Effect of dislocations on electrical and electron transport properties of InN thin films. I. Strain relief and formation of a dislocation network

J. Appl. Phys. 100, 094902 (2006); https://doi.org/10.1063/1.2363233 100, 094902 © 2006 American Institute of Physics. Effect of dislocations on electrical and electron transport properties of InN thin films. I. Strain relief and formation of a dislocation network Cite as: J. Appl. Phys. 100, 094902 (2006); https://doi.org/10.1063/1.2363233 Submitted: 19 February 2006 . Accepted: 21 September 2006 . Published Online: 07 November 2006 V. Lebedev, V. Cimalla, J. Pezoldt, M. Himmerlich, S. Krischok, J. A. Schaefer, O. Ambacher, F. M. Morales, J. G. Lozano, D. González, et al. ARTICLES YOU MAY BE INTERESTED IN Effect of dislocations on electrical and electron transport properties of InN thin films. II. Density and mobility of the carriers Journal of Applied Physics 100, 094903 (2006); https://doi.org/10.1063/1.2363234 Indium nitride (InN): A review on growth, characterization, and properties Journal of Applied Physics 94, 2779 (2003); https://doi.org/10.1063/1.1595135 Threading dislocations in In-polar InN films and their effects on surface morphology and electrical properties Applied Physics Letters 90, 151901 (2007); https://doi.org/10.1063/1.2720717 Effect of dislocations on electrical and electron transport properties of InN thin films. I. Strain relief and formation of a dislocation network V. Lebedev,a兲V. Cimalla, J. Pezoldt, M. Himmerlich, S. Krischok, J. A. Schaefer, and O. Ambacher Center for Micro- and Nanotechnologies, Technical University Ilmenau, D-98684 Ilmenau, Germany F. M. Morales,b兲J. G. Lozano, and D. González Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica y Química Inorgánica, Facultad de Ciencias, Universidad de Cádiz, 11510 Puerto Real Cádiz, España 共Received 19 February 2006; accepted 21 September 2006; published online 7 November 2006兲 The strain-relaxation phenomena and the formation of a dislocation network in 2H-InN epilayers during molecular beam epitaxy are reported. Plastic and elastic strain relaxations were studied by reflection high-energy electron diffraction, transmission electron microscopy, and high resolution x-ray diffraction. Characterization of the surface properties has been performed using atomic force microscopy and photoelectron spectroscopy. In the framework of the growth model the following stages of the strain relief have been proposed: plastic relaxation of strain by the introduction of geometric misfit dislocations, elastic strain relief during island growth, formation of threading dislocations induced by the coalescence of the islands, and relaxation of elastic strain by the introduction of secondary misfit dislocations. The model emphasizes the determining role of the coalescence process in the formation of a dislocation network in heteroepitaxially grown 2H-InN. Edge-type threading dislocations and dislocations of mixed character have been found to be dominating defects in the wurtzite InN layers. It has been shown that the threading dislocation density decreases exponentially during the film growth due to recombination and, hence, annihilation of dislocations, reaching ⬃109cm−2 for ⬃2200 nm thick InN films. © 2006 American Institute of Physics.关DOI: 10.1063/1.2363233兴 I. INTRODUCTION Recent studies of the narrow band gap InN compound semiconductor 共0.6–0.7 eV兲show an increasing interest in investigating its fundamental properties and evaluate its potential applications.1–5InN was predicted to have the lowest effective mass for electrons among all III-nitride semiconductors,6resulting in high mobility and high saturation velocity. The theoretical maximum mobility calculated for wurtzite 共2H兲InN at 300 K is ⬃4400 cm2V−1 s−1, while at 77 K the limits are beyond 30 000 cm2V−1 s−1.7It was found that InN exhibits an extremely high peak drift velocity at room temperature and achieves the highest steady-state peak drift velocity of 4.23⫻107cm/s among the III nitrides.8,9It was also concluded that the transport characteristics of InN are superior to those of GaN and GaAs, over a wide range of temperatures from 150 to 500 K and doping concentrations up to ⬃1019 cm−3. It was predicted that InN- based field-effect transistors 共FETs兲may have a cut-off frequency of over 1 THz for 0.1 ␮ m gates. All the superior electronic properties of this material make InN a highly potential material for the fabrication of high-frequency heterojunction FETs. Another important discovery about wurtzite InN is its very strong surface charge accumulation on the order of 1013 cm−2.4The accumulated electrons, which