Download R-107_WangCY.pdf

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
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
2
Email: [email protected], [email protected], [email protected]
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
⎯ 848 ⎯
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
⎯ 849 ⎯
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
. The electric field quantities include the electric potential φ, charge q, displacement vector Dj,
eigenstrain tensor
. We shall adopt the usual compact notations of using capital-letter subscripts
field vector Ej, and eigenfield vector
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.
be the extended eigenstrain of the QD, which has a domain V bounded by its boundary ∂V. The extended stress
Let
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)
⎯ 850 ⎯
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
force is defined in the QD domain V.
(Mura, 1987 [25]; Pan, 2002 [30]). Notice, the equivalent body
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
⎯ 851 ⎯
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,
are complex conjugates of .
(θ ) are roots of D = 0 with positive imaginary parts (i.e,
Now, based on Cauchy’s residue theorem, integration in Eq. (24) with respect to η yields residues at
result is
> 0), and
. The
(26)
where
(27)
and
(28)
In the above expression,
⎯ 852 ⎯
(29)
Following the procedure presented above, we can also obtain
(30)
and
We notice the integrand of the line integral (26) is given by the product of two terms,
. 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
⎯ 853 ⎯
(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
with the unit vectors
and
and n, where
Figure 4: Local coordinate system
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
⎯ 854 ⎯
use (q1, q2, n) as the unit orthogonal base vectors for local coordinates
given by
. The coordinate transformation is
(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)
⎯ 855 ⎯
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
and
are regular, when imaginary part of does not vanish. From
(48) and (53) are regular functions and hence
Eq. (50) we find that
only when
. Therefore we conclude, when
, i.e. when y is not on the
and
are regular functions of β and thus so are the integrands of
plane of the triangular surface A,
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
and
, where
the integrand in the line integral (34) is given by the products of functions
are simple algebraic functions of location and geometry.
are functions of the material properties only, while
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
. We select the reference values for the stress, strain, electric induction, and the electric fields to be
given by
8
2
σ0 = 10 (N/m ), γ 0 = 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
) shown in Fig. 8. The centroid of the cube coincides with the coordinate origin.
the unit cube (
,
and
on the xz-plane
Figs. 9(a), 10(a) and 11(a) show the variation of the normalized strain components
. The corresponding plots of the analytical solution [25] are given in Figs. 9(b), 10(b) and 11(b) for
where
⎯ 856 ⎯
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.
and
. Notice
comparison. In each figure, the inclusion region is defined by a rectangle,
that the results obtained by the proposed technique agree perfectly with the analytical results, including those for
,
and
not shown here. Especially notable is the accurate representation of the
threeother strain components
logarithmic singularity, as predicted by the analytical solution, at the corners of the cuboid as shown in Fig. 10.
⎯ 857 ⎯
(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.
⎯ 858 ⎯
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 are given inside the pyramid.
= 1 and
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
⎯ 859 ⎯
Figure 13: Gallium Arsenite (001): Distribution of
the vertical plane VPQ of Fig. 12.
on
Figure 14: Gallium Arsenite (001): Distribution of
on the vertical plane VPQ of Fig. 12.
Figure 15: Gallium Arsenite (001): Distribution of
the vertical plane VPQ of Fig. 12.
on
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.
⎯ 860 ⎯
on
Figure 19: Gallium Arsenite (001): Distribution of
the vertical plane VPQ of Fig. 12.
on Figure 20: Gallium Arsenite (001): Distribution of
the vertical plane VPQ of Fig. 12.
Figure 21: Gallium Arsenite (001): Distribution of
the vertical plane VPQ of Fig. 12.
on
Figure 22: Gallium Arsenite (001): Distribution of
on the
horizontal plane through the centroid G of pyramid in Fig. 12
on the
Figure 23: Gallium Arsenite (001): Distribution of
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
⎯ 861 ⎯
on the
Figure 25: Gallium Arsenite (001): Distribution of
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
on the Figure 28: Gallium Arsenite (001): Distribution of
Figure 27: Gallium Arsenite (001): Distribution of
on the
horizontal plane through the centroid G of pyramid in Fig. 12 horizontal plane through the centroid G of pyramid in Fig. 12
Figure 29: Gallium Arsenite (001): Distribution of
on the Figure 30: Gallium Arsenite (001): Distribution of
on the
horizontal plane through the centroid G of pyramid in Fig. 12 horizontal plane through the centroid G of pyramid in Fig. 12
⎯ 862 ⎯
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
, which is more costly to compute than the
geometry. Notice the location and geometry independent term
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).
