Lattice Strain Relaxation and Compositional Control in As-Rich GaAsP/(100)GaAs Heterostructures Grown by MOVPE

The fabrication of high-efficiency GaAsP-based solar cells on GaAs wafers requires addressing structural issues arising from the materials lattice mismatch. We report on tensile strain relaxation and composition control of MOVPE-grown As-rich GaAs1−xPx/(100)GaAs heterostructures studied by double-crystal X-ray diffraction and field emission scanning electron microscopy. Thin (80–150 nm) GaAs1−xPx epilayers appear partially relaxed (within 1−12% of the initial misfit) through a network of misfit dislocations along the sample [011] and [011−] in plane directions. Values of the residual lattice strain as a function of epilayer thickness were compared with predictions from the equilibrium (Matthews–Blakeslee) and energy balance models. It is shown that the epilayers relax at a slower rate than expected based on the equilibrium model, an effect ascribed to the existence of an energy barrier to the nucleation of new dislocations. The study of GaAs1−xPx composition as a function of the V-group precursors ratio in the vapor during growth allowed for the determination of the As/P anion segregation coefficient. The latter agrees with values reported in the literature for P-rich alloys grown using the same precursor combination. P-incorporation into nearly pseudomorphic heterostructures turns out to be kinetically activated, with an activation energy EA = 1.41 ± 0.04 eV over the entire alloy compositional range.


Introduction
Multi-junction (tandem) semiconductor solar cells in the form of stacked singlejunction cells, each absorbing a different interval of the solar spectrum, allow for external quantum efficiencies beyond the Shockley-Queisser limit for single-junction cells [1][2][3]. Tandem solar cells based on a crystalline silicon (Si) bottom junction are very attractive due to the relative low cost of Si. A dual-junction cell with a 1.7 eV top junction based on GaAsP and a Si (1.12 eV) bottom cell raises the theoretical power conversion efficiency (PCE) of the tandem cell above 30%. A common approach to the fabrication of such tandem cells is the direct monolithic growth of the III-V cell onto the Si bottom cell. However, structural constraints between III-V compounds and Si (e.g., the combined effects of lattice, thermal, and crystal polarity mismatches) limit the performances of such cells; despite the tremendous improvements in III-V/Si heteroepitaxy over recent years [4,5], solar PCEs remain far from theoretical figures [6,7].
Alternative approaches are being studied to overcome these limitations. Among others, the combination of a top cell based on free-standing III-V nanowires with a planar Si bottom

Materials and Methods
GaAs 1−x P x thin epilayers were grown on device-quality vertical gradient freeze (100)GaAs substrates (Wafer Technology, Milton Keynes, UK) by low (50 mbar) pressure MOVPE using an Aix 200RD reactor (Aixtron, Herzogenrath, Germany). Trimethylgallium (Me 3 Ga), tertiary-butyl-arsine ( t BuAsH 2 ), and tertiary-butyl-phosphine ( t BuPH 2 ) (Dockweiler Chemicals, Marburg, Germany) were employed as Ga, As, and P precursors, respectively. Before loading into the reactor, the substrates were cleaned in isopropanol vapors for 1 h, etched in a H 2 SO 4 :H 2 O 2 :H 2 O (4:1:2) solution for 8 min at around 40 • C, thoroughly rinsed in de-ionized water, and finally dried under pure N 2 . In-situ annealing of the substrates was then performed for 10 min at 625 • C under a H 2 + t BuAsH 2 atmosphere to desorb oxides, organic residuals, and other contaminants from the GaAs surface. A thin (7 nm) GaAs epilayer was grown at the same temperature to reconstruct the substrate surface before GaAsP growth. Upon completion of the GaAs epilayer growth, the reactor temperature was lowered to 600 • C in a H 2 + t BuAsH 2 flow. GaAsP epilayers were grown under a fixed Me 3 Ga molar flow of 12.3 µmol/min to ensure the same growth rate (~0.085 nm/s) for all samples. Different concentrations of V-group elements in the vapor were adopted to study the effects on GaAsP composition: the vapor stoichiometry x v = [ t BuPH 2 ]/([ t BuPH 2 ] + [ t BuAsH 2 ]) was varied between 0.46 and 0.60, while the V:III ratio was fixed at 20:1 or 40:1. The growth time was in the 15-30 min range, so to obtain GaAsP thickness around one hundred nm.