stay within 4 nm from the top InN surface, can be modulated by external treatments contributing considerably to the lateral conductivity of thin InN films. Recently reported results reveal the promising applications of InN epilayers for chemical and biological sensors.10 The most valuable problem of InN heteroepitaxy is related to the large lattice mismatches between the epilayer and any available substrate material. In particular, due to the mismatch in the InN/AlxGa1−xN heterosystem, the strain relaxation plays a crucial role in achieving high-quality heterostructures. The lattice mismatches in 2HInN/GaN and InN/AlN systems are −0.11 and −0.14, respectively, and the disregistry of the interfacial atomic arrangements between the substrate and the epilayer will be accompanied by structural defects or by strain. This raises the question of strain relaxation even for very thin epilayers since the misfit dislocation 共MD兲density in InN is initially ⬎1011 cm−2.11 However, in order to be of a high optical or electronic quality, misfit relaxation defects should be avoided in these heterostructures as soon as possible within the constituting layers. Extended defects, such as dislocations and stacking faults, are known to adversely affect III-nitride device properties causing nonradiative recombination,12,13 carrier scattering effects,14 and diffusion of dopants and impurities.15 Moreover, threading dislocation 共TD兲densities in the range of 108–1010 cm−2 arising as a result of enormous strains may ultimately limit III-nitride device performance. Thus, in the investigation of strain relaxation it is important to understand the structural, electronic, and optical properties of InN-based heterostructures. Strain evolution has been intensively studied for the growth of binary III nitrides 共GaN and AlN兲,16–21 a兲Electronic mail: [email protected] b兲Also at Center for Micro- Nanotechnolgies of TU Ilmenau. JOURNAL OF APPLIED PHYSICS 100, 094902 共2006兲 0021-8979/2006/100共9兲/094902/13/$23.00 © 2006 American Institute of Physics100, 094902-1 and their heterostructures,22–26 with a few studies being published on strain relief in InN/AlxGa1−xN heterostructures.11,30–33 Most of the published contributions are focused on the measurement of the strain and strain relaxation with only a few reports upon the actual mechanisms involved in relaxation.16,27–29 The purpose of this article is to study those very mechanisms of strain relief, including the formation and elimination of dislocations, in wurtzite 共0001兲InN/AlN and InN/GaN heterostructures grown by plasma induced molecular beam epitaxy 共PIMBE兲. The type of misfit defects involved in strain relief at the interface between InN and the template on which it is deposited is characterized in detail for the case of AlN epitaxial templates. The relaxation of a lattice mismatch-induced strain is analyzed in situ during PIMBE by reflection high-energy electron diffraction 共RHEED兲,ex situ by high resolution x-ray diffraction 共XRD兲, and transmission electron microscopy 共TEM兲in both image and diffraction modes. The following phenomena will be discussed in the framework of a model that emphasizes the decisive role of the coalescence process as revealed by RHEED, TEM, and XRD measurements: 共1兲plastic relaxation of the primary strain by the introduction of geometric misfit dislocations, 共2兲elastic strain relief during the island growth phase, 共3兲 formation of threading dislocations induced by the coalescence of domains, 共4兲relaxation of elastic strain by the introduction of secondary misfit dislocations, and 共5兲and the mechanisms by which various dislocation slip systems are formed. II. EXPERIMENTAL DETAILS The samples were grown in a Balzer’s PIMBE system with a base pressure of ⬃1⫻10−10 mbar. Standard Knudsen cells were used for Ga, Al, and In evaporations. An Oxford Applied Research MDR21 radio-frequency nitrogen plasma source has been used to supply active N radicals from purified 6N5N 2gas. The substrate temperatures were calibrated by an infrared pyrometer. The growth process was monitored by digitized patterns of RHEED. Molecular fluxes were monitored in situ by a quadrupole mass spectrometer and calibrated using RHEED oscillations. The samples consisted of epitaxial templates overgrown by a 2H-InN共0001兲layer. The template structures and the epitaxial relationship in the heterosystems are summarized in Fig. 1. More details on the carbonization of Si surfaces34 and on the growth of AlN on SiC/Si共111兲共Ref. 