Analytical approach for strain and piezoelectric potential in conical self-assembled quantum dots (original) (raw)
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Computational Methods for Electromechanical Fields in Self-Assembled Quantum Dots
Communications in Computational Physics, 2012
A detailed comparison of continuum and valence force field strain calculations in quantum-dot structures is presented with particular emphasis to boundary conditions, their implementation in the finite-element method, and associated implications for electronic states. The first part of this work provides the equation framework for the elastic continuum model including piezoelectric effects in crystal structures as well as detailing the Keating model equations used in the atomistic valence force field calculations. Given the variety of possible structure shapes, a choice of pyramidal, spherical and cubic-dot shapes is made having in mind their pronounced shape differences and practical relevance. In this part boundary conditions are also considered; in particular the relevance of imposing different types of boundary conditions is highlighted and discussed. In the final part, quantum dots with inhomogeneous indium concentration profiles are studied in order to highlight the importance of taking into account the exact In concentration profile for real quantum dots. The influence of strain, electric-field distributions, and material inhomogeneity of spherical quantum dots on electronic wavefunctions is briefly discussed.
Two-step strain analysis of self-assembled InAs/GaAs quantum dots
Semiconductor Science and Technology, 2006
Strain effects on optical properties of self-assembled InAs/GaAs quantum dots grown by epitaxy are investigated. Since a capping layer is added after the self-assembly process of the quantum dots, it might be reasonable to assume that the capping layer neither experiences nor affects the induced deformation of quantum dots during the self-assembly process. A new two-step model is proposed to analyse the three-dimensional induced strain fields of quantum dots. The model is based on the theory of linear elasticity and takes into account the sequence of the fabrication process of quantum dots. In the first step, the heterostructure system of quantum dots without the capping layer is considered. The mismatch of lattice constants between the wetting layer and the substrate is the driving source for the induced elastic strain. The strain field obtained in the first step is then treated as an initial strain for the whole heterostructure system, with the capping layer, in the second step. The strain from the two-step analysis is then incorporated into a steady-state effective-mass Schrödinger equation. The energy levels as well as the wavefunctions of both the electron and the hole are calculated. The numerical results show that the strain field from this new two-step model is significantly different from models where the sequence of the fabrication process is completely omitted. The calculated optical wavelength from this new model agrees well with previous experimental photoluminescence data from other studies. It seems reasonable to conclude that the proposed two-step strain analysis is crucial for future optical analysis and applications.
Elastic fields and physical properties of surface quantum dots
Physics of the Solid State, 2011
Elastic fields in a system consisting of a surface coherent axisymmetric quantum dot island on a massive substrate have been theoretically studied using the finite element method. An analysis of the influ ence of the quantum dot shape (form factor) and relative size (aspect ratio) δ on the accompanying elastic fields has revealed two critical quantum dot dimensions, δ c1 and δ c2 . For δ > δ c1 , the fields are independent of the quantum dot shape and aspect ratio. At δ ≥ δ c2 , the quantum dot top remains almost undistorted. Variation of the stress tensor component σ zz (z is the quantum dot axis of symmetry) reveals a region of tensile stresses, which is located in the substrate under the quantum dot at a particular distance from the interface. Using an approximate analytical formula for the radial component of displacements, model electron microscopy images have been calculated for quantum dot islands with δ > δ c1 in the InSb/InAs system. The possibility of stress relaxation occurring in the system via the formation of a prismatic interstitial dislocation loop has been considered.
Elastic fields of quantum dots in subsurface layers
Journal of Applied Physics, 2001
In this work, models based on conventional small-strain elasticity theory are developed to evaluate the stress fields in the vicinity of a quantum dot or an ordered array of quantum dots. The models are based on three different approaches for solving the elastic boundary value problem of a misfitting inclusion embedded in a semi-infinite space. The first method treats the quantum dot as a point source of dilatation. In the second approach we approximate the dot as a misfitting oblate spheroid, for which exact analytic solutions are available. Finally, the finite element method is used to study complex, but realistic, quantum dot configurations such as cuboids and truncated pyramids. We evaluate these three levels of approximation by comparing the hydrostatic stress component near a single dot and an ordered array of dots in the presence of a free surface, and find very good agreement except in the immediate vicinity of an individual quantum dot.