The sample surface morphology was investigated by field-emission scanning electron microscopy (FESEM) in-plan observations using a Sigma VP (Zeiss, Oberkochen, Germany) microscope equipped with a Gemini-1 electron column and a primary electron beam energy of 20 keV. To this end, secondary (SE) or backscattered electron (BSE) signals were employed.
The microstructural properties of GaAs 1−x P x alloy epilayers were investigated by X-ray diffraction. The measurements were carried out using an Empyrean diffractometer (Malvern-Panalytical, Malvern, UK) in a high-resolution double-crystal (HRDC) configuration. A Cu-target was employed as an X-ray source, and a 4-bounce Ge-crystal Bartels monochromator-collimator (symmetrical (220)-reflections) with an angular divergence of about 14 arcsec was employed as X-ray incidence optics. All measurements were carried out using a "wide open" PIXcel detector. The strain state and alloy composition of GaAs 1−x P x epilayers were determined by recording symmetrical and asymmetrical HRDC measurements in the vicinity of the (400) and (422) lattice points, respectively. In order to account for a possible offcut of the substrate surface or a tilt/rotation of the epilayer with respect to the substrate, all measurements were performed at four azimuthal angle settings (i.e., sample rotations about the surface normal), namely ϕ = 0, π/2, π, and 3/2π, corresponding to the X-ray scattering plane along the in-plane 110 directions. For the asymmetric (422) reflection, the geometrical configuration with high incidence and glancing exit angles, i.e., direction cosines γ 0 > |γ h |, was chosen because this configuration is very sensitive to in-plane strain components and lattice relaxation.

Results and Discussion
3.1. Determination of Lattice Strain and Alloy Composition of GaAs 1−x P x Epilayers from HRDC Figure 1 shows the HRDC patterns around the symmetrical (400) and asymmetrical (422) reflections of two GaAs 1−x P x /(100)GaAs heterostructures having different phosphorous mole fractions x. Patterns measured for the different azimuthal angles (not reported here) demonstrate that all epilayers are nearly pseudomorphic. Measured values of GaAs 1−x P x lattice strain parallel to the (100) interface plane ( ) are summarized in Table 1 for the studied samples. The lattice mismatch (f ) and mole fraction x in the Table were calculated by using the relations of the second-order approximation for 100 -oriented zinc-blende heterostructures [26]. For the calculations, the lattice parameters and elastic constants of GaAs and GaP reported by Adachi [27] were used. The samples show different amounts of strain and thus different degrees of plastic relaxation δ ≡ f − /f within the 1−12% range. Figure 2a shows a FESEM surface micrograph of Sample C (δ = 11.3%) obtained using SE imaging (sensitive to surface morphology); it shows the presence of mutually perpendicular (i.e., along the sample [011] and [011] in-plane directions) undulations of the epilayer surface, so-called cross-hatch morphology, observed in low-misfit GaAsP layers grown at relatively low temperatures [28]. The presence of cross-hatch is explained as the combination of strain relaxation by dislocation nucleation at surface steps (and their subsequent glide into the epilayer) and growth by surface step flow, which tends to smooth down the steps [29]. FESEM observations of the same surface region using BSE imaging show dense patterns of well-resolved mutually perpendicular dark lines (Figure 2b), corresponding (both in position and alignment) to the surface undulations in Figure 2a. As the samples are compositionally homogeneous and BSE imaging is less sensitive to surface features, these lines originate most likely from electron channeling contrast imaging (ECCI) associated with crystal defects (e.g., dislocations) [30][31][32]. Besides the surface cross-hatch, we observed the seldom occurrence of faceted trenches (FTs) (inset of Figure 2a) aligned along the [011] direction with lengths varying between a few microns and several hundred microns. The latter have been associated with the formation of a micro-twin at the FT cusp [25]; interestingly, a strong BSE imaging contrast is observed in Figure 2b at a FT location. In comparison, BSE micrographs of Sample A (not reported here) show a negligible (although not null) density of dark lines, in agreement with the reduced plastic relaxation of this sample (δ = 1.78%).   