ∂U = gijklmn ηlmn (x), ∂ηijk

(1)

Di (x) = −

where ui and ϕ are displacements, and the electric potential, respectively, The total strain is given as the sum of elastic and eigestrains

εije = εij − εij∗, εij∗ = f (x) χ

hQD

where n is the power law index. For n = 1, the linear variation in the QD is considered. The strain-gradient tensor is defined as

A periodic distribution of QDs (InAs) in the matrix (GaAs) is considered and a representative volume element (RVE) for the quantum nanostructure solar cell is illustrated in Fig. 1. The misfit strains induce electro-mechanical fields in this nanostructure solar cell. This eigenstrain is created by the lattice mismatch, which are occured only at the interface of the QD and matrix. Inside of the QD functional variations should be considered with vanishing misfit at the center. A quasistatic electro-mechanical theory is sufficient to describe the system realistically. The strain tensor εij , and the electric field vector Ej are defined as

1 (ui, j + uj, i ); 2

n

/ hQD ⎤ for α = 1, 2 ⎦

f3 (x3) = ⎡2 x3 − b + ⎣

2. The gradient theory for electro-mechanical fields

εij =

b 2

(7)

(8)

∂U = eijk εjk (x) + hij Ej (x) ∂Ei

(9)

The Voigt notation is used for constitutive Eqs. (7)–(9) with cubic symmetry of material in the following matrix form:

(2)

c11 c12 c12 0 0 0 ⎤ σ ⎧ σ11 ⎫ ⎡ c12 c11 c12 0 0 0 ⎥ ⎢ 22 ⎪ ⎪ ⎢ ⎪ σ33 ⎪ 0 0 0⎥ c c c = ⎢ 12 12 11 ⎥ ⎨ σ23 ⎬ ⎢ 0 0 0 c44 0 0 ⎥ ⎪ σ13 ⎪ ⎢ 0 0 0 0 c44 0 ⎥ ⎪ σ12 ⎪ ⎢ ⎩ ⎭ ⎣ 0 0 0 0 0 c44 ⎥ ⎦

where εij∗ is the eigenstrain tensor, am , aQD are lattice constants of the matrix and QD and f(x) expresses a functional variation of the initial misfit strains with respect to the center of QDs [22], respectively. The lattice mismatch occurs only at the interface of the QD and the matrix. Vanishing values of the lattice mismatch are considered at the center of the QD. Then, the functional variation for three directions can be given as

e ⎧ ε11 ⎫ e ⎪ ⎡0 0 0⎤ ⎪ ε22 ⎪ e ⎪ ⎢ 0 0 0 ⎥ E1 ⎪ ε33 ⎪ ⎢ 0 0 0 ⎥ ⎧ ⎫ E2 , − e ⎬ ⎢ e14 0 0 ⎥ ⎨ ⎬ ⎨ 2ε23 ⎪ e ⎪ ⎢ 0 e14 0 ⎥ ⎩ E3 ⎭ ⎪ 2ε13 ⎪ ⎢ 0 0 e ⎥ 14 ⎦ e ⎪ ⎣ ⎪ 2ε12 ⎭ ⎩

(10)

⎧ D1 ⎫ ⎡ 0 0 0 e14 0 D2 = ⎢ 0 0 0 0 e14 ⎨ ⎬ ⎢ ⎩ D3 ⎭ ⎣ 0 0 0 0 0

ε ⎧ ε11 ⎫ 22 ⎪ 0 ⎤ ε ⎪ ⎡ h11 0 0 ⎤ ⎧ E1 ⎫ ⎪ 33 ⎪ 0 ⎥ 2ε + ⎢ 0 h11 0 ⎥ E2 ⎨ 23 ⎬ ⎢ ⎥⎨ ⎬ e14 ⎥ ⎦ ⎪ 2ε13 ⎪ ⎣ 0 0 h11⎦ ⎩ E3 ⎭ ⎪ 2ε ⎪ ⎩ 12 ⎭

