We are briefly reviewing the current status of elastic strain field determination in III–V heteroepitaxial nanostructures, linking finite elements (FE) calculations with quantitative nanoscale imaging and atomistic calculation techniques. III–V semiconductor nanostructure systems of various dimensions are evaluated in terms of their importance in photonic and microelectronic devices. As elastic strain distribution inside nano-heterostructures has a significant impact on the alloy composition, and thus their electronic properties, it is important to accurately map its components both at the interface plane and along the growth direction. Therefore, we focus on the determination of the stress-strain fields in III–V heteroepitaxial nanostructures by experimental and theoretical methods with emphasis on the numerical FE method by means of anisotropic continuum elasticity (CE) approximation. Subsequently, we present our contribution to the field by coupling FE simulations on InAs quantum dots (QDs) grown on (211)B GaAs substrate, either uncapped or buried, and GaAs/AlGaAs core-shell nanowires (NWs) grown on (111) Si, with quantitative high-resolution transmission electron microscopy (HRTEM) methods and atomistic molecular dynamics (MD) calculations. Full determination of the elastic strain distribution can be exploited for band gap tailoring of the heterostructures by controlling the content of the active elements, and thus influence the emitted radiation.
Nanoscale properties of matter lead the innovation and world’s technological developments towards the future. It is well established that understanding the physical and chemical properties of inorganic matter, it is crucial to observe and interpret nanoscale ensembles of atoms. To this end, quantum-scale control of modern semiconductor systems is reduced to defining the properties of their internal interfaces and crystal defects and to apprehend the correlated structural mechanisms at the atomic scale. Transmission electron microscopy (TEM) is a powerful experimental technique that utilizes electron diffraction to obtain contrast from defects and interfaces in crystalline materials. Moreover, in high-resolution TEM (HRTEM) phase contrast obtained from the interference of multiple diffracted electron beams is employed to visualize atomic columns. Hence, it provides the unique capability of accurately observing, measuring, and probing atoms and their relevant nanostructures. TEM constitutes a steady, time-tested basis for the systematic experimental analysis of the nanostructural features of advanced materials, and can offer reliable feedback on the optimization of their growth process. Equally important in heteroepitaxial growth is the quantitative evaluation of the strain field distribution inside nanostructures that can affect the alloy composition, and thus their electronic properties, which can be achieved by the geometric phase analysis (GPA) experimental method . For GPA strain measurements, phase variations in HRTEM images are utilized to obtain the reduced relative variation of the lattice fringes spacing with respect to an unstrained reference area, usually the substrate. In this respect, the elastic strain differs from the GPA strain, with the former being the reduced relative variation of the lattice parameters in a strained lattice regarding its own relaxed counterpart. Hence, a strain-free epilayer exhibits non-zero GPA strain equal to its effective mismatch with the substrate, while zero in-plane GPA strain signifies a full elastically strained epilayer. The GPA method is applied in reciprocal space.
However, in modern understanding of a thorough structural analysis, experimental information should be mutually combined with complex image simulation and computational techniques in an integrated manner. In this respect, several nanostructures simulation techniques have been developed, which can be classified in three major categories: the first comprises atomistic calculation methods, such as molecular dynamics (MD) and ab initio density functional theory (DFT), the second one includes the analytical continuum approximation, and the third is the finite elements (FE) simulation method. The latter constitutes the most versatile method of calculating the stress-strain fields of nanostructures, since it is suitable for simulating complex nanostructure geometries, in contrast to the analytical continuum approximation, while its needs for computing power is much less compared with the atomistic calculations. The FE method is a numerical method for the calculation of approximate solutions of partial differential equations in the framework of the linear continuum elasticity (CE) approximation. Analyzing nanostructures with FE, involves the construction of a two- or three-dimensional (2D or 3D) geometric model, which subsequently is solved numerically for the extraction of results that interpret its physical properties, or the influence of certain conditions , , . In contrast to the experimental GPA method, the FE method is applied in direct space.