Figure 2a shows a FESEM surface micrograph of Sample C ( ∥ = 11.3%) obtaine using SE imaging (sensitive to surface morphology); it shows the presence of mutuall perpendicular (i.e., along the sample 011 and 011 in-plane directions) undulations o the epilayer surface, so-called cross-hatch morphology, observed in low-misfit GaAs layers grown at relatively low temperatures [28]. The presence of cross-hatch is explaine as the combination of strain relaxation by dislocation nucleation at surface steps (an their subsequent glide into the epilayer) and growth by surface step flow, which tends t smooth down the steps [29]. FESEM observations of the same surface region using BS imaging show dense patterns of well-resolved mutually perpendicular dark lines (Figur 2b), corresponding (both in position and alignment) to the surface undulations in Figur 2a. As the samples are compositionally homogeneous and BSE imaging is less sensitive t surface features, these lines originate most likely from electron channeling contrast im aging (ECCI) associated with crystal defects (e.g., dislocations) [30][31][32]. Besides the su face cross-hatch, we observed the seldom occurrence of faceted trenches (FTs) (inset o Figure 2a) aligned along the 011 direction with lengths varying between a few micron and several hundred microns. The latter have been associated with the formation of micro-twin at the FT cusp [25]; interestingly, a strong BSE imaging contrast is observed i  Table 1. Composition (x) and elastic strain parallel to the hetero-interface ( ) measured through HRDC for the investigated GaAs 1−x P x /(100)GaAs heterostructures. Values of the epilayer thickness (h) and calculated lattice misfit (f ) are also reported for each sample.

Sample
GaAs * Defined as f = a GaAs − a /a and = a − a /a, where a GaAs is the GaAs bulk lattice parameter, whilst a and a are the GaAs 1−x P x bulk and strained lattice parameters in the direction parallel to the heteroepitaxial interface, respectively.   Figure 3a reports the GaAs1−xPx thickness (Table 1) along with values of the critical thickness ( ℎ ) for strain relaxation calculated based on the equilibrium theory of Matthews-Blakeslee [33] (Appendix A) as a function of alloy compositions. It appears that the epilayer thickness is beyond the corresponding ℎ value, in qualitative agreement with the sample partial relaxation observed by HRDC. However, more compelling information on strain relaxation behavior in present heterostructures can be obtained by comparing the calculated Matthews-Blakeslee residual strain ∥ for ℎ ℎ (Equation (A2)) with that measured in our samples as a function of the epilayer thickness, as shown in Figure 3b. The diagram clearly shows that the present epilayers are less plastically relaxed (metastable) than expected based on the equilibrium theory. This is a common experimental finding in mismatched heterostructures grown on high-crystalline-quality substrates (VGF-grown GaAs in our case), i.e., whenever the substrate TD density is not large enough to generate the required amount of plastic relaxation; new misfit dislocations must be then nucleated during the growth, a process limited by energy balance or kinetic barriers. The first case was proposed by People and Bean [34], who estimated the energy threshold for the generation of screw dislocations in a strained epilayer (Appendix A), despite the fact that such dislocations cannot relax elastic strain. Figure 3a reports the critical thickness for strain relaxation as a function of GaAs1−xPx composition based on the People-Bean model (Equation (A3)): the as-estimated values of ℎ appear indeed much larger than those calculated from Matthews-Blakeslee theory and well beyond the thickness of our partially relaxed epilayers. Plastic relaxation has been described by Marée et al. [35] in terms of surface nucleation and expansion into the epilayer of dissociated half-loops, taking into account the work done by the elastic stress field acting on expanding loops. This model was found to agree fairly well with experimental strain relaxation data in compressively-strained heterostructures [36].  