τ ⎡ c11 c12 c12 0 0 0 ⎤ ⎧ τ11k ⎫ ⎢ c12 c11 c12 0 0 0 ⎥ 22 k ⎪ ⎪ ⎢c c c ⎪ τ33k ⎪ 0 0 0⎥ 2 = l ⎢ 12 12 11 ⎥ c 0 0 0 0 0⎥ ⎨ τ23k ⎬ 44 ⎢ ⎪ τ13k ⎪ ⎢ 0 0 0 0 c44 0 ⎥ ⎪ τ12 k ⎪ ⎢ 0 0 0 0 0 c44 ⎥ ⎩ ⎭ ⎦ ⎣

η ⎧ 11k ⎫ ⎪ η22k ⎪ ⎪ η33k ⎪ , ⎨ 2η23k ⎬ ⎪ 2η13k ⎪ ⎪ ⎪ ⎩ 2η12k ⎭

(11)

(12)

In Eq. (12) it is assumed that higher-order elastic parameters gjklmni are proportional to the conventional elastic stiffness coefficients cklmn by the internal length material parameter l [25–27]. A more compact form is convenient for Eqs. (10)–(12)

σ = Cεe − ZE,

D = ZT ε + GE.

(13)

Recently, the authors have derived the governing equations for strain gradient theory in piezoelectricity [28]:

Fig. 1. A unit quantum nanostructure solar cell. 2

Composite Structures 229 (2019) 111460

J. Sladek, et al.

σij, j (x, t ) − τijk, jk (x, t ) = 0, Dk, k (x, t ) = 0.

(14)

ε ⎡ ∂1 0 ⎡ ε11 ⎤ ⎢ 0 ∂2 ⎢ 22 ⎥ ∂1 0 ⎤ ⎢ 0 0 ⎢ ε33 ⎥ ⎡ ε = ⎢ 2ε ⎥ = ⎢ 0 ∂3 ⎥ ⎢ 23 0 ∂3 ⎢ ⎢ ⎥ ⎢ 2ε ⎥ ∂3 ∂1⎦ ⎢ ∂3 0 ⎢ 13 ⎥ ⎣ ⎢ ⎣ 2ε12 ⎦ ⎣ ∂2 ∂1 ⎡ ∂1 ⎤ = ⎢ ∂2 ⎥ ϕ = Bϕ (ξ1, ξ2, ξ3) q ϕ. ⎢∂ ⎥ ⎣ 3⎦

Essential and natural boundary conditions (b.c.) can be prescribed in the present theory: Essential b.c.: ui (x) = u¯ i (x) on Γu , Γu ⊂ Γ (15)

si (x) = s¯i on Γs , Γs ⊂ Γ ϕ (x) = ϕ¯ (x) on Γϕ , Γϕ ⊂ Γ Natural b.c.: ti (x) = t¯i (x) on Γt , Γt ∪ Γu = Γ , Γt ∩ Γu = ∅ Ri (x) = R¯ i (x) on ΓR , ΓR ∪ Γs = Γ, ΓR ∩ Γs = ∅

(16)

S (x) = S¯ (x) on ΓS , ΓS ∪ Γϕ = Γ , ΓS ∩ Γϕ = ∅where si: =

∂ui , Ri : =nk nj τijk . ∂n

∂ρi + ∂π

u u u ⎡ 1⎤ ⎡ 1⎤ ⎡ 1⎤ s = ∂1 ⎢u2 ⎥ n1 + ∂2 ⎢u2 ⎥ n2 + ∂3 ⎢u2 ⎥ n3 = Bs (ξ1, ξ2, ξ3) qu. ⎣u3 ⎦ ⎣u3 ⎦ ⎣u3 ⎦

(17)

∑ ∥ρi (x c) ∥ δ (x − x c), c

S: =nk Dk ,

(26)

(18)

In the mixed FEM formulation the strains are approximated independently. This approximation in local coordinates can be written as [20]:

(19)

ε̂ In = A (ξ1, ξ2, ξ3 ) α

where ρi : =nk πj τijk , δ(x) being the Dirac delta function and πi is the Cartesian component of the unit tangent vector on boundary Γ. The jump at a corner (xc) on the oriented boundary contour Γ is defined as

∥ρi (x c)∥: =ρi (x c + 0) − ρi (x c − 0).

