In this article, the analysis of the strained semiconductor nanostructures, self-assembled quantum dots (SAQDs), are divided into three parts. First, a linear elastic finite element calculation is performed to determine the strain distribution in and around SAQD nanostructures. Second, the carrier confinement potential in SAQD is modified by the strain distribution.
The deformation potential theory is used to estimate the strain-induced potential superimposed to the energy bands of SAQD. Third, a steady-state three- dimensional Schrödinger equation is solved numerically to obtain the energy levels and wave function spectra in the SAQD. Finally, the energy of interband optical transitions is obtained in the numerical experiments.
2.1 Continuum Model
The strain fields inside and in the neighborhood of SAQD strongly affect the electronic properties, and hence the optoelectronic properties, in vicinity of the dots [9,10]. For the optoelectronic properties in III-V semiconductors (such as the materials are InAs and GaAs), there are two predominated strain effects, namely, changes of the conduction band level and the valence band level. The conduction band is only affected by the hydrostatic strain, often referred to as the dilatation or trace of the strain tensor. The valence level can change both with hydrostatic and biaxial strains. In addition, for zinc blende material structures, deviatoric strains give rise to piezoelectrically induced electric fields [11].
To understand the strain effects on optoelectronic properties of QD nanostructures, it is necessary to determine the elastic strain field in the quantum dots and surrounding matrix. There have been various methods in three different main categories: (1) Analytical continuum method [12~14], (2) Continuum finite element method (FEM) [15~18], and (3) Atomistic modeling [19~21]. Analytical continuum method leads to integral expressions for the elastic fields which can be integrated in closed forms for some simplest inclusion shapes, e.g., cylindrical or spherical quantum dots. On the other hand, the interactions between the quantum dot and the surrounding material might have not been fully encountered. FEM is a very versatile and effective numerical method, which have capabilities to easily accommodate various theories and model on QD nanostructures to different levels.
Atomistic models might be more reasonable to model systems in nano-scale provided that accurate interatomic potentials are available. Moreover, it requires a huge computing capacity to model quantum dots and the surrounding matrix. In the following sections, we first brief analytical continuum method and continuum finite element method for the QD problems, respectively. Then continuum finite element method will be used to study the SAQD nanostructures for the optoelectronic properties.
2.1.1 Analytical Continuum Method
Analytical continuum methods for calculating the strain and stress distributions attributed to a uniformly strained inclusion were originally developed by Eshelby [22]. From viewpoint of Eshelby’s result, a quantum dot can be regarded as an inclusion embedded in an infinite matrix. Due to different lattice constants of the quantum dot and the matrix, the inclusion can be considered as initially strained to accommodate the mismatch in lattice constants. For general cases, having closed forms expression on analytical solutions of stress and strain distributions are not possible, except for some simple geometric shapes such as cubic, cylindrical, spherical, etc. Analytical solutions of the strain field for a buried pyramidal QD structure, including a truncated pyramidal quantum dot and a surrounding matrix, were presented by Pearson [23].
The problem of an initially uniformly strained QD of arbitrary shape embedded in an infinite isotropic medium can be solved analytically by theory of inclusion, under the simplification that both the QD and the matrix have the same elastic moduli. Based on previous works [24,25], a set of vectors has been chosen such that the divergence of each vector gave the Green’s function for the stress component σij. First, six vectors are defined as follows,
6 Bulletin of the College of Engineering, N.T.U., No. 91, June 2004 modulus, respectively, of the QD and matrix. The misfit strain, ε0, of the quantum dot is taken with respect to the surrounding matrix; and the quantum dot is assumed to be initially strained by ε0 in all three directions, namely x-, y-, z-directions. This initial strain is taken as negative for dots, since it is under compression.
The vectors in Eq. (1) are chosen so that ∇⋅A leads to the Green’s function stress components σsph corresponding to a spherical point inclusion. Moreover, the above expressions for A is not unique because other choices of vectors A can also yield the Green’s function stress components σsph when taking divergence. For an arbitrary QD shape, the resultant integral to determine the stress σ at the point (x, y, z) is repeated index has been assumed through this article.
Therefore, the solution for a spherical inclusion can be used as a Green’s function for the three-dimensional problem. This Green’s function was then integrated over the volume of the QD which gives rise to the stress field of the QD.
