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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Analysis of Quantum Dots Induced Strain and Electric-Field in Piezoelectric Semiconductor Substrate of General Anisotropy

C.-Y. Wang 1*, M. Denda 2, E. Pan 3 1 Mathematics and Modelling Department, Schlumberger-Doll Research, Old Quarry RD, Ridgefield, CT 06877-4108, USA 2 Mechanical and Aerospace Engineering Department, Rutgers University, 98 Brett Road, Piscataway, NJ 08854-8058, USA 3 Advanced Material and Structure Center, College of Engineering, University of Akron, Akron, OH 44325-3905, USA Email: wang20@slb.com, denda@jove.rutgers.edu, pan2@uakron.edu Abstract Characteristics of the self-organized quantum dots (QDs) such as electron and hole energy levels and wave functions are dependent to the state of strain and electric field produced during the growing process of QDs in a semiconductor substrate. The calculation of the strain and electric field is one of the most challenging components in the QDs simulation process. It involves material anisotropy induced coupling between the elastic and electric fields and it must include the full three-dimensional and usually intricate shapes of the QDs. Numerical simulations are often performed by finite difference, finite element, or atomistic techniques, all require substantial computational time and memory. In this paper, we present a new Green’s function approach which takes into account QDs of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezoelectricity. Following the literature of micromechanics, the problem is formulated as an Eshelby inclusion problem of which the solution can be expressed by a volume-integral equation that involves the Green’s functions and the equivalent bodyforce of eiegenstrain. The volume integral is subsequently reduced to a line integral based on exploring a unique structure of the Green’s functions. The final equations are cast in a form that most of the computation results can be repeatedly used for QDs at different locations - a very attractive feature for simulating large systems of QD arrays. The proposed algorithm has been implemented and validated by comparison with analytical solutions. Numerical simulations are presented for pyramidal QDs in the substrates of Gallium Arsenite (GaAs) [001] and GaAs [111].

Key words: quantum dots, semiconductor, strain and electric field, material anisotropy INTRODUCTION

The QDs growth technology has advanced to a stage that the self organized QD structures can be produced with high quality in a wide range of semiconductor compounds; see Figs. 1 and 2. That has translated into immense interests in research to understand the fundamental physics, as well as an enormous investment in the development of nanotechnology. The quantum effects due to confinement on electronic motion are completely foreign phenomena to natural materials. It is widely anticipated that the quantum effects and other special electronic and optical properties of QDs will render novel devices with fascinating functionalities and applications. One technology particularly attractive to the oil service industry is the application in semiconductor light-emitting and laser diodes. Nanotechnology based laser has demonstrated very good temperature characteristics - potentially excellent light sources for downhole optical telemetry and sensing [12, 14, 22, 23, 37]. Obviously mathematical and numerical simulations have an important role in the study of QDs. At the research stage of exploring the physics, computational simulation is instrumental in interpreting data, and guiding future developments. At the engineering stage of optimization, it provides quantitative analysis quickly and economically that experiment or other means cannot. Analysis of the electronic structure of QDs includes calculations of the electron and hole energy levels and wave functions in arbitrary-shaped QD structures. In addition, it is necessary to take into account effects of the strain and piezoelectric fields induced during the process of QDs growth. Often the growing of semiconductor quantum dots is achieved by the controlled coarsening of a thin film that is strained with

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Figure 1: Micrograph of pyramid-shaped quantum dots grown from indium, gallium, and arsenic

(Each dot is about 20 nanometers wide and 8 nanometers in height. Image courtesy NIST)

respect to the substrate. This self-assembled coarsening/roughening is a result of misfit-lattice-induced strains. The dots are often capped by the substrate material, thus extending the strain around the dot to all angular directions. Not surprisingly, the electronic structure of QDs is profoundly affected by their strain profile. Furthermore, should the semiconductor be piezoelectric, a misfit-lattice-induced electric potential will also contribute to the properties of the QDs. Indeed one could control/optimize the electronic and optical properties of QDs by altering the elastic strain and electric field [4, 21, 23, 24, 26]. The calculation of the strain tensor and electric field is one of the most challenging components in the QDs simulation process. It involves material anisotropy induced coupling between the elastic and electric fields and it must include the full three-dimensional and usually intricate shapes of the QDs. Various computational methods with different degrees of sophistication and limitations have been proposed in the last few years. The finite element method (FEM) and finite difference method (FDM) are the most frequently used [2, 15, 20, 35]. Other methods include the two-dimensional Fourier transform method [1, 16], the atomistic quantum-mechanics model [18, 33], and the Green’s function methods [11]. Among them, the Green’s function method appears particularly attractive. Its analytical nature provides insights to the understanding of the physics involved. It also produces very accurate numerical solutions with a speed much faster than the FEM and FDM [5, 11, 32]. Recently, Pan and his co-workers have employed the equivalent body-force concept in the literature of Eshelby inclusion problem [10, 25]. They succeeded in extending the Green’s function approach to anisotropic semiconductor substrates [17, 28, 29, 30, 31, 43].