(25)

The normal derivative of displacement is approximated by

The traction vector, and the electric charge are defined as

ti: =nj (σij − τijk, k ) −

0⎤ 0⎥ u1 ⎡ E1 ⎤ ∂3 ⎥ ⎥ ⎡u2 ⎤ = Bu (ξ1, ξ2, ξ3) qu, −E = −⎢ E2 ⎥ ⎢ ⎥ ∂2 ⎥ u3 ⎢ E3 ⎥ ⎣ ⎦ ⎣ ⎦ ∂1 ⎥ ⎥ 0⎦

(27)

where α are undetermined coefficients. The polynomial function matrix A(ξ1, ξ2, ξ3) for 3D 8-node brick element can be selected as:

A(ξ1, ξ2, ξ3) = [1 ξ1 ξ2 ξ3 ξ1 ξ2 ξ1 ξ3 ξ2 ξ3 ξ1 ξ2 ξ3 ].

(20)

In order to preserve the number of degrees of freedom, both the approximations (22) and (27) for the strain should be equal at clever selected internal points of the element to get the unknown coefficient sα . The Gauss quadrature points are selected for collocation

A finite element formulation is developed to our piezoelectric problem in the next section. 3. The mixed finite element method (FEM)

ε̂ In (ξ c , α) = ε̂ (ξ c , qu),

(28)

The principle of virtual work is applied to derive the FEM equations for a boundary value problem in gradient theory of piezoelectricity. It means that variation of the electro-elastic enthalpy has to be equal to the work of the external generalized forces on generalized displacements

where ξ c = (ξ1c , ξ2c , ξ3c ) are collocation points. The right hand side in (27) is replaced by approximation from (25) at the 8 collocation points and one can write:

∫Ω (σij δui,j + τijk δui,jk + Dk δϕ,k ) dΩ = ∫ t¯i δui dΓ + ∫ R¯ i δsi dΓ + ∫ Γ Γ Γ

The independent strains can be expressed by nodal displacements if Eqs. (27) and (28) are utilized

t

R

S

¯ Γ, Sδϕd

A (ξ c) α = Bu (ξ c) qu.

ε̂ In = A (ξ1, ξ2, ξ3) Lcqu,

(21)

where overbar is used for prescribed boundary values. The order of derivatives in governing equations in gradient theory of piezoelectricity is higher than in conventional theory. Then, C1 continuous interpolations of displacements are required in the FEM. However, it is a difficult task to derive C1 continuous elements for 3-D problems. It seems to be easier to develop the mixed FEM, where C0 continuous interpolation is applied independently for displacement and displacement gradients. The kinematic constraints (1) between strains and displacements are satisfied by collocation at some cleverly chosen internal points in elements [20,29]. Both mechanical displacements and electric potential are approximated by the nodal values and shape functions

u = Nu qu,

(22)

ϕ = Nϕ q ϕ,

(23)

= Nε q *ε ,

(30)

where Lc = A−1 (ξ c) Bu (ξ c) . Analogically one can get an approximation of the independent strain gradients in the local coordinate system

⎡ ∂1 ⎤ ⎡ ∂1 ⎤ η̂ In = ⎢ ∂2 ⎥ ε̂ I¨n = ⎢ ∂2 ⎥ A (ξ1, ξ2, ξ3 ) α = A∗ (ξ1, ξ2, ξ3 ) α = A∗ (ξ1, ξ2, ξ3 ) Lcqu. ⎢∂ ⎥ ⎢∂ ⎥ ⎣ 3⎦ ⎣ 3⎦ (31) The gradient of eigenstrains εij∗, k are approximated similarly as the electric intensity vector in (25):

⎡ ∂1 ⎤ η∗ = ⎢ ∂2 ⎥ ε∗ = Bε (ξ1, ξ2, ξ3 ) q *ε , ⎢∂ ⎥ ⎣ 3⎦

(32)

where Bε = Bϕ . The variations of the primary field variables can be also expressed in terms of the variations of the nodal values δqu , and δq ϕ . Thus, the variational statement (21) can be rewritten as

where qu , q ϕ , and Ni (ξ1, ξ2, ξ3) are nodal displacements and electric potential shape functions, respectively. Similarly the nonhomogeneous eigenstrains εij∗ on the QD are approximated by C0 continuous interpolation

εij∗

(29)