The analytical stress field of a pyramid QD and a truncated pyramid QD, as illustrated in Fig. 2, can be obtained by evaluating Eq. (2) and expressing in closed form, especially with the aid of any available algebraic software package. The volume of a rectangular-base,
Fig. 2 The geometries of the pyramid and truncated pyramid QD
truncated pyramid QD is defined by,
0 0 absence of truncation and f represents the degree of truncation,0≤ f ≤1. Once each stress component σij
has been calculated, substituting σij into the Hooke’s law, of isotropic elastic solids, gives the strain fields:
εij 1 [ (1 ) σij δij σ ]kk
E ν ν
= + − (4)
Equations (2) and (4) enable us to calculate the stress and the strain fields for a single uniformly strained pyramid QD of arbitrary truncation embedded in an infinite medium. For systems containing more than one QD, the principle of superposition of linear problems allows the stresses (strains) at a point to be determined by simply summing the stresses (strain) field of each single QD in the arrays.
Notice that in this analysis, both quantum dots and matrix are assumed as isotropic elastic solids with the same elastic moduli, though it is far away from the reality. Moreover, the QD is assumed to be deformed initially only without any further elastic deformation.
The full interaction between QD and the matrix may have not been taken into account properly. Besides, the matrix is assumed to be of unbounded extent; hence effects of free surfaces have not yet been included in this formulation.
2.1.2 Continuum Finite Element Method
Another important category of methods on (0,0,0)
Lin．Kuo．Liao．Hung：Mechanical and Optical Properties of InAs/GaAs self-Assembled Quantum Dots 7 computing the stress and strain fields induced by QD is
the continuum finite element method (FEM). FEM gives only the numerical results for quantum dots. On the other hand, it works even for rather complex geometries, and provides a complete picture of the strain distribution of a three-dimensional model. In addition, assumptions of isotropic as well as same elastic moduli as in previous methods are not needed here. Furthermore, with the improvement of computer technology, it becomes much easier even to handle heavy tasks, especially 3D (three dimension) problems with FEM. Benabbas [26] and Muralidharan [16] gave a 2D (two dimension) axisymmetric model to analyze the strain distribution inside and in the neighborhood of QD. A 3D model for strain field was presented in [27].
Notice that the mismatch of lattice constants will induce further elastic deformation in the whole nanostructure system, namely, in both the substrate and the quantum dot island. In other words, the lattice mismatch will not be the only strain in the quantum dot island. In these works, the lattice mismatch between dot and substrate is simulated under the framework of the theory of thermo-elasticity. For these calculations, both QD and substrate are assigned to have corresponding hypothetical coefficients of thermal expansion, in order to produce a desired misfit across the interface matching the initial strain from lattice-mismatched. For example, growing InAs on (001) GaAs substrate by molecular beam epitaxy (MBE) results in pyramid-like islands (dots) [28~30]. The lattice mismatch between InAs and GaAs is about 6.7%. The initial strain field induced by lattice mismatch can then be simulated by setting the hypothetical coefficient of thermal expansion of the DQ and the substrate to be as 0.067°K−1and 0, respectively, with a raised temperature by 1°K.
In the numerical experiments of this article, a buried pyramidal InAs/GaAs SAQD nanostructure, as depicted schematically in Fig. 3, is considered. The InAs dot is self-assembled under certain conditions during heteroepitaxy on (001) GaAs substrates on which a thin wetting layer is grown first, followed by coherent island formation. The dot is subsequently covered by additional substrate materials. Epitaxially grown semiconductor heterostructures, such as SAQDs, often consist of several materials with different lattice constants. The mismatch of lattice constants gives rise to the strain field in a QD nanostructure, which will then affect the optoelectronic properties of the quantum dots. The model based on the linear elasticity theory will be developed to evaluate the strain distribution in and around the InAs/GaAs SAQD nanostructure.
Fig. 3 Schematics of (a) the buried InAs/GaAs QD nanostructure
：B = 30nm, H = 30nm, d = 0.5nm, and (b) the island (InAs dot) b = pyramid width, h = pyramid height
In-plane (the base plane of QD) lattice mismatch parameter is usually defined as0 0 material and the QD, respectively. Notice that the mismatch of lattice constants will induce further elastic deformation in the whole nanostructure system, namely, in both the substrate and the QD island. In other words, the lattice mismatch parameter defined in Eq. (5) will not be the only strain in the quantum dot island, it will instead be treated as in-plane initial strain in the latter finite element analysis. On the other hand, the out-of-plane initial strain has never yet been as conclusive as its in-plane counterparts. In this article, the out-of-plane initial strain is set to zero such that the initial strain tensor is given as
0
Based on the theory of linear elasticity, the stress- strain relation of anisotropic solids with initial strains can be written as:
is the initial strain tensor defined in Eq. (6). Here both GaAs and InAs are treated as cubic materials with 3 independent elastic moduli each. A list of material properties used, including lattice constants and elastic moduli, is summarized in Table 1.