Here, we are briefly reviewing the up-to-date status of strain field determination in III–V heteroepitaxial nanostructures, combining FE calculations with advanced experimental and theoretical techniques. In Section 2, the III–V semiconductor nanostructure systems of various dimensions are reviewed in terms of their importance in modern technological applications in optoelectronic and microelectronic devices. Subsequently, in Section 3, we focus on the determination of the stress-strain fields in III–V heteroepitaxial nanostructures by a variety of experimental and theoretical methods with emphasis on the FE method. Finally, in Section 4, we share our contribution to the field by coupling FE simulations on certain III–V nanostructures with HRTEM imaging and atomistic MD calculations. Our study is related with recent results, where we have thoroughly explored the morphology, strain properties and chemical composition of InAs quantum dots (QDs) grown on (211)B GaAs substrate, and GaAs/AlGaAs core-shell nanowires (NWs) grown on (111) Si, using quantitative HRTEM techniques and calculations , . As strain inside nano-heterostructures has a substantial impact on the alloy composition and their optical response, it is imperative to accurately map its distribution at the interface plane and along the growth direction. The results can then be exploited for manipulating the content of the active elements during growth, to tailor the band gap of the heterostructures and gain control of the emitted radiation.
2 III–V semiconductor nanostructures
A nanostructure is defined as a low-dimensional structure that has at least one dimension at the nanometer scale. While most materials in micrometer scale have similar properties with their bulk counterparts, their nanostructures exhibit very different physicochemical properties. This is due to the fact that nanomaterials are related with a high amount of surface atoms, high surface energy, as well as less defects compared to a bulk material. Nanostructures are usually classified according to the number of reduced dimensions. In view of this classification, the main types of nanostructures are: (a) nanocomposites [3D], (b) quantum wells [2D], which include coatings and thin-film-multilayers, (c) NWs, nanorods and nanotubes [1D], and (d) nanoparticles, QDs and nanodots [0D]. Nanostructures synthesized from different type of materials like metals, semiconductors or oxides attracted the scientific and industrial interest for their exceptional optical and electrical properties. The key difference between the most compound semiconductors and other type of materials is the direct band gap that is essential for excitonic radiative recombinations in optoelectronic devices, such as light-emitting diodes (LEDs), laser diodes (LDs) and solar cells.
One of the most important compound semiconductors system is the III–V one, comprising materials such as GaAs, InAs, GaP, InP and their alloys. The high electron mobility, the wide energy direct band gap, and the ability to easily form ternary and quaternary compounds are some of the significand advantages offered by III–V compound semiconductor devices over silicon-based technology. Heteroepitaxy of III–V semiconductor systems on foreign substrates has dominated quantum semiconductor technology over the past years. Two of the most interesting heteroepitaxially grown groups of III–V nanostructures are the self-assembled QDs and NWs that drawn immense scientific attention, due to their potential applications as active regions in electronic and photonic devices. These nanostructures, consisting of a wide range of III–V materials, have become a mainstay in modern technology.
InAs QDs grown on GaAs substrates have been investigated over the past decades both for the heteroepitaxial phenomena, as well as for the structural and optical properties suitable for technological applications , , , . The specific QD system is one of the most important in semiconductor nanostructures, due to the single-photon emission in a cavity, which can be applied to quantum cryptography and quantum computing , , . Another, competitive QD system is the InAs QDs buried in InP matrix, since it can emit radiation within the telecommunications-window wave lengths. In particular, single-photon emission from the InAs/InP QD system at wavelengths longer (1.2–2.0 μm) than can be achieved by the InAs/GaAs QD system (0.9–1.3 μm) has been reported , , . Significant alternatives in the field of QD systems are the group III-nitrides, such as GaN/AlN QDs, due to their commercial success in optoelectronic devices , , . The pyroelectric and piezoelectric fields, as well as, the wide band gap of the GaN/AlN QDs make them prime candidates in many technological applications, for example, in quantum information processing , .