∫Ω {δ qu}T (Lc)T AT [C ALc {qu} + ZBϕ {q ϕ}] dΩ + l2 ∫Ω {δ qu}T (Lc)T A∗TCA∗Lc {qu } dΩ + ∫ {δ q ϕ }T BTϕ [ZT Bu {qu} − GBϕ {q ϕ }] dΩ Ω ¯ dΓ + {δ qu }T = ∫ {δ qu }T (Lc)T AT CNε {q∗ε } dΩ + {δ qu }T ∫ NTu T Ω Γ ∫Γ BTs R¯dΓ + {δ q ϕ}T ∫Γ NTϕ S¯dΓ, (33)

(24)

where q *ε are nodal values of known eigenstrains. The approximation of strain and electric intensity vector can be expressed from the kinematic Eq. (1) and approximation (22) and (23) as:

t

R

3

S

Composite Structures 229 (2019) 111460

J. Sladek, et al.

where C , Z and G are matrices of elastic, piezoelectric and dielectric coefficients, respectively. Since the variations δqu and δq ϕ can be arbitrary in the variation statement (33), the following systems of algebraic equations are obtained:

∫Ω (Lc)T AT [C ALc {qu} + ZBϕ {q ϕ}] dΩ + l2 ∫Ω (Lc)T A∗TCA∗Lc {qu} dΩ ¯ dΓ + ∫ BTs R ¯ dΓ, = ∫ (Lc)T AT CNε {q∗ε } dΩ + ∫ NTu T Ω Γ Γ

(34)

∫Ω BTϕ [ZT Bu {qu} − GBϕ {q ϕ}] dΩ = ∫Γ

(35)

t

R

S

NTϕ S¯dΓ.

4. Numerical results The representative volume element (RVE) in Fig. 1 is assumed to have a cubic geometry with side length of b = 40 nm, and a cubic QD with side length of hQD = 4nm is embedded in it. The top surface of the InAs QD is 4 nm from the top surface of the matrix. Material properties of the QD correspond to InAs [30]. The substrate is made of GaAs, whose properties are

c11 = 118.8GPa, c12 = 54GPa, c44 = 59.4GPa, e14 = −0.16 C/m2, h11 = 0.1346·10−9C2/Nm2. The four side faces of the matrix are fixed along their normal direction, the bottom face is fixed along its normal direction, and the upper surface is free of tractions. All surfaces have vanishing normal components of electric displacements, except for the bottom side, where a vanishing electric potential is prescribed. One can write the following boundary conditions for the InAs/GaAs quantum dot structure (Fig. 1):

e Fig. 2. Elastic strain component ε11 along X1-direction (passing the QD center) (upper) and the potentials ϕ along X3 at X1 = 18 nm and X2 = 22 nm (bottom).

On surfaces BCGF and ADHE: u1 = 0 , t2 = t3 = 0 , D1 = 0 ; ABFE and DCGH: u2 = 0 , t1 = t3 = 0 , D2 = 0 ; ABCD: u3 = 0 , t1 = t2 = 0 , ϕ = 0 ; EFGH: t3 = 0 , t1 = t2 = 0 , D3 = 0 . On the interface between InAs and GaAs, the continuity of primary fields is considered

uim = uiQD ,

ϕm = ϕQD ,

(36)

as well as the reciprocity of traction vectors and electric displacements

tim + tiQD = 0,

Dim + DiQD = 0,

(37)

e Fig. 3. Elastic strain component ε11 along X1-direction (passing the QD center) for different misfit function exponents n.

4.1. Nonhomogeneous lattice mismatch The lattice mismatch occurs only at the interface of the QD and ∗ ∗ ∗ = ε22 = ε33 = 0.07 . To test the commatrix with prescribed values: ε11 puter code we have analyzed the QD structure under uniform eigenstrains in the QD and in the framework of the classical theory with l = 0. A 3D FEM model was set up by constructing the geometry and meshing it using 195,696 elements corresponding to 339,117 nodes. The variation of the strain component ε11 along X1-direction (passing the QD center) and electric potential ϕ along X3 (aligned with the edge of the QD at X1 = 18 nm and X2 = 22 nm) is presented in Fig. 2, where results are obtained by classical piezoelectric theory. One can observe excellent agreement between the mixed FEM and ANSYS results with errors less than 1% for both the strain and electric potential. Next a nonhomogeneous lattice mismatch is considered with vanishing values at the center of the QD. The functional variation for three directions is given by Eq. (3). The gradient theory is applied since the material length scale parameter is considered as l = 0.5 × 10−9m [31]. To illustrate the influence of the functional variation of eigenstrains along three directions, three power law exponents n are considered in e Eq. (3). Elastic ε11 and total strain ε11 variations along x1 are presented in

Fig. 4. Total strain component ε11 along X1-direction (passing the QD center) for different misfit function exponents n.