35兲and sapphire15 can be found elsewhere. All template epilayers were grown in the twodimensional 共2D兲Frank–van der Merwe mode at substrate temperatures of ⬃900 and ⬃710 °C for AlN and GaN, respectively. The III/N flux ratio was kept constant through the whole template epitaxy process to maintain slightly metalrich conditions for AlN and GaN growths. For all samples, an almost fully relaxed AlN layer was deposited as a buffer, typically 200–250 nm thick. In particular, XRD reciprocal space maps around the symmetric 共0002兲and the asymmetric 共202 ¯ 5兲reflexes of 2H-AlN reveal a small increase of the in-plane stress 共a decrease of the c/aratio兲for the AlN layers grown on Si and sapphire. c/a⬃1.592 for 2H-AlN共0001兲/ Al2O3and ⬃1.578 for 2H-AlN共0001兲/3C-SiC/Si共111兲have been deduced 共nominal c/a⬃1.601兲. The templates were overgrown in situ bya2H-InN epilayer at Tsub⬃380 °C. The InN epilayers were grown under stoichiometric 共1:1兲conditions to prevent surface metal accumulation. The thickness of the InN layer has been varied from 0.35 to 2.2 ␮ m. TEM experiments were performed in two different microscopes: 共i兲TECNAI 20S-TWIN 共FEI兲and 共ii兲JEM-2011 共JEOL兲, both operating at 200 kV. Mechanical thinning and ion milling in a Gatan PIPS system were carried out in order to prepare the samples for cross-section 共XTEM兲and planview 共PVTEM兲inspections. Selected area electron diffraction 共SAED兲patterns and micrographs obtained in brightfield 共BF兲and dark-field 共DF兲modes by conventional twobeam 共2B兲conditions and high resolution TEM 共HRTEM兲 were analyzed. Structural analyses were performed by high resolution x-ray diffraction 共HRXRD兲using a Bruker D8 diffractometer. Rocking curves at the symmetric 共0002兲and reciprocal space maps at the symmetric 共0002兲and the asymmetric 共202 ¯ 5兲InN reflections were taken to evaluate the crystal quality and the residual strain, respectively. To study the surface properties, the samples were characterized in another ultrahigh vacuum 共UHV兲system with a base pressure of ⬃1⫻10−10 mbar described in more detail in Ref. 36. Due to the high reactivity of the InN surface the transfer to the analytical UHV chamber was performed with an absolute minimum time window of 30 s. The surface morphology was measured in vacuo by atomic force microscopy 共AFM兲using contact mode 共Fig. 2兲. A typical root mean square surface roughness of an InN epilayer having a thickness of ⬃1 ␮ m does not exceed 2 nm. The surface chemical composition and electronic properties 共valence band兲were studied by x-ray photoelectron spectroscopy 共XPS兲and ultraviolet photoelectron spectroscopy 共UPS兲using monochromated Al K ␣ 共h ␯ =1486.7 eV兲,HeI共h ␯ =21.2 eV兲,orHeII FIG. 1. The structure of the templates used in this work and the epitaxial relationship in the grown heterosystems. 094902-2 Lebedev et al. J. Appl. Phys. 100, 094902 共2006兲 共h ␯ =40.8 eV兲radiation, respectively. All spectra are recorded using a hemispherical electron analyzer 共EA125兲. III. STRUCTURAL AND SURFACE PROPERTIES OF InN The crystal quality of thick InN was analyzed by XRD. Rocking curves at the 共0002兲InN reflex show a homogeneous decrease of the full width at half maximum 共FWHM兲 with the film thickness. The best FWHM of 0.277° was found at about 1 ␮ m indicating the continuous improvement of the InN layers up to this thickness and a change in the relaxation mechanism thereafter. Thick InN layers of ⬃1 ␮ m have only a small residual stress of ⬍0.1%. These results agree very well with those previously reported for epitaxial InN layers grown at Cornell University.31 In particular, the dependence of ⌬c/⌬ais in excellent agreement, and thus the determined biaxial strain coefficient RB=2C13/C33= −Izz/Ixx will be used to calculate the strain components Ixx and Izz in plane and in growth direction, respectively 共see Table I兲. For the calculation of Izz the experimental value of the biaxial strain coefficient RB⬃0.6 was used31 which differs from the most accepted theoretical value, RB=0.82, derived using the elastic coefficients c13 and c33 provided by Wright.37 The results of XPS and UPS measurements are presented in Fig. 3. After direct loading to the UHV system a small amount of oxygen and carbon contaminants were detected on the surface. From the XPS peak intensities a thickness of the contaminated layer, of d⬃0.5 nm, was calculated. As conventional sputtering leads to the formation of metallic indium on the InN surface,36 the samples were annealed at T =500 K to reduce the amount of adsorbed molecules, resulting in d⬃0.1 nm. AFM scans after heating showed that this procedure had no influence on the surface morphology, but the photoelectron spectra revealed some changes. The relevant core level peaks of the InN surface are shown in Fig. 3. All binding energies are given with respect to the Fermi level of a Ag reference