Strain analysis for the prediction of the preferential nucleation sites of stacked quantum dots by combination of FEM and APT
 Jesús HernándezSaz^{1}Email author,
 Miriam Herrera^{1},
 Sébastien Duguay^{2} and
 Sergio I Molina^{1}
DOI: 10.1186/1556276X8513
© HernándezSaz et al.; licensee Springer. 2013
Received: 24 September 2013
Accepted: 23 November 2013
Published: 5 December 2013
Abstract
The finite elements method (FEM) is a useful tool for the analysis of the strain state of semiconductor heterostructures. It has been used for the prediction of the nucleation sites of stacked quantum dots (QDs), but often using either simulated data of the atom positions or twodimensional experimental data, in such a way that it is difficult to assess the validity of the predictions. In this work, we assess the validity of the FEM method for the prediction of stacked QD nucleation sites using threedimensional experimental data obtained by atom probe tomography (APT). This also allows us to compare the simulation results with the one obtained experimentally. Our analysis demonstrates that FEM and APT constitute a good combination to resolve strain–stress problems of epitaxial semiconductor structures.
Background
In the last decades, semiconductor quantum dots (QDs) have been extensively investigated because they are attractive structures for electronic and optoelectronic advanced devices[1–3]. The characteristics of these QDs can be modified by controlling the growth parameters in order to fulfil the requirements of each device. Often, wellordered and similarsized QDs are required in order to take advantage of their discrete energy levels for intermediate band solar cells[4], lasers[5], and photodetectors[6]. This order can be achieved by stacking several layers of QDs forming a QD matrix or superlattice. During the epitaxial growth, the strain fields of the buried QDs have a large influence in the formation of the subsequent layer as it determines the nucleation sites of the incoming stacked QDs[7, 8]. The complex strain fields around a QD can produce vertical or inclined alignments[9, 10], antialignments[11], or random distributions of the QDs[12], having a strong effect on the optoelectronic behaviour[13].
The simulation of the strain–stress fields in a semiconductor material in order to predict the location of stacked QDs lead to a better understanding of the behaviour of these complex nanostructures. The finite elements method (FEM) is a widespread tool to calculate the strain and stress fields in semiconductor nanostructures, and it has been used in the study of QDs[11, 14, 15], QRings[16], or QWires[17]. In order to obtain reliable predictions by FEM, the simulations should be based in experimental composition data, because of the large impact of the concentration profile of the QD systems in the strain of the structure[18]. However, because of the difficulties in obtaining threedimensional (3D) composition data with atomic resolution, many authors use theoretical compositions[11, 19], or twodimensional (2D) experimental composition data (obtained by electron energy loss spectroscopy[20] or extrapolating composition concentration profiles measured by the lattice fringe analysis technique[21]). This makes a direct correlation between the predictions and the experimental results unfeasible, and prevents from verifying the accuracy of FEM in predicting the nucleation sites of QDs. To solve this, 3D composition data with atomic resolution should be collected. One of the most powerful techniques to obtain 3D composition data is atom probe tomography (APT). APT is an analytical technique that has the unique ability to identify and map out the positions of individual atoms from a nanostructure with an almost 3D atomic resolution, allowing the analysis of different semiconductor structures[22, 23], such as QDs[24] and QRings[25].
In this paper, we have performed a strain analysis using FEM based on APT experimental data of a sample of InAsstacked QDs. We have used the 3D compositional data obtained by APT from a layer of QDs to predict the nucleation site of the next layer of QDs, and we have compared the predictions obtained by FEM with the experimental observations by APT. Our results show that the combination of FEM with APT constitutes a powerful methodology for the analysis of the nucleation sites in stacked semiconductor QDs.
Methods
The sample used to exemplify the study consists of InAs/GaAsstacked QDs covered by a 2nm In_{0.2}Al_{0.2}Ga_{0.6}As layer grown by molecular beam epitaxy. A specimen with the needleshaped geometry required for APT has been milled using a dualbeam FEI Quanta200 3D focused ion beam (FIB) instrument (FEI Company, Eindhoven, Netherlands) equipped with an in situ OMNIPROBE micromanipulator (Dallas, TX, USA), and following the procedure described in HernándezSaz et al.[26]. The needle has been milled in such a way that the needle axis coincides with the [001] direction in the sample (the growth direction). In order to obtain a sharp nanometric tip (radius of about 50 nm), a sample cleaning process has been carried out with a Nvision 40 Zeiss FIB instrument (Oberkochen, Germany) using a Ga beam at 2 kV, which also reduces implantation damages. The atomic scale characterization by APT has been performed using a CAMECA LAWATAP instrument (Gennevilliers Cedex, France). About the FEM analysis, the 3D model has been defined, taking into account the composition of the structure obtained by APT using the structural mechanics module of the COMSOL software. To include the atom concentrations in the software, a discrete function of the three space variables was added. This function contains the value of the atomic concentrations of every 3 Å in the region of interest. To ensure the continuity of the data, a linear interpolation between the nearest data points is used. In order to have a negligible influence of the domain boundaries on the strain close to the QD, the Barettin et al.