REFERENCES
1. Andreev AD, Downes JR, Faux DA, O’Reilly EP. Strain distribution in quantum dots of arbitrary shape. J. Appl.
Phys., 1999; 86: 297-305.
2. Benabbas T, Francois P, Androussi Y, Lefebvre A. Stress relaxation in highly strained InAs/GaAs structures as
studied by finite element anlaysis and transmission electron microscopy. J. Appl. Phys., 1996; 80: 2763-2767.
3. Bimberg D, Grundmann M, Ledentsov NN. Quantum Dot Heterostructures. John Wiley & Sons Ltd, New York,
USA, 1998.
4. Constantin C, Martinet E, Leifer K, Rudra A, Lelarge F, Kapon E, Gayral B, Gerard JM. Strained V-groove
quantum wires in multidimensional microcavities. Materials Science and Engineering B, 2000; 74: 158-164.
5. Davies JH. Elastic and piezoelectric fields around a buried quantum dot: A simple picture. J. Appl. Phys., 1998;
84: 1358-1365.
6. Daruka I, Barabasi AL, Zhou SJ, Germann TC, Lomdahl PS, Bishop AR. Molecular-dynamics investigation of
the surface stress distribution in a Ge/Si quantum dot superlattice. Phys. Rev. B, 1999; 60: R2150-R2153.
7. Deeg WF. The Analysis of Dislocation, Crack and Inclusion Problems in Piezoelectric Solids. PhD Thesis,
Stanford University, 1980.
8. Dunn ML. Electroelastic Green’s functions for transversely isotropic piezoelectric media and their application to
the solution of inclusion and inhomogeneity problems. Int. J. Eng. Sci., 1994; 32: 119-131.
9. Eshelby JD. Elastic inclusions and inhomogeneities. in Sneddon IN, Hill R. eds. Progress in Solid Mechanics, 2,
North-Holland, Amsterdam, Holand, 1961, pp. 89-140.
10. Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc.
Lond., 1957; A241: 376-396.
11. Faux DA, Pearson GS. Green’s tensors for anisotropic elasticity: application to quantum dots. Phys. Rev. B,
2000; 62: R4798-R4801.
12. Fu Y, Zhao QX, Ferdos F, Sadeghi M, Wang SM, Larsson A. Strain and optical transition in InAs quantum dots
on (001) GaAs. Superlattices and Microstructures, 2001; 30: 205-213.
13. Gel’fand IM, Graev MI, Vilenkin NY. Generalized Functions. Vol. 5, Academic Press, New York, USA, 1966.
14. Grundmann M, Stier O, Bimberg D. Symmetry breaking in pseudomorphic V-groove quantum wires. Physical
Review B, 1994; 50: 14187-14192.
15. Grundmann M, Stier O, Bimberg D. InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons,
and electronic structure. Physical Review B, 1995; 52: 11969-11981.
16. Holy V, Springholz G, Pinczolits M, Bauer G. Strain induced vertical and lateral correlations in quantum dot
superlattices. Phys. Rev. lett., 1999; 83: 356-359.
17. Jogai B, Albrecht JD, Pan E. The effect of electromechanical coupling on the strain in AlGaN/GaN
heterojunction field effect transistors. Journal of Applied Physics, 2003; 94: 3984-3989.
18. Kikuchi Y, Sugii H, Shintani K. Strain profiles in pyramidal quantum dots by means of atomistic simulation. J.
Appl. Phys., 2001; 89: 1191-1196.
19. Kogan L, Hui CY, Molkov V. Stress and induction field of a spheroidal inclusion or a penny-shaped crack in a
transversely isotropic piezoelectric material. Int. J. Solids Struct., 1996; 33: 2719-2737.
20. Kret S, Benabbas T, Delamarre C, Androussi Y, Dubon A, Laval JY, Lefebvre A. High reolution electron
microscope analysis of lattice distortions and In segregation in highly strained In0.35Ga0.65As coherent islands
grown on GaAs (001). J. Appl. Phys., 1999; 86: 1988-1993.
⎯ 863 ⎯
21. Lelarge F, Constantin C, Leifer K, Condo A, Lakovlev V, Martinet E, Rudra A, Kapon E. Effect of indium
segregation on optical properties of V-groove InGaAs/GaAs strained quantum wires. Applied Physics Letters,
1999; 75: 3300-3302.