Figure 3a reports the GaAs 1−x P x thickness (Table 1) along with values of the critical thickness (h c ) for strain relaxation calculated based on the equilibrium theory of Matthews-Blakeslee [33] (Appendix A) as a function of alloy compositions. It appears that the epilayer thickness is beyond the corresponding h c value, in qualitative agreement with the sample partial relaxation observed by HRDC. However, more compelling information on strain relaxation behavior in present heterostructures can be obtained by comparing the calculated Matthews-Blakeslee residual strain for h > h c (Equation (A2)) with that measured in our samples as a function of the epilayer thickness, as shown in Figure 3b. The diagram clearly shows that the present epilayers are less plastically relaxed (metastable) than expected based on the equilibrium theory. This is a common experimental finding in mismatched heterostructures grown on high-crystalline-quality substrates (VGF-grown GaAs in our case), i.e., whenever the substrate TD density is not large enough to generate the required amount of plastic relaxation; new misfit dislocations must be then nucleated during the growth, a process limited by energy balance or kinetic barriers. The first case was proposed by People and Bean [34], who estimated the energy threshold for the generation of screw dislocations in a strained epilayer (Appendix A), despite the fact that such dislocations cannot relax elastic strain. Figure 3a reports the critical thickness for strain relaxation as a function of GaAs 1−x P x composition based on the People-Bean model (Equation (A3)): the as-estimated values of h c appear indeed much larger than those calculated from Matthews-Blakeslee theory and well beyond the thickness of our partially relaxed epilayers. Plastic relaxation has been described by Marée et al. [35] in terms of surface nucleation and expansion into the epilayer of dissociated half-loops, taking into account the work done by the elastic stress field acting on expanding loops. This model was found to agree fairly well with experimental strain relaxation data in compressively-strained heterostructures [36].

Analysis of Epilayer Strain Relaxation
For sufficiently thick epilayers, the observed dependence of the residual strain with thickness can be fitted by the semi-empirical power-law function where m = 1 for the Matthews-Blakeslee theory and m = 1/2 for the energy balance models [34,35] (Appendix A). The latter value is in good agreement with relaxation data for compressively strained metastable heterostructures [36,37]. Best-fitting of experimental data in Figure 3b with Equation (1) returned instead, a value m = 0.671 ± 0.046 (i.e., m~2/3). This finding suggests a relaxation rate behavior intermediate between that of Matthews-Blakeslee and the half-loop nucleation models; indeed, a larger proclivity toward plastic relaxation is expected for tensile-strained epilayers with respect to compressive ones.  For sufficiently thick epilayers, the observed dependence of the residual strain with thickness can be fitted by the semi-empirical power-law function where m = 1 for the Matthews-Blakeslee theory and m = 1/2 for the energy balance models [34,35] (Appendix A). The latter value is in good agreement with relaxation data for compressively strained metastable heterostructures [36,37]. Best-fitting of experimental data in Figure 3b with Equation (1) returned instead, a value m = 0.671 ± 0.046 (i.e., m~ 2/3). This finding suggests a relaxation rate behavior intermediate between that of Matthews-Blakeslee and the half-loop nucleation models; indeed, a larger proclivity toward plastic relaxation is expected for tensile-strained epilayers with respect to compressive ones. Finally, we estimate the apparent critical thickness (ℎ ) for strain relaxation of GaAsP/(100)GaAs heterostructures as a function of alloy composition by imposing the pseudomorphicity condition ∥ to the quantity ∥ ℎ (Equation (1)) best fitting our experimental data. Figure 3a shows that the Matthews-Blakeslee ℎ values lie below the ℎ curve, while all experimental points lie above it. Clearly, the ℎ curve represents an upper bound to epilayer pseudomorphicity in reason of the limited resolution (1 × 10 −4 ) of HRDC strain measurements; indeed, the absence of measurable strain does not imply that misfit dislocations are not present, as they would be generated as soon as the energy conditions allow it, that is, well before strain relaxation becomes appreciable. In this sense, electron microscopy observations (e.g., through ECCI) of individual dislocations are necessary to verify whether the onset of relaxation coincides with that of Matthews-Blakeslee or if it occurs at a larger thickness.