[27] criteria were followed. For this, we have considered the APT data corresponding to the lower QD layer and the barrier layer above it, and we have added simulated data around it in the growth plane and below it, in order to obtain a larger model to increase the distance from the QD to the boundaries of the model. Thus, the total simulated volume has a size of 120 × 120 × 45.5 nm, where the APT data is located in the centre, having a cylinder shape (because of the needleshaped specimen) with a diameter of 46 nm and a height of 25 nm. The distribution of the domains in the model has been made based on the mesh density and kind of composition (experimental or simulated). For example, the base of the model consists of a subdomain with simulated data with a coarse mesh; the WL and QD are an entire subdomain with a very fine mesh; and the three last nanometers close to the surface form a subdomain of simulated data with a fine mesh. The mesh generator is based on the Delaunay algorithm, and the mesh has been designed to have higher density in the volume of the APT data and in the surface of the full domain because these are the regions of interest. Anisotropic linear elastic behaviour has been considered. Vegard's law has been assumed for the determination of the In_{ x }Al_{ y }Ga_{1xy}As elastic constants and the lattice parameters; it is based on the atomic concentration obtained from the APT data (consequently we only import the In and Al distribution from the APT data, considering all the rest is GaAs). Initial strain was assumed to be ϵ_{0} = (a_{Inx Aly Ga1xy As}  a_{GaAs})/a_{GaAs} in all subdomains except in the base, where a_{ i } denotes the lattice parameter of i. The elastic properties have been taken from[28]. The elastic strain energy density (SED) can be expressed as SED = σ_{ ij }ϵ_{ ij }/2, where σ_{ ij } (ϵ_{ ij }) with i,j = x,y,z are the components of the stress (strain) matrix (the Einstein summation convention is assumed). The normalized SED is expressed as SED/SED_{max}, where SED_{max} is the maximum value of SED at the top layer surface.
Results and discussion
Figure 1b,c shows two perpendicular In composition slices of the APT data corresponding to the lower QD layer. The APT data in this region is the input data for the FEM analysis that will be performed next. As it can be observed in the figure, both images show an inhomogeneous In distribution, where the dark blue area indicates the higher In concentration, corresponding to the core of the QD. The absence of a uniform composition gradient from the centre of the QD in different directions prevents from the accurate theoretical simulation of the QD composition required to perform a FEM simulation that approaches the real situation. This proves that atomic scale experimental data such as those obtained from APT are essential in order to obtain realistic predictions of the QD nucleation sites from FEM analysis that can be used in the design of QD heterostructures for advanced devices.
In order to predict the nucleation site of the QD in the second layer, the chemical potential of the material during growth should be considered. In this case, the chemical potential has two major contributions: the one related to the surface energy and the one corresponding to the elastic strain. With regard to the first one, a previous analysis of the structure by transmission electron microscopy has shown that the structure grows with a flat surface, as no undulations have been observed in the wetting layers or in the surface of the structure. Because of this, the surface energy is not expected to have a major effect in the chemical potential of the structure in the prediction of the nucleation sites because prior to the formation of the second layer of QDs, the wetting layer is flat, therefore this term is neglected. As a result, the elastic strain is expected to be the determining factor for the growth process. This parameter will be calculated in this work using FEM based in the APT data.
On the other hand, our results have shown that the upper QD does not grow vertically aligned with the lower QD, but there is some deviation. Previous theoretical analyses have shown that this misalignment is, in part, related to the elastic anisotropy in the material[14], where the increase in the degree of anisotropy favours the anticorrelated island growth[19]. It has also been reported that the QD base size and density have a strong influence on this misalignment[11], although the QD shape (truncatedpyramidal or lensshaped) may not have a major effect in the strain at the surface of the capping layer[14]. These theoretical analyses are very useful for understanding the parameters that influence the QD nucleation sites. However, they have been developed considering ideal structures, for example including perfectly symmetric QDs. Our results have shown that real QDs are far from symmetric, and small composition variations can change the strain distribution of the structure. It has been found that the strain in semiconductor structures such as QRings has a significant importance in its optoelectronic characteristics[16]. This shows that in order to understand the functional properties of real semiconductor nanostructures, it is indispensable considering real compositional data for the FEM calculations, as the APT experimental data considered in the present work.
Conclusions
In conclusion, we have evaluated the accuracy of strain calculations by FEM using 3D atomic scale data obtained by APT for the prediction of the preferential nucleation sites of InAs stacked QDs. Our results by FEM have shown a very good agreement with our experimental observations, showing that this is a very useful tool for the analysis of the strain distribution in semiconductor systems. The combination of APT with FEM opens up the possibility of understanding the behaviour of complex semiconductor systems where strain plays a major role.
Authors' information
JHS is a PhD student at the Universidad de Cádiz. MH is an Associate Professor at the Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica y Química Inorgánica, Universidad de Cádiz. SD holds an Associate Professor at Université et INSA de ROUEN and he is the responsible of the Matériaux de la Microélectronique et de la Photonique (ER2MP) group. SIM is a full professor at the Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica y Química Inorgánica, Universidad de Cádiz and the head of the Materials and Nanotechnology for Innovation group (INNANOMAT). This group belongs to the Institute of Electron Microscopy and Materials (interim stage) of the University of Cádiz.
Abbreviations
 APT:

Atom probe tomography
 FEM:

Finite elements method
 FIB:

Focused ion beam
 GaAs:

Gallium arsenide
 InAs:

Indium arsenide
 QDs:

Quantum dots
 SED:

Strain energy density.
Declarations
Acknowledgements
This work was supported by the Spanish MINECO (projects TEC201129120C0503 and Consolider Ingenio 2010 CSD200900013), the Junta de Andalucía (PAI research group TEP946 INNANOMAT), and METSA project. The authors greatly acknowledge J. Houard for discussion and help in APT analyses and Prof. C. R. Stanley from University of Glasgow for QD sample fabrication.
Authors’ Affiliations
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