Figure 2: STM of quantum dot formed by self-assembling (Ge ’pyramid’)

(Courtesy Hewlett-Packard; image acquired by G. Medeiros-Ribeiro, Hewlett-Packard Labs)

In this paper, we present a new Green’s function approach which takes into account QDs of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezo-electricity. Following Pan [29], the problem is formulated as a Eshelby inclusion problem of which the solution can expressed by a volume-integral equation that involves the Green’s functions and the equivalent body-force of eiegenstrain. The volume integral is subsequently reduced to a line integral based on exploring a unique structure of the Green’s functions. The final

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equations are cast in a form that most of the computation results can be repeatedly used for QDs at different locations ⎯ a very attractive feature for simulating large systems of QD arrays. This work can be considered as an improvement of the Green’s functions approach in computation time and in the inclusion of three-dimensional QDs of arbitrary shapes in a semiconductor substrate of general anisotropy and piezoelectricity. The proposed algorithm has been implemented and validated by comparison with analytical solutions. Numerical simulations are presented for pyramidal QDs in the substrates of Gallium Arsenite (GaAs) [001] and GaAs [111].

EQUIVALENT BODY FORCE OF EIGENSTRAIN

The ‘eigenstrain’ in an elastic material and its representation by the ’equivalent body-force’ are well established in the classical micromechanics [9, 25]. They have been found useful in the study of QD induced elastic-electric field in a piezoelectric substrate [29, 30]. In this section we follow the same train of arguments to derive the equivalent body-force of a QD in a piezoelectric substrate of the most general class of anisotropy, in a three dimensional Cartesian coordinate system x = (x1, x2, x3). Before getting into the subject matter, we shall first introduce compact notations for the coupled electric-elastic fields. The elastic field is described by its displacement vector uj , body force vector fj , stress tensor σij , strain tensor γ ij and

eigenstrain tensor . The electric field quantities include the electric potential φ, charge q, displacement vector Dj, field vector Ej, and eigenfield vector . We shall adopt the usual compact notations of using capital-letter subscripts

for the ’extended’ displacement uJ, body force fJ, stress σiJ, strain γ Ji and egenstrain strain defined as follows [40],

(1)

and

(2)

As shown above, when a capital-letter subscript appears, it indicates this is an ‘extended’ elasticelectric field quantity with four components. This notation may appear a little awkward, but it has proven efficient in putting equations in compact forms.

Let be the extended eigenstrain of the QD, which has a domain V bounded by its boundary ∂V. The extended stress is determined by the constitutive equations

(3)

where the material constants ciJKl are given by

(4)

in which cijkl, eijk and κ ik are the elasticity tensor, the piezoelectric tensor and the dielectric permittivity tensor,

respectively. The symmetry of the extended fields leads to the symmetry of cijkl, eijk and κ ik. Furthermore, a positive

extended strain energy requires that cijkl and κ ik be positive definite. Upon substitution of the following relationships

between uj -γ kl and between φ-El:

(5)

the constitutive equations (3) yield

(6)

where the following symmetry conditions of the material constants ciJKl have been used.