To date, studies on the structural properties of step-graded GaAs1−xPx buffer layers for solar cell applications have predominantly focused on the evaluation of TD and FT densities as functions of the step compositional height and grading rate (i.e., the compo- of HRDC strain measurements; indeed, the absence of measurable strain does not imply that misfit dislocations are not present, as they would be generated as soon as the energy conditions allow it, that is, well before strain relaxation becomes appreciable. In this sense, electron microscopy observations (e.g., through ECCI) of individual dislocations are necessary to verify whether the onset of relaxation coincides with that of Matthews-Blakeslee or if it occurs at a larger thickness.
To date, studies on the structural properties of step-graded GaAs 1−x P x buffer layers for solar cell applications have predominantly focused on the evaluation of TD and FT densities as functions of the step compositional height and grading rate (i.e., the compositional change per unit thickness of the grown alloy): a lower grading rate was shown necessary with increasing x to maintain the density of FTs low and reduce the TD density [24,25]. However, no particular attention was paid to the actual strain relaxation within each of the buffer grading steps in those studies, despite the fact that the actual degree of plastic relaxation would affect the distribution of TDs throughout the final buffer layer. The present findings will help in further optimizing the structural properties of such step-graded GaAsP buffer layers, as well as in properly engineering strain-balanced InGaAs/GaAsP multiple quantum well structures as current-matched light-absorbing medium in monolithic triplejunction InGaAs/GaAs/Ge solar cells [38,39], ultimately leading to better performance III-V tandem solar cells.
Tensile-strained GaAsP layers also find applications in the fabrication of InGaAsP quantum well-based laser diode heterostructures on GaAs for NIR photon emission [40,41]. In this case, the use of an InGaAsP/InGaAsP/GaAsP active region allows for an effective reduction of non-radiative recombination within the device and suppression of carrier leakage with respect to conventional AlGaAs/GaAs heterostructure laser diodes [42]. As the mechanism of degradation in a laser diode is related to the development of dark line defects associated with the generation and multiplication of misfit dislocations within the heterostructure active region, understanding GaAsP relaxation behavior is therefore critical in ensuring suppression/reduction of plastic relaxation within the proposed laser device structures. Figure 4a reports the solid-vapor distribution diagram for the analyzed GaAsP epilayers. It can be clearly observed that the P-composition x in the solid alloy is always below the corresponding content in the vapor (x v ) during the sample growth. The relative distribution of As and P between the two phases is described by the so-called segregation coefficient η defined as [43]

Determination of the Solid-Vapor Segregation Coefficient for GaAs 1−x P x
where N As /N P represents the As to P anion concentration ratio in the GaAsP alloy and [tBuAsH 2 ]/[tBuPH 2 ] is the corresponding precursor concentration ratio in the vapor. η has been shown to depend on the nature of the employed precursors and the growth temperature [43,44]. Furthermore, preferential As (P) segregation was observed for tensile (compressive) strained GaAsP epilayers with respect to fully relaxed ones, a compositional latching phenomenon ascribed to the different radii of As and P anions [28,45].

Conclusions
We reported on tensile strain relaxation and composition control of MOVPE-grown GaAs1−xPx/(100)GaAs heterostructures studied by HRDC X-ray diffraction measurements and FESEM observations. The strain values and alloy P-compositions were measured by HRDC, while FESEM observations proved the presence of misfit dislocations and their effect on the epilayer surface morphology. Thin (80-150 nm) GaAs1−xPx epilayers appear partially relaxed through a network of misfit dislocations along the sample 011 and 011 in-plane directions, giving rise to a cross-hatch surface morphology.