The existence of enormous strain fields in self-assembled quantum dots has led to the expectation of dramatic effects of piezoelectricity. However, only linear piezoelectric tensors were used in all previous calculations. We calculate the piezoelectric properties of self-assembled quantum dots using the linear and quadratic piezoelectric tensors derived from first-principles density functional theory. We find that the previously ignored quadratic term has similar magnitude as the linear term and the two terms tend to cancel each other. We show the effect of piezoelectricity on electron and hole energy levels and wave functions as well as on correlated absorption spectra. The engineering of stain-induced self-assembled semiconductor quantum dots relies on a mismatch a = a − a 0 between the lattice constant of the dot material a and the lattice constant of the substrate a 0 on which the dots are grown. 1 The ensuing strain inside the quantum dots can be significantly larger than in ordinary i.e., flat, parallel interfaces semiconductor heterostructures because in the latter case large strains must be avoided to prevent dislocations a / a 2% whereas in nonflat geometries of InAs quantum dots grown on GaAs even a / a 7% can be tolerated. In zinc-blende quantum well superlattices having parallel interfaces the piezoelectric field appeares only if grown on specific sub-strate directions, strongest for 111, while it vanishes by symmetry for 100. However, in three-dimensional lens-shaped quantum dots even the conventional 100 growth direction leads as a result of curvature to all types of strains diagonal and off diagonal, which give rises to piezoelectric behavior. Indeed, large piezoelectric effects were suggested for 100-grown quantum dots a decade ago. 2 Since then, a number of electronic structure calculations of quantum dots using k · p Refs. 2-5 and atomistic methods 6 have been performed taking the hitherto known linear piezoelectric effect into account and demonstrating important electronic consequences of piezoelectricity such as splitting of P levels, rotation of wave function lobes, and a strong decay of fine-structure splitting with increasing gaps. 7 Recently, 8 we calculated the piezoelectric tensors of GaAs and InAs from first principles and found surprisingly large nonlinear components. 8 Applications to conventional quantum wells revealed a piezoelectric field with very strong contributions from the second-order piezoelectric terms that were previously neglected. In this paper we study the effect of nonlin-earities of the piezoelectric effect on electronic and optical properties of quantum dots and find that the quadratic and linear piezoelectric effects tend to oppose each other. In fact, neglecting the piezoelectric effect is a better approximation than using only the linear term. We present a simple procedure for accurately incorporating both linear and nonlinear piezoelectric effects in all non-self-consistent calculations e.g., effective mass, k · p or tight binding. A self-consistent calculation of the electronic structure of a deformed solid naturally includes the field generated by piezoelectric displacements. However, such a calculation requires the inclusion of all occupied energy levels and their response to strain and is thus limited to small systems. When considering large 10 3 atoms nanostructures, it is often impractical to compute all occupied levels, and one prefers to concentrate on only a few 100 states in the physical range of interest. By necessity, in such cases effective mass, k · p, or any few-band calculation the calculation is no longer self-consistent, and piezoelectricity does not arises naturally, but has to be added as an external potential V piezo. In the pseudopotential representation the total potential used for the single-particle Schrödinger equation is V tot r = V PS r + V piezo r where V PS is the superposition of 2 10 6 screened atomic potentials, including spin-orbit interaction. The calculation of the piezoelectric potential V piezo is performed in four steps. In the first step we calculate the linear and nonlinear piezoelectric coefficients e ˜ and B ˜ of strained bulk GaAs and InAs Ref. 9 using linear response density functional theory. This was done in Ref. 8. In the second step, the polarization is calculated to second order in strain as p = j e ˜ j 0 j + 1 2 jk B ˜ jk j k , 1 where is the strain in Voigt notation, and e ˜ and B ˜ are the piezoelectric tensors. In the third step, the piezoelectric density is calculated from the divergence of the polarization: piezo r = − e a 0 2 · p; 2 and in the final step, the potential V piezo r is obtained from the solution of the Poisson equation: piezo r = 0 · s r V piezo r. 3 More details of the calculations are given in Ref. 8. With the piezoelectric potential added to V PS , the ensuing Schrödinger equation is solved within a basis constructed from a linear combination of strained bulk bands LCBB PHYSICAL REVIEW B 74, 081305R 2006