(7)

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In this paper, ∂if or f,i are used to indicate partial derivative of f(x) with respect to xi. The summation convention over double indices is applied. Now, substitution of Eq. (6) into the equilibrium equations

(8)

leads to

(9)

The right side term

(10)

is the equivalent body force of the eigenstrain (Mura, 1987 [25]; Pan, 2002 [30]). Notice, the equivalent body force is defined in the QD domain V. VOLUME AND BOUNDARY INTEGRAL EXPRESSIONS Eq. (9) lends itself directly to the volume integral representation

(11)

where gJP(x, y) is the Green’s function, or the singular displacement solution produced by a point source. Specifically, gJP(x, y) corresponds to the J-th component of the extended displacement at location x produced by the P-th component of an extended point force applied at y. If we assume that the eigenstrain is uniform within the QD domain V, the volume integral (11) can be transformed into a surface integral on the boundary ∂V, following the framework of Mura [25] or its extension to the piezoelectricity of Pan [30]. The result is

(12)

where ni(x) is the outward normal to the boundary ∂V.

In what follows, we shall first derive a line integral expression for the Green’s function gJP following the framework of Wang and Achenbach 1995 [38], and then evaluate the surface integral (12) analytically. GREEN’S FUNCTIONS The Green’s functions gPM(x) is the solution of the singular partial differential equation

(13)

where ΓJP is the differential operator

(14)

and δ JM is the Kronecker delta. Using the identity of delta function δ(x):

(15)

it is easily verified that Eq. (13) is satisfied by

(16)

where

(17)

Let (e1, e2, n) be a set of unit orthogonal base vectors (for now, we consider n as a constant unit vector, it will be used

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later as the outward normal to the boundary ∂V). We express the unit sphere |d| = 1 by

(18)

and change integral (16) to

(19)

In deriving the above equation, symmetry with respect to θ and φ has been used. A subsequent substitution of leads to

(20)

where

(21)

Since [13]

(22)

where indicates the principle part,

(23)

Hence, we have

(24)

The positive definiteness of cijkl and κ ik guarantees that the system of partial differential equations (13) is strictly elliptical, and that roots of D = 0 cannot be real-valued [7, 8]. That together with the fact that D(s) is a polynomial with real coefficients permit us to write

(25)

where di(θ ) are the real-valued coefficients, (θ ) are roots of D = 0 with positive imaginary parts (i.e, > 0), and are complex conjugates of .

Now, based on Cauchy’s residue theorem, integration in Eq. (24) with respect to η yields residues at . The result is

(26)

where

(27)

and

(28)

In the above expression,

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(29)

Following the procedure presented above, we can also obtain

(30)

We notice the integrand of the line integral (26) is given by the product of two terms, and . The first term given by Eq. (28) is a function of the material properties only;

or in other words, independent to location vectors x and y. The second term , is a simple algebraic function. It is in fact a ‘plane wave function’ which is in essence a one-dimensional function. The simple structure of the integrands in Eqs. (26) and (30) are very useful in many applications.

EVALUATION FOR QUANTUM DOTS OF ARBITRARY SHAPES 1. Exchange of integration order We proceed to evaluate the boundary integral (12) using the line integral expressions for the Green’s functions (26) and (30). We consider a QD of arbitrary shape, but we assume the QD surface can be effectively represented or approximated by a number of triangular surfaces. It is therefore sufficient to consider integral (12) over a triangular surface A:

(31)

where n is the outward normal to A. From Eqs. (5) and (31), it is straightforward to derive the QD induced strain field

(32)

and the QD induced electric filed

(33)

Substituting solutions (26) and (30) into Eqs. (31), (32) and (33), and exchanging integration orders, we obtain

(34)

(35)

and

(36)

where

(37)

and

(38)

The above two surface integrals (37, 38) over A can be integrated analytically. As a result, only the line integrals over

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(0, π /2) in Eqs. (34- 36) need to be computed numerically.

2. Geometry and local coordinates of triangular A To carry out the integration of integrals (37- 38), we shall first set up local coordinates for the triangular A. Referring to Figs. 3 and 4, let us number the three nodes of the triangular (x(1), x(2), and x(3)) anti-clock wisely across the outward normal vector n, such that

(39)

where Δ is the area of the triangular A. Let

Figure 3: Geometry of triangular A. Three nodes x(1)x(2)x(3) and the local orthogonal base vectors (e1, e2, n),

where e1 e2 are an arbitrary pair of orthogonal vectors on in the plane of A and n is the unit normal to A

Figure 4: Local coordinate system with the unit vectors and and n, where n = 3 in this figure. The axis and the unit normal n, perpendicular to the element, are not shown

(40)

Fig. 4 illustrates the geometry of q1 and q2. Notice q1 is in the direction of the side line facing to the 3d node x(3). We

