Critical thickness of atomically ordered III-V alloys

Atomic ordering in III-V epitaxial alloys is known to affect physical and optoelectronic properties such as the interatomic bond lengths, band gap, and band splitting.^1,2 However, the influence of ordering on dislocations is less considered. In III-V alloys, the ordered structure is typically metastable and is a result of the surface reconstruction during epitaxy.³ Dislocation glide through the ordered planes disrupts the order pattern, lowering metastability and releasing bulk energy. We have previously calculated the energy of the anti-phase boundary (APB) created in the ordering for standard glissile 60°, (a/2)〈101〉{111}, dislocations in several III-V materials.⁴ Because energy is released through the creation of the APB, dislocations that create APBs receive an additional driving force, enhancing glide. We have also shown that glide in ordered materials results in a skewed distribution of dislocations in metamorphic buffers, with dislocation glide through ordered planes highly preferable.⁵ In addition, reducing the order parameter alters glide energetics and can result in a change in the distribution of active dislocations and formation of new dislocations.

Here, we investigate the implications of atomic ordering on the stability of dislocations in III-V pseudomorphic materials. First, we add a general boundary energy term to the standard critical thickness model to describe the implications of atomic ordering on dislocation stability using a simple model. Then, we experimentally validate that ordering impacts dislocation stability by studying GaInP with low misfit and various degrees of ordering.

The basic critical thickness model, often called the Matthews-Blakeslee critical thickness model,⁶ determines the stability of a dislocation to glide in a strained epitaxial layer by balancing the increase in elastic energy surrounding a dislocation, termed the line energy, with the decrease in misfit energy as a dislocation glides. Both the line energy and misfit energy are functions of the thickness of the growing epilayer, and there is a critical thickness at which the relief of misfit energy begins to dominate and glide is energetically favorable. The model is useful for determining the coherency limits of strained epitaxial layers as a function of the strain and thickness of the epilayer. Several modifications and versions of the model exist,^6–10 and here, we use the derivation from classical elasticity shown in Ref. 10.

As a dislocation moves in an atomically ordered material, it can disrupt the ordering by producing an anti-phase boundary. This change can either raise or lower the bulk energy of the solid, depending on the sign of the mixing enthalpy. Here, we modify the critical thickness model with a general boundary energy for the APB formation. The model is simple, but highlights the implications of the additional boundary energy. Models for more complicated phenomena, such as surface half-loop formation or other critical thickness models, could similarly be modified.

The total work done (W_T) as a dislocation glides is the sum of the line energy (W_d), misfit energy (W_m), and the boundary energy (W_b), per unit misfit dislocation line length

where

$b∥$ and $b⊥$ are the edge components of the Burgers vector parallel and perpendicular to the mismatched interface, respectively, and b_s is the screw component. h is the thickness of the strained layer, r_o is the dislocation core cutoff radius, ε_m is the misfit strain, M_f is the biaxial modulus of the film, v_f is the Poisson ratio of the film, E_b is the boundary energy per unit area in the glide plane, and θ is the angle between the glide plane and mismatched interface. The critical thickness (h_c) for dislocation stability occurs at the value of h, where W_T = 0. The resulting implicit equation for the critical thickness is

In general, the boundary energy can be positive or negative. Figure 1(a) shows the calculated critical thickness for an arbitrary isotropic material with varied E_b, where v_f = 0.3, M_f = 0.87 eV/ ų, and for 60°, (a/2) 〈101〉{111} dislocations with a = 5.65 Å. We use r_o = a/√2 = b, the Burgers vector length. The absolute value of r_o is not well established, and has an impact on the absolute value of h_c but not the relative trends shown here. The anti-phase boundary energy created by glide of a 60° dislocation through perfectly CuPt-ordered GaInP is −9.9 meV/Ų,⁴ near the lower limit of E_b in Figure 1(a). Figure 1(a) shows that the critical thickness is reduced for ordered alloys with negative E_b, and increased for alloys with positive E_b. For negative E_b, a critical thickness still remains for very low values of strain; $hc(εm=0)=−Wd/Ebcscθ$⁠. For positive E_b, the critical thickness approaches infinity with non-zero strain, when $εm=−Ebcscθ/b∥Mf$⁠.