sample. The binding energies 共BEs兲of the In3d5/2 and N1sare 443.9 and 396.2 eV, respectively. Upon annealing, changes in the chemical composition of the surface were observed. The O1slevel consists of an adsorbate peak at 532.1 eV and a small feature at 530.1 eV originating from In–O bonds which is in good agreement with previous measurements.36 The oxygen is almost completely removed after annealing at 500 K 共see Fig. 3兲. The amount of adsorbed carbon is only slightly affected by this procedure. The annealing leads to a change of the In3d5/2 signal. The shoulder at ⬃445 eV originating from the oxidized indium state is strongly reduced, matching the change in the O1ssignal. The same behavior was observed for the In4dpeak, also measured by UPS 共He II兲, which has its maximum intensity at 17.8 eV. From both the In3dand In4dfeatures, no formation of metallic indium was detected. The N1speak width is also reduced after heating. This can be explained by the removal of NOxspecies at ⬃397.5 eV BE. From the UPS measurements the work function ␾ was obtained using the onset of secondary electrons when a bias of −5 V was applied to the sample. The initial value for ␾ of 4.3 eV changed to 4.1 eV after annealing. The initial surface exhibits a structure at 5.3 eV in the valence band which is most likely related to adsorbed species because after annealing this feature is removed and peaks at 6.8 and 3.1 eV are observed 共see Fig. 3兲. These structures origi- FIG. 2. 共a兲3⫻3 ␮ m2AFM scan of a 0.8 ␮ m thick InN epilayer grown on AlN共0001兲template. The inset shows a typical triangular hole created by the set of 共1 ¯ 100兲and 共112 ¯ 0兲planes of the islands due to the partial character of the coalescence phase. 共b兲250⫻250 nm2AFM scan demonstrates the domain structure of an ⬃30 nm thick 2H-InN nucleation layer. Bilayer-step morphology of the surface proves a 2D growth within nucleated islands. TABLE I. Calculated residual stresses in the InN/AlN heterosystem during the nucleation and coalescence. Ixx is the biaxial strain, Izz is the strain in the zdirection, ␴ xx is the biaxial stress 共GPa兲,and ␴ zz is the stress in the z direction 共GPa兲. Method used HRTEM RHEED RHEED XRD Growth phase Nucleation Nucleation After coalescence Averaged Ixx −0.0125 −0.0411 −0.0011 −0.001 Izz 0.008 0.023 0.0006 ¯ ␴ xx −1.75 −4.99 −0.15 ¯ ␴ zz 0.65 1.85 0.05 ¯ 094902-3 Lebedev et al. J. Appl. Phys. 100, 094902 共2006兲 nate from nitrogen electron states in indium nitride and have been previously observed experimentally38 and predicted by theoretical calculations.39 The valence band 共VB兲maximum at the surface is found to be 1.6 eV below the Fermi edge. This strong downward band bending can be explained by the observed high electron concentration of InN surfaces. IV. GROWTH DYNAMICS In all the experiments, the growth of InN 共0001兲was initiated in situ on the epitaxial template 关i.e., 共0001兲AlN or GaN, see Figs. 4共a兲and 4共b兲at 380 °C兴. The growth parameters were not varied during the process and similar growth conditions were used for PIMBE of InN on both AlN and GaN templates. Identical changes observed in the RHEED patterns during the growth indicate that in both cases 共i.e., AlN and GaN templates兲the film growth is affected by similar mechanisms. The only difference was observed in the characteristic relaxation time 共thickness兲of the in-plane lattice parameter. Moreover, differences in electron transport properties of InN epilayers grown on both AlN and GaN templates were not observed. Thus, in the following sections, we restrict our considerations to the case of InN共0001兲/AlN共0001兲hetero- system, which, due to the larger lattice misfit, is more illustrative to show the relaxation phenomena. The in situ RHEED observations allowed us to distinguish at least three different growth phases characteristic for any highly mismatched heterosystem. These stages are schematically shown in Fig. 4共c兲.共1兲共A兲Three-dimensional 共3D兲 nucleation and subsequent growth of slightly misoriented 2H-InN islands up to a thickness of ⬃10–15 nm 共a pure Volmer-Weber mode兲,共2兲共B兲a partial coalescence of the islands into a continuous film, and 共3兲共C兲a long-term epitaxy of 2H-InN followed by the gradual improvements of the surface morphology. To analyze the growth dynamics, we consider a classic growth model applied in our earlier work for the case of 2H-AlN/3C-SiC/Si共111兲growth.35 In the simplest case, the growth mode is determined by the balance between the formation energies of the film surface ␴ film, the template surface ␴ sub, and the interfacial energy Eint. One can write the conditions for the 2D growth of a strained thin film as ␴ sub ⬎共 ␴ film −⌬EM兲+Eint −⌬ ␮ /2A共hkl兲+nlatEstr,共1兲 where nlat=3 for the hexagonal cell, Estr is the macroscopic stress energy, A共hkl兲is an intersection of the elemental cell FIG. 3. Photoelectron spectra of the InN surface, 关共a兲– 共c兲兴 In3d5/2,N1s, and C1score levels measured using monochromated Al K ␣ radiation 共h ␯ =1486.7 eV兲and 共d兲valence band spectra measured using He II radiation h ␯ =40.8 eV. The spectra marked by arrows correspond to the as loaded sample, in comparison to the surface after annealing at 500 K. FIG. 4. 