III–V semiconductor NWs grow mainly under the vapor–liquid–solid (VLS) mechanism  and can be tailored with either axial, or radial orientation of heterointerfaces (core-shell configuration). The GaAs/InGaAs core-shell or core-multishell NW system is of paramount importance in applications such as long wavelength optical transmission and integrated photonics , . Furthermore, the GaAs/InGaAs core-shell system now penetrates also in the field of photovoltaics (PV) . Another example of core-shell NWs is the GaAs/AlGaAs system, which is among the systems with great interest for high-performance infrared lasers up to room temperature , , . Additionally, GaAs/AlGaAs core-shell NWs are more familiar in high-efficiency PV devices and solar cell technology, due to the existence of axial piezoelectric fields, while simultaneously having a very small lattice mismatch , . Because of the excellent properties of the III-nitride semiconductor materials, InGaN/GaN core-shell and multi-quantum well core-shell NWs prevail in next generation LEDs and solar cells , , , .
3 Strain field determination in quantum nanostructures
For many years, it was believed that long-lived laser devices could only be constructed from the deposition of different lattice-matched epitaxial materials , , . However, the attention has been shifted from unstrained to strained heterostructures, due to more recent theoretical and experimental investigations. Strained quantum-well devices in early 1990s exhibited improved characteristics compared with corresponding lattice-matched devices , . The enhanced performance observed in strained lasers arose from the influence of the strain on the band structure and laser gain process , . Therefore, it is imperative to apprehend how strain influences the optoelectronic properties of nanostructures and to provide quantitative accuracy of the stress-strain distribution inside nanostructures.
The strain in heteroepitaxial growth, due to lattice and thermal mismatch between the involved materials, plays an important role in the band structure of the active region because it induces changes in the levels of the valence-band maximum (VBM) and conduction-band minimum (CBM), and hence modifies the QW depth and the confinement of electrons and holes. While hydrostatic strain shifts the energy levels of a band, uniaxial and biaxial strain split the degeneracy of a band . Another important effect of strain in semiconductor nanostructures, is that it induces significant piezoelectric fields in polar crystalline orientations, such as c-axis (0001)-oriented wurtzite and (111)-oriented zinc-blende structures , . The piezoelectric field can reduce the electron-hole wave function overlap and leads to low internal quantum efficiency , . Moreover, strain can also change the band curvature, leading to effective masses of charge carriers and density of states changes , .
A variety of experimental and theoretical methods have been developed in order both to understand how the strain field affects the properties of semiconductor nanostructures, and to analyze the corresponding strain fields in III–V semiconductor QDs, NWs, thin films and superlattices. Most of the research presented in this review has been performed by the numerical method of FE, which is based on the CE assumption. While there is also the atomistic approach utilizing atomic potentials that provide more accurate results, the application in complex geometries with less computational resources renders the FE method more versatile to model nanostructures.
3.1 Buried and free-standing InAs Stranski-Krastanov islands
This heteroepitaxial system includes pyramidal InAs QDs grown on GaAs (001) substrate, whose shape has been experimentally evidenced by HRTEM . Numerical simulations of the elastic and plastic behavior of these QDs have been performed using the FE method coupled with TEM techniques to determine the stress and strain distribution within truncated and non-truncated pyramidal In(Ga)AsQDs, showing that stress relaxation occurs at the top area of the islands, while the stress concentration occurs at their edges , , , . The QDs size as well as the thickness of the capping layer above the islands is found to play an important role in stress-strain distributions of the QDs. Furthermore, the elastic anisotropy was found to have significant influence on the stress and strain field, either in truncated pyramidal or lens shaped QDs , , . Jonsdottir et al. studied dots with different island shapes such as spherical, cylindrical or pyramidal shapes, showing that regardless of the shape, the same strain trend exists for all of them . Latini et al. have studied InAs QDs multilayers on GaAs (001) and showed that the strain field of buried QD layers depends on the QD mutual in-line distances, while the alignment of a single QD chain depends on the surface morphology . Recently, Reyes et al. studied the influence of annealing on InAs/GaAs QDs embedded in a strain reduction layer (SRL) of GaAsSb by TEM and FE simulations. They found that annealing improves the crystalline quality of the SRL, possible enhancing the photoluminescence (PL) emission of the system . The local composition has also been investigated with the experimental method CELFA . In order to evaluate the effect of strain on the single-electron charging properties of InAs QDs, researchers presented a 3D simulation program based on DFT and the local density approximation (LDA) . Furthermore, major attention has been given to the GaN/AlN QDs system. Young et al. presented a study with FE analysis of different geometries of GaN QDs using a piezoelectric model, to determine the strain state and the electrostatic potential of the particular geometries  while, a coupled model of thermo electroelastisity was applied in the same system .