This work is motivated by ordered III-V alloys, which are typically metastable and have a negative E_b. In this case, isolated misfit dislocations glide to the epilayer/substrate interface. This is because the ordered structure is favorable at the epitaxial growth front due to the surface reconstruction, but metastable in the bulk. The other alternative would be a stable ordered structure, such as many bulk metal alloys, where dislocations have a positive E_b. When E_b is positive, the anti-phase boundary created by the glide of a dislocation is unstable. A subsequent dislocation gliding on the same plane then has a negative E_b and is energetically favorable. This causes a pair of dislocations to glide through the material on the same glide plane, a phenomenon termed a superdislocation in ordered metal alloys which has been extensively investigated elsewhere.^11,12 The remainder of the paper will focus on the negative E_b case for ordered III-V alloys.

Figure 1(b) shows the critical thickness for the specific cases of 60° dislocations that create APBs in CuPt-ordered Ga_xIn_1−xP and Ga_xIn_1−xAs grown on (001) GaAs and (001) InP substrates, respectively, and with different relative order parameters, η_rel. We have assumed no substrate offcut for simplicity, and note that only half of the Burgers vectors for typical 60° glissile dislocations create APBs in these systems (half of α-dislocations and half of β-dislocations).¹³ In these cases, the absolute order parameter, η_abs, is a function of the composition, x, and thus strain. For x ≤ 0.5, $ηabs=2ηrelx$⁠; for x > 0.5, $ηabs=2ηrel(1−x)$⁠. The boundary energy in perfectly ordered GaInAs and GaInP, termed $Ebmax$⁠, is −7.5 and −9.9 meV/Ų, respectively.⁴ E_b is modified by η_abs for a given η_rel using $Eb=Ebmaxηabs2$⁠. Standard biaxial moduli and Poisson ratios of the ternary alloys were used.¹⁴ 

The model predicts a significant reduction in the critical thickness of highly ordered III-V alloys. For GaInP/GaAs and GaInAs/InP, the reduction is most significant at low strain, where the order parameter is greatest and the misfit energy is low. Thus, the stability of dislocations to glide in CuPt-ordered GaInP/GaAs and GaInAs/InP with low misfit is significantly reduced. Indeed, a critical thickness exists at very low strains. At larger strains, W_b/W_m decreases, and so the critical thickness reduction related to the ordering is less significant. In addition, the absolute order parameter decreases for GaInP(GaInAs) at higher strains, as the composition approaches InP(InAs), and so the absolute value of W_b also decreases.

It is worth pointing out that we are using a simple critical thickness model to illustrate the essential physics of the implications of atomic ordering in III-V alloys on the strain relaxation. The model is isotropic, ignores dislocation interactions,¹⁵ kinetics,⁷ and does not include a surface step energy associated with glide,¹⁶ each of which can significantly impact the critical thickness. Nonetheless, h_c for each of these models is reduced by a negative boundary energy. However, the extent of the reduction depends on its relative importance. Other material properties can also influence glide: notably phase separation and other microstructure.^17,18 Thus, whether atomic ordering has a real impact on glide behavior and the critical thickness needs to be experimentally tested.

To experimentally observe the effect of atomic ordering on the critical thickness, we have grown Ga_0.46In_0.54P epilayers with a low strain, 0.3%, on (001) GaAs substrates miscut 4° to (111)B using atmospheric-pressure metalorganic vapor phase epitaxy (MOVPE). The order parameter is systematically varied by using Sb surfactant, which is known to disrupt the surface dimerization that leads to ordering.^19,20 The growth temperature, V/III ratio, and growth rate were held constant for all growths at 675 °C, 400, and 3 μm/h, respectively, and are conditions that naturally lead to strong ordering in MOVPE-grown GaInP. To determine the order parameter, a set of lattice-matched Ga_0.5In_0.5P with identical growth conditions and surfactants was analyzed by comparing the intensity of the diffracted (113) and ½(113) skew symmetric diffraction spots from high resolution XRD.²¹ Coherent, lattice-matched samples were used to determine the relative order parameter because dislocation glide in ordered lattice-mismatched samples may lower the order parameter. The [110] relaxation, corresponding to α-dislocation glide, was monitored with in situ wafer curvature using a k-Space Associates multibeam optical sensor and techniques described previously.²² 

Figure 2 shows the relaxation throughout growth of three Ga_0.46In_0.54P epilayers. The relative order parameters are 0.4, 0.3, and 0.1, leading to boundary energies of −1.4, −0.8, and −0.1 meV/Ų, respectively. The relaxation process is not smooth, but a clear difference in the relaxation is observed. Epilayers with higher ordering and more negative boundary energies relax sooner during growth, at a higher rate, and have less residual strain after 3 μm of growth. Significant relaxation occurs after only 200 nm of growth for the sample with the strongest ordering, while the onset of significant relaxation is delayed over 1 μm in the sample with the lowest order parameter. In situ wafer curvature is not sensitive enough to directly determine the critical thickness, but qualitatively confirms the modified critical thickness model: strain relaxation occurs earlier in growth when strong ordering is present.