具21 ¯ 1 ¯ 0典RHEED patterns of 共a兲a 200 nm thick AlN共0001兲and 共b兲 220 nm thick GaN共0001兲epilayers. 共c兲Schematic representation of the growth of 2H-InN on highly mismatched templates: 共A兲early formation of the slightly misoriented flat InN platelets with elastic relaxation at the edges, 共B兲partial coalescence of the platelets, and 共C兲further growth and formation of a continuous layer. 094902-4 Lebedev et al. J. Appl. Phys. 100, 094902 共2006兲 with the surface plane 共hkl兲, and ⌬EMstates for the reduction in the total energy of the system due to the periodical undulations at the heterointerface.40 The two last terms in the Eq. 共1兲represent the formation energy of the epilayer. ⌬ ␮ = ␮ − ␮ bulk=kBTln共S兲is the difference in the chemical potentials of the bulk and gas phases of the species representing the actual growth conditions being a function of the supersaturation Sand temperature T. The actual growth conditions can be described by the following relation: ␮ In = ␮ InNbulk − ␮ Nbulk +⌬Hf InN +⌬ ␮ In,共2兲 where ⌬Hf InN is the heat of formation of the binary compound. The conditions ⌬ ␮ In=−⌬Hf InN and ⌬ ␮ In=0 imply N-rich and In-rich growths, respectively. As it follows from Eq. 共1兲, the supersaturation is the only parameter capable of changing the growth mode at the nucleation stage. At the flat surface and low supersaturation used for the stoichiometric growth, the term ⌬ ␮ /2A共hkl兲does not affect the overall energy balance. As a result, the stoichiometric nucleation and growth have a pure 3D character due to an enormous value of Estr, until the coalescence phase. The use of a higher supersaturation for the nucleation can shift the energy balance towards the Stranski-Krastanov growth mode. In particular, predeposition of 2–3 ML 共monolayer兲of In on the flat 2H-AlN surface before the epitaxy results in the formation of a very thin 2D wetting layer of InN, due to the presence of activated nitrogen in the chamber volume. However, the experimentally observed characteristic transition into 3D growth takes place after deposition of 1–2 ML of InN. Consequently, as InN growth conditions remain constant, the last term in Eq. 共1兲,Estr, plays the dominant role in growth dynamics controlling the mechanisms involved in the lattice relaxation. For a particular bicrystal consisting of a semi-infinite substrate and an epilayer of thickness d, the energy per unit area Eel of a grid of edge dislocations having Burger’s vectors blaying on the growth plane, with dislocation spacing p is given approximately by Eel =nlat G兩b兩2 4 ␲ 共1− ␯ 兲pln 冉 ed r0 冊 .共3兲 The elastic strain remaining is given by I=f−兩b兩/p. The total strain energy per unit area in Eq. 共1兲,Estr, is then given by Estr =nlat G共1+ ␯ 兲 4 ␲ 共1− ␯ 兲p I2d+nlat G兩b兩共f−I兲 2 ␲ 共1− ␯ 兲ln 冉 ed r0 冊 ,共4兲 where Gand ␯ are the interfacial shear modulus and Poisson’s ratio of the epilayer, respectively, Iis the biaxial stress remaining in the epilayer, and r0is the inner core radius of dislocation.41–43 In Eq. 共4兲, the first term represents the elastic strain energy, which is the driving force for the relaxation of the lattice during growth. The second term is the energy accumulated by the dislocations network and corresponds to the plastic relaxation associated with dislocation formation. In our experiments, the in-plane lattice relaxation was monitored in situ by recording the intensity profile of RHEED patterns. A bitmap image, shown in Fig. 5共a兲, presents a time dependence of the RHEED intensity profile during the nucleation and coalescence phases of the InN growth. Every horizontal line 共1 pixel thick兲in the image represents the intensity profile averaged over 1 s of the growth. The height of the image equals 90 s of growth time. The lateral resolution of 752 lines per 6/acorresponds to the real space resolution of ⬃0.02 Å/pixel. The corresponding variations of the in-plane lattice parameter were derived from the continuous variations of the spacing between planes of the type 兵011 ¯ 0其perpendicular to the basal plane. It is presented in Fig. 5共b兲as a function of the epilayer thickness showing the general trends of the misfit accommodation between AlN template