3.2 Core-shell and core-multishell NWs
It is well known that the quality of heteroepitaxy in one-dimensional NW systems strongly affects their structural and optoelectronic properties, but, up to now, only few studies demonstrated the critical dimensions for heteroepitaxial growth of axial or radial NW heterostructures. Particularly, Ertekin and Sands  and Glas  found the critical diameters below which the heteroepitaxial interface appears defect-free, while Ye et al. developed a 3D FE model to simulate axial NWs heterostructure, in order to calculate the critical thickness with respect to the lateral dimensions of the nanostructure . Another FE study referred in critical dimensions for axial InGaAs/GaAs NWs is from Geng et al. , who simulated the stress field of NWs. In addition to axial InGaAs/GaAs NWs, theoretical studies on critical dimensions, using FEM, have been made for core-shell InGaAs/GaAs NWs heterostructures. Their results showed that these structures can reduce the strain in the shell and increase the critical shell thickness . The key of a detailed characterization of the strain distribution and chemical composition of GaAs/(InGa)As core-shell NWs is the combination of experimental and numerical methods. Grandal et al. performed TEM, scanning TEM, GPA and FE calculations to examine the local strain, and electron energy loss spectroscopy (EELS) to determine the local chemical composition of the core-shell structure . A general description of the strain distribution, in core-shell NWs of zinc-blende structure along the -direction for cylindrical and hexagonal geometries, has been discussed simultaneously in Refs. , , . FE combined with other experimental and/or theoretical methods have been applied to GaN and InGaN/GaN core-shell NWs to calculate the piezoelectric response, the in-plane strain field and the composition profile , , .
3.3 Other complex III–V semiconductor heterostructures
To date, 2D and 3D FE method is used to study the evolution of the stress and strain of complex III–V semiconductor systems. A simple system has been presented from Pramanik and Zhang, where they studied the residual stress in silicon thin film on sapphire (SOS) during cooling . The 3D FE analysis in this anisotropic system has provided great understanding of the residual stress generation mechanism. Furthermore, 2D FE calculations have been used to explain the strain distribution of vertically stacked multi-quantum well (MQW) structures and superlattices. In the case of MQW, the results show that the strain filed has a significant influence in the thermal expansion coefficient of the structure, while in the case of superlattices strain distribution promote the formation of alloy separation effect , . Due to the increasing interest, more complex systems have also been investigated by FE calculations. Specifically, InAs quantum wires multilayer embedded in InGaAlAs barriers grown on InP substrate were studied using FE simulations to calculate the chemical potential distribution . Also, analytical and FE calculations have provided useful mechanical information during fabrication on Si nanomembranes for fingertip electronics .
4 Coupling FE calculations with experimental and theoretical methods
We have reviewed and discussed various reasons explaining the major importance of stress-strain fields determination in III–V semiconductor nanostructures. We have recently studied the morphology, strain properties and chemical composition of InAs QDs grown on (211)B GaAs substrate, and GaAs/AlGaAs core-shell NWs grown on (111) Si, using quantitative HRTEM techniques , . The aim of this research is to numerically simulate the elastic strain fields, directional deformation and stress in the above nanostructures, using the FE calculation package ANSYS. FE simulations are compared with the experimental results obtained from GPA of experimental HRTEM images and the theoretical results obtained from MD simulations.
The shape of the InAs QDs grown on (211)B GaAs was modeled as truncated pyramidal, elongated along <111> direction, according to the morphological characterization performed by TEM-HRTEM (Figure 1). In the case of uncapped QDs, quantitative measurements have been performed using GPA on experimental HRTEM images. It was shown that large surface QDs were almost unstrained, due to plastic relaxation. This was attributed to the introduction of misfit dislocations (MDs) at the interface . A residual in-plane elastic strain was observed that progressively decreased from the base to the top of QDs, where they were found to be in an almost relaxed state .