22. Lin G. Studies on Semiconductor Quantum Structure Lasers. Doctoral Dissertation, National Chiao Tung
University, Taiwan, China, 2001.
23. Martinet E, Dupertuis MA, Reinhardt F, Biasiol G, Kapon E, Stier O, Grundmann M, Bimberg D. Separation of
strain and quantum-confinement effects in the optical spectra of quantum wires. Physical Review B, 2000; 61:
4488-4491.
24. Medeiros-Ribeiro G. Epitaxial growth of strained nanocrystals. Physica Status Solidi, 2002; 230: 443-450.
25. Mura T. Micromechanics of Defects in Solids. Second, Revised Edition, Kluwer Academic Publishers, 1987.
26. Notomi M, Hammersberg J, Weman H, Nojima S, Sugiura H, Okamoto M, Tamamura T, Potemski M.
Dimensionality effects on strain and quantum confinement in lattice-mismatched InAsxP1-x/InP quantum wires.
Physical Review B, 1995; 52: 11147-11158.
27. Nozaki H, Taya M. Elastic fields in a polyhedral inclusion with uniform eigenstrains and related problems. J.
Appl. Mech., 2001; 68, 441-452.
28. Pan E, Yang B. Elastostatic fields in an anisotropic substrate due to a buried quantum dot. Journal of Applied
Physics, 2001; 90: 6190-6196.
29. Pan E. Elastic and piezoelectric fields around a quantum dot: Fully coupled or semi-coupled model? Journal of
Applied Physics, 2002; 91: 3785-3796.
30. Pan E. Elastic and piezoelectric fields in substrates GaAs (001) and GaAs (111) due to a buried quantum dot.
Journal of Applied Physics, 2002; 91: 6379-6387.
31. Pan E, Yang B. Elastic and piezoelectric fields in a substrate AlN due to a buried quantum dot. Journal of
Applied Physics, 2003; 93: 2435-2439.
32. Pearson GS, Faux DA. Analytical solutions for strain in pyramidal quantum dots. J. Appl. Phys., 2000; 88:
730-736.
33. Pryor C, Kim J, Wang LW, Williamson AJ, Zunger A. Comparison of two methods for describing the strain
profiles in quantum dots. J. Appl. Phys., 1998; 83: 2548-2554.
34. Rahman M. On the Newtonian potentials of heterogeneous ellipsoids and elliptical discs. Proc. R. Soc. Lond. A,
2001; 457: 2227-2250.
35. Romanov AE, Beltz GE, Fischer WT, Petroff PM, Speck JS. Elastic fields of quantum dots in subsurface layers.
J. Appl. Phys., 2001; 89: 4523-4531.
36. Sharma P, Sharma R. On the Eshelby’s inclusion problem for ellipsoids with nonuniform dilatational Gaussian
and exponential eigenstrains. J. Appl. Mech., 2003; 70: 418-425.
37. Shim HW, Choi RJ, Jeong SM, Vinh LV, Hong CH, Suh EK, Lee HJ, Kim YW, Hwang YG. Influence of the
quantum-well shape on the light emission characteristics of InGaN/GaN quantum-well structures and
light-emitting diodes. Applied Physics Letters, 2002; 81: 3552-3554.
38. Wang CY, Achenbach JD. Three-dimensional time-harmonic elastodynamic Green’s functions for anisotropic
solids. Proc. R. Soc. Lond. A, 1995; 449: 441-458.
39. Wang CY. Stress fields produced by dislocations in anisotropic solids. J. Mech. Phys. Solids, 1996; 44(3):
293-305.
40. Wang CY. General formalism for piezoelectric crystals of general anisotropy. Appl. Math. Letters, 1996; 9(4):
1-7.
41. Wang CY. Elastic fields produced by a point source in solids of general anisotropy. J. Engng. Math., 1997; 32(1) :
4152.
42. Yang B, Pan E. Elastic analysis of an inhomogeneous quantum dot in multilayered semiconductors using a
boundary element method. Journal of Applied Physics, 2002; 92: 3084-3088.
43. Yang B, Pan E. Elastic fields of quantum dots in multilayered semiconductors: A novel Green’s function
approach. ASME Journal of Applied Mechanics, 2003; 70: 161-168.
⎯ 864 ⎯