and InN epilayer. As one can see, the different stages of the relaxation of the InN in-line lattice parameter correspond well to the three growth phases shown in Fig. 4共c兲. While the phases Aand Bare characteristic by the 3D growth mode, which is typical for highly stressed epilayers, phase Ctypically occurs in the 2D mode resulting in a slow relaxation rate and in a smooth epilayer surface 关Fig. 6共a兲兴. Moreover, it is obvious from the order of the variation in the lattice parameter that both plastic as well as elastic mechanisms are involved in the primary misfit accommodation between InN epilayers and 2H-AlN templates. The biaxial strain due to the thermal mismatch arising during the cooling of the films is out of the scope of this work. It should be mentioned that the major portion of this stress arises at temperatures below the brittle-ductile transition temperature of FIG. 5. In-plane lattice parameter evolution during the nucleation and growth phases of InN on AlN template. 共a兲A bitmap image presents a time dependence of 具21 ¯ 1 ¯ 0典RHEED intensity profiles. The height of the image equals 90 s of growth time. 共b兲In-plane lattice parameter extracted by the profile analyses vs thickness of the InN layer. 094902-5 Lebedev et al. J. Appl. Phys. 100, 094902 共2006兲 2H-InN. As a result, some cracks may be introduced into the film. V. LATTICE RELAXATION IN 2H-InN It was generally found that the initial relaxation in the highly mismatched wurtzite III-nitride heterosystem occurred via a three-step growth mechanism consisting of the following: the formation of planar platelets, their partial coalescence, and, finally, the introduction of dislocations, most likely at the edges of the platelets.16,24,30,44,45 A similar model was used successfully to explain 2D PIMBE of 2H-AlN on Si共111兲共Ref. 46兲and 共001兲共Ref. 47兲substrates. It is also important to note that, in the case of 2H-InN/AlxGa1−xN heterosystems, all the steps presume different mechanisms of strain relaxation. The growth of 2H-InN involves deposition of a film, which is lattice mismatched with respect to any available template. Assuming two perfectly rigid relaxed crystals, the film/template misfit parameter fis equal to −0.11 and −0.14 for InN/GaN and InN/AlN heterosystems, respectively. Such systems cannot be grown pseudomorphically.48 In systems with large misfit parameters 共depending on the elastic moduli of the film, f⬎4%–5%兲most of the lattice mismatch is accommodated by misfit dislocations 共MDs兲, which are physically distinct from the classical van der Merwe–Matthews-type MDs.49 In InN/AlxGa1−xN, one may observe an approximate matching of m兵011 ¯ 0其InN planes of the film with n兵011 ¯ 0其template planes of the template. Extra half planes define geometrical misfit dislocations 共GMDs兲, 60° dislocations with band line parallel to 具21 ¯ 1 ¯ 0典directions and lying on the 共0001兲basal plane, which exist in a high percentage from the very start of the film growth 关Fig. 6共b兲兴. Besides the accommodation of the initial misfit by GMDs, there is another way to relax misfit strain destabilizing the growth front by producing surface modulation or eventually forming islands. Moreover, on the planar top surface of each platelet, the layer-by-layer growth mode can be achieved if the conditions in Eq. 共1兲are satisfied, similar to what has been observed for AlN/GaN and GaN/AlN 2H heterosystems.16 The corresponding term in Eq. 共1兲,⌬EM, describes the reduction in the total energy of the system due to periodic undulations at the growing surface.39 It was also reported16,44 that the nucleation step results in a second-order elastic relaxation of the epilayer within the nucleated nanoplatelets of slightly different orientations 关see Figs. 4共c兲and 5共b兲, phase A兴. In the case of the InN/AlN共0001兲heterosystem, their lateral sizes reach diameters of ⬃50 nm with heights of 10–15 nm before the coalescence takes place 共see Fig. 7兲. As we will discuss later on, due to observed relaxation, the residual biaxial strain in the nanoplatelets is not large enough to introduce dislocations into the growing islands on the stage of 3D growth. Addi- FIG. 6. Epitaxial relationship and atomic-plane relationship at the AlN/InN heterointerfaces: 共a兲Bright-field XTEM micrograph of a 2.2 ␮ m2H-InN layer grown on an AlN/Al2O3template 共general view兲; inset: 具21 ¯ 1 ¯ 0典SAED pattern showing sapphire, AlN and InN related aligned reflection spots; the following epitaxial relationships were observed: 具011 ¯ 0典Al2O3/具21 ¯ 1 ¯ 0典AlN/ 具21 ¯ 1 ¯ 0典InN and 具0001典Al2O3/具0001典AlN/具0001典InN. 