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use (q1, q2, n) as the unit orthogonal base vectors for local coordinates . The coordinate transformation is given by

(41)

where the coordinate origin x0 is given by (see Figs. 3 and 4)

(42)

We shall also introduce the following geometrical quantities (see Fig. 4)

(43)

In the local coordinates, the location vector y is expressed by

(44)

where

(45)

Recall that (e1, e2, n) and (q1, q2, n) are both unit orthogonal base vectors. Henceforth, using Eqs. (27), (41) and (44), we get

(46)

where α, the angles of q1 measured positive counter-clockwise relative to e1, is determined by

(47)

ANALYTICAL SOLUTIONS OF φ AND ψ

Substitution of Eq. (46) into integral (37) yields

(48)

where

(49)

and

(50)

In writing expression (48), ξ3 = 0 for x2∈A has been considered. By a straightforward integration of Eq. (48), we obtain

(51)

where

(52)

Similarly, upon substitution of Eq. (46), integral (38) becomes

(53)

Thus,

(54)

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Now, with the analytical solutions given by Eqs. (51) and (54), the only concerns remained are the computation of the three line integrals (34-36), which can be achieved by numerical integration. We observe the integrands of expressions (48) and (53) are regular functions and hence and are regular, when imaginary part of does not vanish. From Eq. (50) we find that only when . Therefore we conclude, when , i.e. when y is not on the plane of the triangular surface A, and are regular functions of β and thus so are the integrands of the line integrals (34-36). Eqs. (51) and (54) together with the three line integral equations (34-36) are the key results of this paper. We observe the integrand in the line integral (34) is given by the products of functions and , where are functions of the material properties only, while are simple algebraic functions of location and geometry. Notice the location and geometry independent term , which is more costly to compute than the simple functions , can be repeatedly used for QDs at different locations. This unique structure of the integrand facilitates the coding procedure and the computation for very large systems.

NUMERICAL RESULTS

The elastic stiffness, the piezoelectric and the dielectric permittivity constants are of the order of 1011 (N/m2), 101 (C/m2) and 10−9(C/(mV)), respectively, and the typical strain and the electric fields are of the order of 10−3

and 107 (V/m), respectively. Let q and q0 represent a dimensional quantity and its reference value, respectively; its normalization is given by . We select the reference values for the stress, strain, electric induction, and the electric fields to be σ0 = 108 (N/m2), γ0 = 10−3, D0 = 10−2 (C/m2), and E0 = 107 (V/m), respectively.

For the verification of the code we have calculated the spherical inclusion problem in an infinite isotropic body. It is known that the strain field inside the ellipsoidal inclusion of constant magnitude is uniform and is given by Eshelby’s S-tensor according to

(55)

Figure 5: (a) Discretization of the unit sphere by 96 triangles, (b) Strain components are plotted

along the x-axis in and outside of the sphere

The values of the S-tensor for the sphere can be found in Mura’s book [25]. Fig. 5(a) shows the discretization of the unit sphere by 96 triangles. The uniform non-dimensional eigen strain components of unit magnitude,

are given inside the sphere. The plots of the strain variation along the x-axis, as shown in Fig. 5(b), are given in Figs. 6 and 7. Note that the strain components inside the unit sphere are constants and perfectly agree with the analytical values given by Eq. (55). Another verification was made with the cuboidal inclusion in an infinite isotropic body. Six components of the uniform non-dimensional eigen strain of the unit magnitude were introduced in the unit cube ( ) shown in Fig. 8. The centroid of the cube coincides with the coordinate origin. Figs. 9(a), 10(a) and 11(a) show the variation of the normalized strain components , and on the xz-plane where . The corresponding plots of the analytical solution [25] are given in Figs. 9(b), 10(b) and 11(b) for

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Figure 6: Variation of strain components ε11, ε22 and ε33 along the x-axis (The analytical values are plotted inside the sphere only)

Figure 7: Variation of strain components ε23, ε31 and ε12 along the x-axis (The analytical values are plotted inside the sphere only)

Figure 8: Cuboidal inclusion with uniform eigenstrain.

comparison. In each figure, the inclusion region is defined by a rectangle, and . Notice that the results obtained by the proposed technique agree perfectly with the analytical results, including those for threeother strain components , and not shown here. Especially notable is the accurate representation of the logarithmic singularity, as predicted by the analytical solution, at the corners of the cuboid as shown in Fig. 10.