To more closely analyze the formation and glide of initial dislocations, four Ga_0.46In_0.54P samples, grown using the same growth conditions as the above experiment, were analyzed using room temperature cathodoluminescence (CL) on JEOL JSM-7600F and JSM-5800 microscopes equipped with parabolic mirrors for collection and liquid nitrogen cooled germanium detectors. Samples with thicknesses of 300 nm and 750 nm, and boundary energies of −1.4 meV/Ų (η_rel = 0.4) and −0.1 meV/Ų (η_rel = 0.1) were analyzed, where the order parameter was varied using Sb surfactant. Figure 3 shows the cathodoluminescence images. The strong contrast lines are misfit dislocations. Some weak contrast observed across [−110] was observed in Figures 3(a) and 3(c), which was not investigated, but may be due to composition variations or morphology.

A high density of misfit dislocations is observed for both samples with strong ordering, shown in Figs. 3(b) and 3(d), indicating that the critical thickness of nucleation and glide in these samples is less than 300 nm despite the low misfit. The majority of dislocations have [−110] line direction, termed α-dislocations in these compressive III-V materials, and correspond to the same dislocations responsible for relaxing the [110] direction shown in Figure 2. In compressive III-V materials, α-dislocations have group III terminated cores and are commonly found to nucleate and glide prior to β-dislocations, which have [110] line direction. With weak ordering, the 750 nm-thick sample shows a low density of misfit dislocations, and the 300 nm-thick sample shows no misfit dislocations. No dislocations were observed over 0.1 mm² area for the 300 nm-thick sample with Sb surfactant, shown in Figure 3(c). Thus, the critical thickness for the samples with weak ordering is greater than 300 nm and less than 750 nm.

The experimental data fit the relative trends described by the model, but not the absolute values. The modeled critical thickness for disordered Ga_0.46In_0.54P is around 60 nm, whereas misfit dislocations are not observed experimentally until over 300 nm. Indeed, exact correlation is not expected. Discrepancies between the basic critical thickness model and experimental data have been observed previously, and attributed to the experimental resolution in the measurement of critical thickness,²³ assumptions of the model,^9,24 or to kinetic barriers to nucleation and glide.⁷ For instance, dislocation nucleation may have a dependency on E_b or be affected by Sb surfactant. Insertion of E_b into a critical radius model for nucleation would lead to similar trends as the critical thickness model, predicting that nucleation requires more energy in disordered material. Increased activation energy for nucleation in disordered material would also slow relaxation due to the lack of sufficient glissile dislocations, leading to a similar behavior observed in Figure 2. Here, we qualitatively verify the critical thickness model by evaluating both the initial dislocation formation (using CL) and dislocation glide kinetics (using wafer curvature) in order to show that atomic ordering can have a significant impact on dislocation stability in real materials.

The implications of the modified critical thickness are largely for pseudomorphic devices that intend to avoid dislocation formation. The stable thickness of strained III-V alloys is reduced by the presence of atomic ordering. Conversely, intentionally disordering the alloy using surfactants or growth conditions extends the critical thickness and allows access to alloys with greater strain. Interestingly, ordered epitaxial structures such as superlattices or digital alloys, or epilayers with periodic phase separation will also have a boundary energy that may affect the critical thickness. For instance, glide through a superlattice can cause a regular shift in the atomic structure at the superlattice interfaces, and so should be considered. However, we expect that the boundary energy for perfectly ordered GaInP is large in comparison with these other epitaxial structures, since every group-III neighbor is shifted across the GaInP glide plane in perfectly ordered material.

In summary, we modified the critical thickness model with a general boundary energy that can be used to describe the change in bulk energy as a dislocation regularly alters the atomic structure. We evaluated this model using previously calculated values of the boundary energy in metastable CuPt-ordered GaInP and GaInAs. The model predicts a significant lowering of the critical thickness when strong ordering is present, which has implications for pseudomorphic devices. We experimentally verified the model by studying the relaxation of low-misfit GaInP with in situ wafer curvature and ex situ cathodoluminescence imaging, where the order parameter in multiple samples was varied using Sb surfactant. Relaxation indeed occurs earlier in growth in strongly ordered GaInP, and reducing the order parameter using surfactants can be used to extend the critical thickness.

This work was supported by the U.S. Department of Energy under Contract No. DE-AC36-08-GO28308 with the National Renewable Energy Laboratory. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.