The relaxation rate as a function of epilayer thickness was compared with theoretical predictions from equilibrium (Matthews-Blakeslee) and energy balance models. It was shown that present epilayers relax at a slower rate than predicted by the equilibrium model, an effect ascribed to the existence of an energy barrier to the nucleation of new As x ≡ N P /(N P + N As ), its value can be calculated for a given x v composition of the vapor by the following expression: if the actual value of η is known. We best-fitted the experimental points in Figure 4a with Equation (2) in order to determine η for our experimental conditions, which turned out to be 4.76 ± 0.66. As η > 1, a preferential As incorporation in the GaAsP alloy occurs indeed in our nearly pseudomorphic (tensile-strained) samples. Figure 4b allows us to compare our best-fitting η value with those estimated at lower growth temperatures by Chen et al. [44] for the same V-group precursor combination: the Arrhenius plot shows that the 1/η values align almost perfectly (regression coefficient R = 0.9985), indicating that P incorporation into the crystal is kinetically activated (i.e., increases with the growth temperature), with an apparent activation energy E A = 1.41 ± 0.04 eV, not far from that (1.23 ± 0.05 eV) estimated in ref. [44]. Noteworthy is also that very thin (20−40 nm) GaAs 1−x P x (0.91 < x < 1.0) epilayers were employed by those authors for their estimation, indicating that the observed temperature dependence of η in Figure 4b holds across the entire compositional range.

Conclusions
We reported on tensile strain relaxation and composition control of MOVPE-grown GaAs 1−x P x /(100)GaAs heterostructures studied by HRDC X-ray diffraction measurements and FESEM observations. The strain values and alloy P-compositions were measured by HRDC, while FESEM observations proved the presence of misfit dislocations and their effect on the epilayer surface morphology. Thin (80-150 nm) GaAs 1−x P x epilayers appear partially relaxed through a network of misfit dislocations along the sample [011] and [011] in-plane directions, giving rise to a cross-hatch surface morphology.
The relaxation rate as a function of epilayer thickness was compared with theoretical predictions from equilibrium (Matthews-Blakeslee) and energy balance models. It was shown that present epilayers relax at a slower rate than predicted by the equilibrium model, an effect ascribed to the existence of an energy barrier to the nucleation of new dislocations. A relaxation rate behavior intermediate between that of Matthews-Blakeslee and the halfloop nucleation models is proposed, although further data over a larger compositional interval are needed to confirm this finding.
The analysis of As-rich GaAs 1−x P x alloy composition as a function of V-group precursors ratio and growth temperature allowed to determine the As/P anion segregation coefficient and compare it with previous reports in the literature. P incorporation into the crystal turned out kinetically activated, with an apparent activation energy E A = 1.41 ± 0.04 eV over the entire alloy compositional range.
The present results will help to optimize the design and growth of metamorphic GaAsP/(100)GaAs heterostructures as virtual substrates for the epitaxy of high-efficiency GaAsP-based solar cells and InGaAsP/InGaAsP/GaAsP-based NIR-emitting laser diodes.

Appendix A
Plastic relaxation in strained heterostructures occurs through the generation of a network of dislocation lines (so-called misfit dislocations) lying on the epilayer/substrate interface plane; each misfit dislocation is supposed to be either (i) generated from the stretching and bending of pre-existing (i.e., in the substrate) TDs by the epilayer elastic stress field or (ii) nucleated anew during the epilayer growth. The two mechanisms have been alternatively employed in heterostructure relaxation models based on energy minimization or energy balance considerations, respectively.
According to Matthews-Blakeslee equilibrium theory [33], the critical thickness (h c ) for the onset of plastic relaxation in mismatched heterostructures can be calculated upon minimization of the epilayer total (elastic and plastic) areal energy density, assuming the misfit dislocations are generated by mechanism (i) above, resulting in the equation where ν is the Poisson ratio of the epilayer crystal (for GaAsP alloys, ν ≈ 0.31 [46,47] Given the heterostructure misfit f, the critical thickness h c and elastic strain (h) can be readily calculated from Equations (A1) and (A2), respectively.
Alternative relaxation models have been proposed to estimate the critical thickness in mismatched heterostructures based on energy balance considerations [34,35]. In the People-Bean model, misfit dislocations are generated when the areal elastic energy density of the strained epilayers exceeds the energy density required for the generation of a screw dislocation at the epilayer/substrate hetero-interface; the critical thickness h c can be then estimated by solving the equation [34].
where a(x) is the epilayer lattice parameter, and the other symbols have the same meaning as above. For h > h c plastic relaxation is expected, and the elastic strain decreases according to the expression For large thickness values, this again leads to (h) ∼ h −1/2 . The same functional dependence is obtained by the half-loop dislocation nucleation model [35].