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(a) (b)

Figure 9: Distribution of on the xz-plane through the centroid of the cuboidal inclusion in an isotropic solid. (a) Numerical and (b) analytical results.

(a) (b)

Figure 10: Distribution of on the xz-plane through the centroid of the cuboidal inclusion in an isotropic solid. (a) Numerical and (b) analytical results.

(a) (b)

Figure 11: Distribution of on the xz-plane through the centroid of the cuboidal inclusion in an infinite isotropic solid. (a) Numerical and (b) analytical results.

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Figure 12: Pyramidal Quantum Dot with uniform eigenstrain.

In order to demonstrate the capability of the technique developed for the application to the quantum dots, we have selected a pyramidal inclusion of uniform strain distribution as shown in Fig. 12. The unit non-dimensional base length

= 1 and height = 1 are specified for the pyramid. Gallium Arsenite [001] (Cubic) with the material constants given by

(56)

is considered. Note that for the stiffness and piezoelectric constants we replace the pair of suffices ij by a single suffix according to the convention (11 → 1), (22 → 2), (33 → 3), (23 → 4), (31 → 5,) (12 → 6). The unit non-dimensional

eigen strain components = 1 and = 1 are given inside the pyramid.

Figs. 13-21 show the distribution of the nine components of the strain in a vertical plane, parallel to the xz-plane, that cut through the vertex V of the pyramid as shown in Fig. 12. Similar results for the distribution of the nine components on the horizontal plane through the centroid G of the pyramid, as shown in Fig. 12, are plotted in Figs. 22-30.

SUMMARY Characteristics of the self-organized quantum dots (QDs) such as electron and hole energy levels and wave functions are dependent to the state of strain and electric field produced during the growing process of QDs in a semiconductor substrate. The calculation of the strain and electric field is one of the most challenging components in the QDs simulation process. It involves material anisotropy induced coupling between the elastic and electric fields and it must include the full threedimensional and usually intricate shapes of the QDs. Numerical simulations are often performed by finite difference, finite element, or atomistic techniques, all require substantial computational time and memory. In this paper, we present a new Green’s function approach which takes into account QDs of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezoelectricity. Following the literature of micromechanics, the problem is formulated as an Eshelby inclusion problem of which the solution can be expressed by a volume-integral equation that involves the Green’s functions and the equivalent body-force of eiegenstrain. The volume integral is subsequently reduced to a line integral based on exploring a unique structure of the Green’s

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Figure 13: Gallium Arsenite (001): Distribution of on

the vertical plane VPQ of Fig. 12. Figure 14: Gallium Arsenite (001): Distribution of

on the vertical plane VPQ of Fig. 12.

Figure 15: Gallium Arsenite (001): Distribution of on

the vertical plane VPQ of Fig. 12. Figure 16: Gallium Arsenite (001): Distribution of

on the vertical plane VPQ of Fig. 12.

Figure 17: Gallium Arsenite (001): Distribution of

on the vertical plane VPQ of Fig. 12. Figure 18: Gallium Arsenite (001): Distribution of

on the vertical plane VPQ of Fig. 12.

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Figure 19: Gallium Arsenite (001): Distribution of on

the vertical plane VPQ of Fig. 12. Figure 20: Gallium Arsenite (001): Distribution of on

the vertical plane VPQ of Fig. 12.

Figure 21: Gallium Arsenite (001): Distribution of on

the vertical plane VPQ of Fig. 12. Figure 22: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

Figure 23: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

Figure 24: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

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Figure 25: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

Figure 26: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

Figure 27: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

Figure 28: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

Figure 29: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

Figure 30: Gallium Arsenite (001): Distribution of on the horizontal plane through the centroid G of pyramid in Fig. 12

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functions. Eqs. (51) and (54) together with the three line integral equations (34-36) are the key results of this paper. We observe the integrand in the line integral (34) is given by the products of functions and , where

are functions of the material properties only, while are simple algebraic functions of location and geometry. Notice the location and geometry independent term , which is more costly to compute than the simple functions , can be repeatedly used for QDs at different locations. This unique structure of the integrand facilitates the coding procedure and the computation for very large systems. Numerical simulations are presented for pyramidal QDs in the substrates of Gallium Arsenite (GaAs) (001).

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