H
8 Bulletin of the College of Engineering, N.T.U., No. 91, June 2004
Table 1 Material properties
Mechanical property a0(nm) C11(GPa) C12(GPa) C44(GPa)
The linear elasticity boundary value problem, arising from the mismatch in lattice constants between the wetting layer, QD island and substrate materials, is analyzed, here, using a commercial finite element package FEMLAB. Appropriate boundary conditions are needed in the analysis: all the nodes of the outer surfaces parallel to the
x
- andy
-outer surfaces are fixed in the normal displacement due to periodic symmetric argument. The bottom outer surface is fixed against displacement inz
-direction to avoid rigid body shift;whiles the upper surface is kept traction free. The displacement compatibility across the interface of island/ substrate is satisfied automatically in the finite element formulation with displacement field as unknowns.
It is worthwhile to point out that the wetting layer and QD island grew on the substrate and embedded within the capping layer, as depicted in Fig. 3. In the conventional analysis, the capping layer is assumed to be deformed together with QD and substrate. On the other hand, in fact, in the fabrication process, the capping layer is deposited after the QD has been deformed to compensate the mismatch of lattice constants. Hence the capping layer should have never been experienced the deformation of QD at all, in fabrication process. To take this sequence of fabrication process into account, in this article, the capping layer is added into the nanostructure system only after the strain analysis of quantum dot and substrate has been completed, and is assumed to be unstrained thereafter. After the strain field has been calculated, the capping layer is added when solving the Schrödinger equation. This is equivalent to assume that the strain distribution in the QD does not to affect the capping layer.
It is also of interest to point out that QDs are grown in a large array instead of a single dot actually.
In a finite element model, it is easy to consider a single dot or a regular array of dots. However, the strain distribution within a disordered array of dots will still
be impractical, if not impossible, to be obtained by using finite element methods. FEM is rather computationally intense compared to analytic solutions.
To obtain results to certain accuracy, the grid size must be kept small enough; however, as the grid size is reduced, the demand on computational memory and time increases rapidly. The choice of the grid size depends on the accuracy required and computer capacity and time available.
2.2 Quantum Model
The band structure is determined by considering the difference in band gap energies of the heterostructure constituent materials. A part of band gap difference is taken up by a conduction band offset, and a portion results in a valence band offset, as shown in Fig. 4, for the cases that QD and substrate are taken as InAs and GaAs, respectively.
The strain effects will induce an extra potential field,
V
strain. For the strained QD nanostructure, the confinement potential can be written as a sum of energy offsets of the conduction band (or valence band),V
band, and the strain-induced potential,V
strain, asband strain
( ) ( ) ( )
V r =V r +V r (8)
The strain contribution to the potential is determined via deformation potential theory, modified by the strain tensor εij. It may then affect the energy band structure for the QD, altering further optoelectronic properties in the devices.
Here, Pikus-Bir Hamiltonian [9,10] together with the computed strain field from above-mentioned strain field calculations (section 2.1.2) are used to determine strain-induced potentials in QD. It is known that the conduction band level will be only affected by the hydrostatic strain [11], and the strain-induced potential for the conduction band can be expressed as [9]
strainc c(εxx εyy ε )zz
V =a + + (9)
Lin．Kuo．Liao．Hung：Mechanical and Optical Properties of InAs/GaAs self-Assembled Quantum Dots 9 On the other hand, the valence level can be
changed with not only hydrostatic strain but also biaxial strain. The strain-induced potential for the valence band can then be written as [10]
strain (ε ε ε ) (ε ε 2ε )
2
v v xx yy zz b xx yy zz
V =a + + − + − (10)
where ac is the deformation potential constant of conduction band; while av and b denote the deformation potential constants of valence band. The data of these three constants for InAs and GaAs are summarized in Table 1. The strain-induced potentials are computed according to Eqs. (9) and (10) and are superimposed to energy band. In our cases, all other potential energy contributions, such as piezoelectric potential energy, are expected to be small and are neglected in the calculations.