Figs. 3 and 4, respectively. One can observe that with increasing values of the power law exponent, the peak values of strains on the interfaces of the QD with the matrix are reduced. The largest peak values are 4

Composite Structures 229 (2019) 111460

J. Sladek, et al.

e Fig. 8. The variation of elastic strain component ε11 in the center of the QD with the size of matrix and uniform lattice misfit.

e Fig. 5. Elastic strain component ε33 along X3 -direction (passing the QD center) for different misfit function exponents n.

that an interaction of the induced electro-mechanical fields for both QDs can be significant. It is the reason why this influence is investigated in this paper. Three various sizes of matrix b are considered here, namely b = 60, 40, 20 nm. e Fig. 8 presents a variation of the elastic strain component ε11 in the center of the QD with the size of matrix and a uniform lattice misfit. One can observe that the absolute value of the elastic strain component in the center of a QD increases if the distance between two QDs is decreasing. The induced electric potential in Fig. 9 increases too if the distance between two QDs is decreasing. However, the influence of distance for two QDs is weak for both fields. 5. Conclusions

A high-density of QDs is widely used within solar photovoltaic cells. Then, the distance between two neigbouring QDs is sufficiently short

The 3D mixed finite element model has been developed for QDs with functionally graded lattice mismatch between the QD and the surrounding matrix. The periodic QD array is replaced by a representative volume element (RVE), which is analyzed by the gradient theory of piezoelectricity. Since governing equations in the gradient theory contain higher order derivatives than in conventional approaches, a continuous approximation of strains is required. The mixed FEM with the C0 continuous interpolation and collocation approach for kinematic constraints between strains and displacements is developed. The largest peak values of strains are observed on interfaces of the QD and the surrounding matrix for a uniform eigenstrain. With increasing value of power law exponent for functionally graded lattice mismatch, the peak values of strains on the interface of the QD with the matrix are reduced. A similar phenomenon is observed for the induced electric potential. The density of QDs in the matrix is analysed too. It is observed that absolute values of the elastic strain component in the center of the QD increase if the distance between two QDs is decreasing. The reported observations can be utilized for an optimal design of QD

Fig. 7. Electric potentials ϕ along X1 at X2 = 22 nm and X3 = 36 nm for different misfit function exponents n.

Fig. 9. The variation of induced electric potential in the center of a QD with the size of matrix and uniform lattice misfit.

Fig. 6. Total strain component ε33 along X3 -direction (passing the QD center) for different misfit function exponents n.

observed for the uniform eigenstrain case. e Similarly, elastic ε33 and total strain ε33 variations along X3 are presented in Figs. 5 and 6, respectively. In this case also the largest peak values of strains on interfaces of the QD and the matrix are observed for a uniform eigenstrain. Finally, the induced electric potential ϕ variations along X1 are presented in Fig. 7. The influence of the functional variation of eigenstrains on the electric potential variation is similar as for both strain components. 4.2. Influence of quantum dot density on strain distribution

5

Composite Structures 229 (2019) 111460

J. Sladek, et al.

nanostructures.