Strain effects on pyramidal InAs/GaAs quantum dot
Strain distribution in a pyramidal InAs/GaAs quantum dot is investigated. The strain field induced by mismatch of lattice constants in heterostructures is analyzed based on theories of linear elasticity and of thermal stress. The strain-induced potential is then incorporated in the steady state Schrödinger equation. Both the strain field and the solution of the steady state Schrödinger equation are found numerically with the aid of a finite element package-FEMLAB. Eigenenergy and the probability density function of conduction band of quantum dot are calculated. Results from two different models, namely anisotropic material model and isotropic material simplification, during the stage of strain analysis are also compared. Numerical results show eigenenergy and the degeneracy of low eigenenergy are affected by strains. On the other hand, the differences between anisotropic and isotropic materials are not large. Therefore, it is suitable to treat InAs/GaAs quantum dot as isotropic materials.
Piezoelectric models for semiconductor quantum dots
Microelectronics Journal, 2008
The importance of fully coupled and semi-coupled piezoelectric models for quantum dots are compared. Differences in the strain of around 30% and in the electron energies of up to 30 meV were found possible for GaN/AlN dots. r
A finite element study of the stress and strain fields of InAs quantum dots embedded in GaAs
Semiconductor Science and Technology, 2002
We report on a stress and strain analysis, using the finite element method, of the heterosystem of InAs quantum dots embedded in GaAs. The methodology of using the finite element method to simulate the lattice mismatch is discussed and a three-dimensional (3D) model of the heterostructure shows the 3D stress distribution in the InAs islands embedded in a matrix of GaAs substrate and cap layer. The initial shape of the InAs islands is pyramidal. The stress and strain distribution calculated corresponds well with the strain induced by the lattice mismatch. Factors such as the height of the spacer layer and the height of the island are found to play an important role in the stress and strain distribution. With the island having the shape of a truncated pyramid, the stress and strain distribution deviates from that of a full pyramidal island showing the effects that a change of shape in the islands has on the stress field. The stress distribution contributes to the driving force for the mechanism of surface diffusion in molecular beam epitaxy. The effects of anisotropy on the strain distribution are also studied.
Effect of wetting layers on the strain and electronic structure of InAs self-assembled quantum dots
Physical Review B, 2004
The effect of wetting layers on the strain and electronic structure of InAs self-assembled quantum dots grown on GaAs is investigated with an atomistic valence-force-field model and an empirical tight-binding model. By comparing a dot with and without a wetting layer, we find that the inclusion of the wetting layer weakens the strain inside the dot by only 1% relative change, while it reduces the energy gap between a confined electron and hole level by as much as 10%. The small change in the strain distribution indicates that strain relaxes only little through the thin wetting layer. The large reduction of the energy gap is attributed to the increase of the confining-potential width rather than the change of the potential height. First-order perturbation calculations or, alternatively, the addition of an InAs disk below the quantum dot confirm this conclusion. The effect of the wetting layer on the wave function is qualitatively different for the weakly confined electron state and the strongly confined hole state. The electron wave function shifts from the buffer to the wetting layer, while the hole shifts from the dot to the wetting layer.