The behavior of individual carrier in QD nanostructures is governed by the three-dimensional steady state Schrödinger equation as:
2
Equation (11) is solved numerically again, in this article, by means of a commercial finite element method package, FEMLAB, to obtain energy levels and the wave functions in a QD nanostructure. The boundary conditions on the quantum dot/substrate interfaces are regions, respectively, n denotes the normal directions of interfaces, and a carrier effective mass is taken along the normal directions of interfaces. In addition, at the top of the capping layer, bottom of the substrate, and their outer surfaces, the wave function is set to zero.
Based on the computed energy and wave function distributions, the energy of interband optical transitions can readily be obtained. The optical conductivity features peaks at particular wavelengths of light that are more strongly absorbed; these wavelengths in turn can be expected to be the strongest emission wavelengths.
2.3 Numerical Results
The geometry of an embedded InAs/GaAs QD nanostructureconsideredhereisdepictedinFig.4;the
0.42eV
Fig. 4 Initial energy band of the unstrained InAs/GaAs heterostructure used for quantum dot calculations
Table 2 Dimensions of the QD nanostructures
Structure I II III IV V width b (nm) 12 12 12 16 20 height h (nm) 3 4 6 6 6 InAs QD is assumed to be as pyramidal shape with square-based of width b. Sizes of QD used in the numericalexperiments of the article are listed in Table 2. The semiconductor materials are treated as anisotropy during the strain analysis. As mentioned in section 2.1.2, since the capping layer does not experience the deformation of QD in the self-assembled process; in the stage of strain analysis, the QD heterostructure system is considered to consist of only quantum dot, wetting layer and substrate but not capping material. Once the strain analyses have been completed, the capping layer is then added into the system. The adding of capping material is only for the purpose of solving the Schrödinger equation to obtain the electronic structure of the nanostructure system.In this model, the capping layer is consequently assumed to be unstrained. The model takes the sequence of fabrication process into account. It is equivalently to neglect the re-distribution of strain in
Fig. 5 Strain components
εxxplotted along the
z- axis for structure I, II, and III
0 20 z(nm) 40 60
10 Bulletin of the College of Engineering, N.T.U., No. 91, June 2004
the QD system during the depositing process of the capping layer, which is expected to be much small compared to the strain distribution in the QD and in the substrate material.
Strain components εxx and εzz, are shown in Figs. 5 and 6, respectively, along z-axis in QD structures for the cases of width b = 12nm with various heights (h = 3, 4, and 6nm). These strain components show ample discontinuities across the interface of InAs wetting layer and the GaAs substrate (i.e. InAs/GaAs interface), as one would expect. Moreover, the strain field εxx
inside the QD is compressive while εzz is tensile.
These reflect the fact that the lattice constant of InAs is larger than that of GaAs. Similarly, strain components εxx and εzz, are shown in Figs. 7 and 8, respectively, along z-axis in QD structures for the cases of height h = 6nm with various widths (b = 12, 16, and
axis for structure III, IV, and V
fields are larger for larger QD size, as one would expect, since a larger QD can be viewed as a larger source of initial-strain field.
The calculated ground state energy level and corresponding transition energy in the QD nanostructure are summarized in Table 3. The transition energy is defined as the energy difference between the electron and the hole. The peak of experimental photoluminescence (PL) spectra is related to the transition energy. From the numerical results in Table 3, it is seen that the energy of interband optical transition becomes smaller as the island size increasing.
This phenomenon agrees well with the previous experimental photoluminescence data [31].
Probability density function profiles, given by square of wave functions, namely |ψ|2, for the ground state energy levels in the QD nanostructures, are shown in Fig. 9. Subfigures 9(a) and 9(b) are the probability density function profiles corresponding to electron and hole, respectively, energy levels for the structure I (h = 3nm). Similarly, subfigures 9(c) and 9(d) are for the case of the structure III, namely h = 6nm. It is easily to see that, from Fig. 9, the probability density function distributions are confined almost entirely to the island region. Moreover, the probability density function distributions of electrons appear to be spherical symmetry within the whole island, subfigures 9(a) and 9(c). On the other hand, the probability density function distributions of hole are likely to be more confined in the bottom of the island, subfigures 9(b) and 9(d).
Similarly, Fig. 10 shows the probability density function profiles of the first excited energy levels of the
Similarly, Fig. 10 shows the probability density function profiles of the first excited energy levels of the