Acta Mech 2018;229:3089–106. [16] Sladek J, Bishay PL, Repka M, Pan E, Sladek V. Analysis of quantum-dot systems under thermal loads based on gradient theory. Smart Mater Struct 2018;27:095009. [17] Aifantis E. On the microstructural origin of certain inelastic models. ASME J Eng Mater Technol 1984;106:326–30. [18] Altan S, Aifantis E. On the structure of the mode III crack-tip in gradient elasticity. Scripta Metall Mater 1992;26:319–24. [19] Askes H, Aifantis EC. Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 2011;48:1962–90. [20] Bishay PL, Atluri SN. High performance 3D hybrid/mixed, and simple 3D Voronoi cell finite elements, for macro- & micro-mechanical modeling of solids, without using multi-field variational principles. CMES. Comput Model Eng Sci 2012;84:41–97. [21] Deng F, Deng Q, Yu W, Shen S. Mixed finite elements for flexoelectric solids. J Appl Mech 2017;84:0810041–810112. [22] Chu HJ, Wang JX. Strain distribution in arbitrarily shaped quantum dots with nonuniform composition. J Appl Phys 2005;98:034315. [23] Toupin RA, Gazis DC. Surface effects and initial stress in continuum and lattice models of elastic crystals. In: Wallis, R.F. (Ed.), Proceedings of the International Conference on Lattice Dynamics held at Copenhagen, Denmark, August 5–9, 1963, 597–605. [24] DiVincenzo DP. Dispersive corrections to continuum elastic theory in cubic crystals. Phys Rev B 1986;34:5450–65. [25] Gitman I, Askes H, Kuhl E, Aifantis E. Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity. Int J Solids Struct 2010;47:1099–107. [26] Liang X, Shen S. Size-dependent piezoelectricity and elasticity due to the electric field-strain gradient coupling and strain gradient elasticity. Int J Appl Mech 2013;5:1350015. [27] Yaghoubi ST, Mousavi SM, Paavola J. Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity. Int J Solids Struct 2017;109:84–92. [28] Sladek J, Sladek V, Stanak P, Zhang Ch, Tan CL. Fracture mechanics analysis of sizedependent piezoelectric solids. Int J Solids Struct 2017;113:1–9. [29] Dong L, Atluri SN. A simple procedure to develop efficient & stable hybrid/mixed elements, and Voronoi cell finite elements for macro- & micromechanics. CMC: Comput, Mater Continua 2011;24:61–104. [30] Glazov VM, Pashinkin AS. Thermal expansion and heat capacity of GaAs and InAs. Inorg Mater 2000;36:225–31. [31] Majdoub MS, Sharma P, Cagin T. Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Physics Review B 2008;77:125424.

Acknowledgment The authors acknowledge the support of the Slovak Science and Technology Assistance Agency registered under number APVV-180004, and VEGA-2/0046/16. References [1] Nozik AJ. Quantum dot solar cells. Physica E 2002;14:115–20. [2] Raffaelle RP, Castro SL, Hepp AF, Bailey SG. Quantum dot solar cells. Prog Photovoltaics Res Appl 2002;10:433–9. [3] Medintz IL, Mattoussi H, Clapp AR. Potential clinical applications of quantum dots. Int J Nanomed 2008;3:151. [4] Wu J, Wang ZM. Quantum dot solar cells. New York: Springer; 2014. [5] Zheng Z, Ji H, Yu P, Wang Z. Recent progress towards quantum dot solar cells with enhanced optical absorption. Nanoscale Res Lett 2016;11:266. [6] Grundmann M, Stier O, Bimberg D. InAs/GaAs pyramidal quantum dots: strain distribution, optical phonons, and electronic structure. Phys Rev B 1995;52:11969. [7] Jogai B. Three-dimensional strain field calculations in multiple InN/AlN wurtzite quantum dots. J Appl Phys 2001;90:699. [8] Pan E. Elastic and piezoelectric fields around a quantum dot: fully coupled or semicoupled model. J Appl Phys 2002;91:3785–96. [9] Pan E. Elastic and piezoelectric fields in substrates GaAs (001) and GaAs (111) due to a buried quantum dot. J Appl Phys 2002;91:6379–87. [10] Bimberg D, Grundmann M, Ledentsov NN. Quantum Dot Heterostructures. New York: Wiley; 1998. [11] Okada Y, Oshima R, Takata A. Characteristics of InAs/GaNAs strain-compensated quantum dot solar cell. J Appl Phys 2009;106:024306. [12] Patil SR, Melnik RVN. Thermoelectromechanical effects in quantum dots. Nanotechnology 2009;20:125402. [13] Duc ND, Seung-Eock K, Quan TQ, Long DD, Anh VM. Nonlinear dynamic response and vibration of nanocomposite multilayer organic solar cell. Compos Struct 2018;184:1137–44. [14] Bishay P, Sladek J, Pan E, Sladek V. Analysis of functionally graded quantum-dot systems with graded lattice mismatch strain. J Comput Theoretical Nanosience 2018;15:542–50. [15] Rashidinejad E, Naderi AA. Analytical study of electro-elastic fields in quantum nanostructure solar cells: the inter-nanostructure couplings and geometrical effects.

6