In our analysis, we have modified the elastic characteristics of any (hkl)-oriented system with respect to the conventional <001> coordinate system. Thus, by determining the affine transformation from one coordinate system to another, we address the calculation of the new stiffness tensor C(hkl), in relation to the independent stiffness constants of the <001>-oriented system. This transformation comprises three successive rotations about the three axes of the <001> coordinate system. Furthermore, we address the calculations of the stiffness tensors in the cubic system for  growth direction, in terms of anisotropic CE theory, while we have considered the plane stress (biaxial) model for the InAs QDs. For the calculations, the elastic properties had been simulated in the framework of thermoelasticity. The calculated in-plane εxx and out-of-plane εzz residual elastic strain components are shown in Figure 2.
FE calculations, in accordance with the GPA analysis , show a small residual elastic strain, almost 0.7%, at the interface that gradually reduces towards the apex and the inclined edges of the QDs to almost 0.3%, while gradual reduction is also observed along the growth direction to almost zero strain at the apex of the dot.
FE simulations of the strain field were also performed in buried InAs QDs grown along  direction. From the TEM-HRTEM characterization it was found that the buried InAs QDs possess the same pyramidal anisotropic shapes with the uncapped QDs but with considerable smaller sizes, due to decomposition of InAs islands from the higher growth temperature of the GaAs cap layer . In Figure 3, both the in-plane εxx and out-of-plane εzz elastic strain components are shown. The compressive elastic strain, along the in-plane direction is almost 6.6% at the interface, implying full registration of the heterostructure, as expected from the absence of MDs. The maximum compressive strain appears at the base of the QD and the smallest occurs at apex part. Moreover, the out-of-plane strain shows a systematic decrease from 3% at the base of QD to less than 1% at the apex region, which is not consistent with experimental GPA measurements, unless we consider alloying variations inside the dot that reverse the strain trend along the growth direction from decreasing to increasing. This can be associated with a gradual increase of the In content from the base to the apex of the QDs, owing to Ga segregation phenomena .
Recent HRTEM observations of GaAs/AlXGa(1−x)As (xAl=35%) core-shell NWs grown on (111) Si showed that both core and shell exhibited a symmetrical hexagonal shape with almost equal core and shell radius, while the stress and strain characteristics were elucidated by MD simulations . By means of the biaxial strain model, it was found that the AlGaAs shell is full elastically registered on the GaAs core with the absence of any plastic deformation. To obtain an accurate description on the induced strain fields, a hexagonal cross-sectional model was constructed, based on the experimentally observed NW geometry and dimensions. Subsequently, FE calculations showed that the maximum relaxation in the shell was observed at the outer hexagon vertices, and the maximum tensile strain occurs at the outer core vertices, while stress concentrators appear at the inner core vertices (Figure 4). In order to investigate the effects of the chemical composition of the active regions and size of the NW heterostructures, FE calculations were extended to include GaAs/AlxGa(1−x)As core-shell NWs with Al concentration varied from 0.2 to 0.65, and shell-to-nanowire (S/NW) relative diameter ratios of 0.45–0.65. The FE simulations showed that strain distribution within the nanostructures is sensitive to both their chemical composition, as well as their size, exhibiting absolute values in line with the MD calculations .
We have reviewed recent developments in the topic of strain field determination in III–V heteroepitaxy, by coupling FE simulations with experimental and theoretical techniques at the nanoscale. The importance of strain distribution in III–V low-dimensional systems, related to quantum phenomena that can tailor their electronic properties, renders crucial a precise mapping of the strain components within the heterostructures. To this end, FE calculations is a versatile tool of obtaining elastic strain fields, using the anisotropic CE approximation. We presented characteristic examples of the application of FE on III–V heteroepitaxy comprising InAs QDs grown on GaAs (211)B and GaAs/AlGaAs core-shell NWs grown on Si (111), combined with experimental GPA strain mapping on HRTEM images and atomistic MD calculations. Provided that the nanostructures are under plane stress, FE calculations provided maps of their strain components at the interface plane and along the growth direction that are in par with both experimental strain measurements, as well as atomistic calculations. Hence, the efficient coupling of all methods can lead to an accurate determination of the strain fields in III–V heteroepitaxy and constitute a credible basis of a targeted exploitation of their exquisite optical properties.
Work supported by the Research Projects for Excellence IKY/Siemens.
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