共b兲HRTEM image in the 具21 ¯ 1 ¯ 0典zone axis; inset: The fast Fourier filtered HRTEM image shows the 共011 ¯ 0兲net planes of AlN and InN demonstrating the match between template and epilayer. FIG. 7. HRTEM image of a 2H-InN island nucleated on an AlN template. 094902-6 Lebedev et al. J. Appl. Phys. 100, 094902 共2006兲 tionally, it has been shown by Cantu et al.50 by means of TEM and XRD analysis that inclined edge TDs also contribute to the initial relaxation process by a TD line redirection at the interface. In our particular case, TEM investigations indicate that the TDs penetrating from the template into the InN layer are indeed inclined providing a MD component 共see Fig. 8兲. The next growth step, the partial coalescence of islands 关see Figs. 4共c兲and 5共b兲, phase B兴, leads to the formation of a high density of either phase boundaries 共PBs兲关Fig. 8共a兲兴共in the case of the occasional existence of a metastable 3C- InN polytype兲or domain boundaries 共DBs兲关Figs. 8共b兲and 8共c兲兴. The efficiency of the coalescence and the surface smoothing depend critically on the stoichiometry of the deposited species and therefore affected by the actual supersaturation. The partial coalescence may cause an additional undesirable effect: the immobilization of incipient dislocations generated from lattice misfit strain relaxation within pinholes between islands. This dislocation “trapping” effect can inhibit dislocation annihilations 关see Fig. 8共d兲兴, which requires dislocation gliding.44 In general, the partial character of the coalescence presumes that 2H-InN epilayers have a pronounced domain structure causing variations in the crystal quality as well as in the carrier transport properties of the film. The coalescence is followed by the gradual introduction of secondary misfit dislocations 共SMDs兲due to the rising biaxial stress energy in the continuous epilayer. These SMD segments with lines propagating parallel to the interface have associated dislocation portions threading towards the surface, namely, “secondary” TDs, whose density changes with the thickness and will be designated in a later section as D2共d兲. Note that the SMD portions will be able to relieve a major component of the actual thickness-dependent residual stress only by the glide of the secondary TD in the most favorable slip systems 共Matthews-Blakeslee mechanism兲.33 This mechanism can play a dominant role, if 2D growth conditions 共layer by layer or step flow growth兲are reached by an appropriate choice of In supersaturation and/or substrate temperature.15 In general, depending on the sign and the order of the remaining strain, other mechanisms could participate to relax biaxial strain in the epilayer, e.g., the propagation of cracks 共tensile stress兲or film decohesion 共compressive stress兲.16 All mechanisms mentioned above are of high importance for epitaxy of high-quality InN layers and will be discussed in the following sections. VI. DISCUSSION A. Plastic relaxation by geometrical misfit dislocations Shown in Fig. 6, TEM and HRTEM images demonstrate a cross-sectional overview, and an atomic-plane relationship at the AlN/InN heterointerface. Fast Fourier filtered HRTEM images 共see inset兲demonstrate the fit between the 兵011 ¯ 0其net planes of InN and AlN displayed along the 具21 ¯ 1 ¯ 0典 direction. Note that similar relationships were found in the InN/GaN interface. The derived approximate ratio, m:n, for the 兵011 ¯ 0其planes InN/AlN is 57:64, as followed from the statistical HRTEM analyses. Consequently, the primary mismatch, mainly accommodated by a series of GMDs, is 兩共n·dAlN-m·dInN兲兩/n·dAlN=−0.013 for InN/AlN. Thus, the FIG. 8. TEM micrographs of typical phases, boundaries, and dislocations types in 具21 ¯ 1 ¯ 0典AlN/共GaN兲/InN heterosystems. 共a兲3C-InN metastable phase inclusion along the 具011典zone axis placed on AlN 关insets: fast Fourier transform 共FFT兲spectra of InN 共top兲InN/AlN heterointerface 共middle兲and InN 共bottom兲are similar to the corresponding SAED patterns and demonstrate the cubic nature of 3C-InN as well as the hexagonal crystallography of 2H-AlN兴.共b兲HRTEM image of a 2D DB between two tilted 2H-InN domains placed on AlN template; 共c兲2D DB network in a 2H-GaN epilayer between AlN and InN structures, after the partial coalescence of domains 共triangles point out to 2D boundaries兲.共d兲Dark-field 共DF兲, two-beam 共2B兲 conditions using the 0002 reflection allow us to image a TD network in the bulk of hexagonal InN grown on AlN. The density of TDs decreases as the layer becomes thicker: 共1兲and 共2兲exemplify TD trapping and/or annihilation. 094902-7 Lebedev et al. J. Appl. Phys. 100, 094902 共2006兲 line density of dislocations in InN near the AlN/InN heterointerface extracted from HRTEM images is DI⬃共4±0.5兲 ⫻106cm−1. The introduction of MDs parallel to 具112 ¯ 0典ac- commodates the main part of the mismatch 共⬃12.2 from the total ⬃13.5%兲in the InN/AlN heterosystem. We propose that those MDs detected by HRTEM in 具21 ¯ 1 ¯ 0典directions should be mainly GMDs plus a minor contribution of the SMDs 共with a line parallel to the 具21 ¯ 1 ¯ 0典direction and b =1/3具112 ¯ 3典兲 formed for relaxing the biaxial strain at the heterointerface.33 The very regular spacing between misfit dislocations only could be explained by a plastic relaxation through GMD generation, since the reaccommodation of misfit dislocations by slip in the basal plane is not very probable. The corresponding values of the calculated residual stress at different growth stages measured by different methods are listed in Table I. Here, Refs. 51 and 52 have been used as a source for InN elastic constants and stress calculation formalism, respectively. As we commented, the plastic relaxation degree measured dynamically by RHEED is smaller than those derived from HRTEM. Three possible reasons can explain this lack of correspondence. 共i兲The atomicplane ratio varies along the interface being a function of the local growth conditions; 共ii兲the initial ratio has been changed due to the recrystallization of the domains during coalescence or postcoalescence phase; and 共iii兲either additional dislocations or cracks form in the epilayer due to the biaxial strain induced by the thermal mismatch arising during the cooling of the sample. On the other hand, the RHEED observations represent the data integrated over, at least, 3 mm2InN surface and can be used as an averaged value. In both cases, however, the magnitude of the residual misfit is small enough to allow the continuation of the second-order elastic relaxation of the epilayer within the nucleated nanoplatelets. Moreover, the influence of the nucleation phase on the electrical and transport properties of the thick 共⬎400–500 nm兲InN film is not significant. As we will discuss later on, the quality of the InN epilayer is determined mainly by the coalescence. B. Elastic relaxation through island growth After the primary relaxation of the InN domains via introduction of GMDs, but before a plastic release of the strain energy via formation of a threading dislocation network, an elastic relaxation of islands takes place. Approximated analytical solutions for the equilibrium stresses and strains have been discussed in the literature for a periodic set of thin deposites.16,51 It was shown that the two facing crystals no longer share the natural misfit, but share a so-called active misfit involving the surface stresses of the substrate. This surface stress may enhance, compensate, or decrease the epitaxial misfit. Moreover, the island relaxes by its free borders so that the lateral stress becomes height dependent. In the frame of a 2D thick ribbon model suggested by Kern and Müller,51 a simple isotropic approximation for the relaxation of the topmost layers of the 2H-InN islands can be obtained. The model assumes that the strain in an infinitely long ribbon of height hand width l, deposited coherently on a substrate having a misfit f0, is given by Iyy共x,N兲=Iyy共x,0兲具M1典N−1,共5兲 具M1典=1−2 ␲ a l+ 冉 1+冑2 ␲ a l 冊 2 exp 冉 −冑2l a ␲ 冊 ,共6兲 where Ndefines the number of monolayers and varies from 0 to h/a—the total number of monolayers included in the ribbon of height h. Based on these approximations, we can interpret the experimental results obtained by AFM and in situ RHEED analyses to calculate a strain relief provided by the 3D growth. Assuming a rigid substrate with f0=−0.0411 共primary misfit is partly relaxed by GMDs兲and a typical island size of ⬃50 nm 关see Figs. 1共b兲and 8兴, which is large compared to atomic distances; an approximate value of ⬃4% for the strain relief through the island formation can be obtained. This value fits well to the RHEED observations 关see Fig. 5共b兲, phase B兴showing similar changes in the in-plane lattice parameter during the 3D growth phase. The elastic strain relief occurring within the island can be reversible16 or might be compensated during the coalescence. However, the observed relaxation has an obvious consequence for the growth. It prevents the formation of a dislocation network inside the growing islands. The residual strain in 2H-InN determines the characteristic size of the growing islands, preventing the growth of large area domains during the nucleation stage. As one can see in Fig. 9, for thin platelets 共⬃10 ML兲, the onset of sig- FIG. 9. Calculated in-plane strain due to the island growth as a function of characteristic island size lfor different thicknesses. 094902-8 Lebedev et al. J. Appl. Phys. 100, 094902 共2006兲