Decomposition, diffusion, and growth rate anisotropies in self-limited profiles during metalorganic vapor-phase epitaxy of seeded nanostructures

We present a model for the interplay between the fundamental phenomena responsible for the formation of nanostructures by metalorganic vapor phase epitaxy on patterned (001)/(111)B GaAs substrates. Experiments have demonstrated that V-groove quantum wires and pyramidal quantum dots form as a consequence of a self-limiting profile that develops, respectively, at the bottom of V-grooves and inverted pyramids. Our model is based on a system of reaction-diffusion equations, one for each crystallographic facet that defines the pattern, and include the group III precursors, their decomposition and diffusion kinetics (for which we discuss the experimental evidence), and the subsequent diffusion and incorporation kinetics of the group-III atoms released by the precursors. This approach can be applied to any facet configuration, including pyramidal quantum dots, but we focus on the particular case of V-groove templates and offer an explanation for the self-limited profile and the Ga segregation observed in the V-groove. The explicit inclusion of the precursor decomposition kinetics and the diffusion of the atomic species revises and generalizes the earlier work of Biasiol et al. [Biasiol et al., Phys. Rev. Lett. 81, 2962 (1998); Phys. Rev. B 65, 205306 (2002)] and is shown to be essential for obtaining a complete description of self-limiting growth. The solution of the system of equations yields spatially resolved adatom concentrations, from which average facet growth rates are calculated. This provides the basis for determining the conditions that yield self-limiting growth. The foregoing scenario, previously used to account for the growth modes of vicinal GaAs(001) and the step-edge profiles on the ridges of vicinal surfaces patterned with V-grooves during metalorganic vapor-phase epitaxy, can be used to describe the morphological evolution of any template composed of distinct facets.

Figure 1

SEM plan view after the growth of QDs on a (111)B substrate patterned with inverted pyramids. (a) Three regions on the substrate: I, a regularly patterned area with partially filled inverted pyramidal recesses (5 $\mu$m pitch); II, a planar unprocessed area with irregular 3D growth; and III, a planar unprocessed area with a mirror-like (no-growth) surface. Inset in (a): Enlarged cross section of region II; (b) SEM image of pyramids located inside the patterned region, and (c) SEM image of pyramids located near the boundary of the patterned region.

Figure 2

Representative AFM cross-sectional image of an AlGaAs/GaAs multilayer grown by MOVPE at the indicated temperatures on a substrate with a3 $\mu$m (from A to B) pitch pattern. The evolution of the boundary between the (100) and (111)A planes is indicated by the dotted line. The contrast between AlGaAs and GaAs is given by the different height signal of the measuring AFM tip, due to a thin oxide layer grown upon AlGaAs.

Figure 3

Arrhenius analysis of the ratio between (100) and (111)A corresponding layers thicknesses of the Al${}_{0.3}$Ga${}_{0.7}$As as from description in the text. The different symbols refer to the two different samples shown in Fig. 4 [triangles refer to sample (a) and circles to sample (b)]. The data point indicated by the open circle was excluded from the fit.

Figure 4

Representative cross-sectional SEM images of the two samples analyzed in Fig. 3 (see text).

Figure 5

One-dimensional V-groove profile showing three facets, the (001) at the top and bottom, and the (111)A along the sidewalls. The lengths of the top, side, and bottom facets are $L_t$, $L_s$, and $L_b$, respectively. The origin of the spatial coordinate $x$ is at the center of the bottom facet. The shaded region is the unit cell of the repeating pattern.

Figure 6

Comparison between the self-limiting width $L_b^{*}$ calculated from Eq. (14) with the parameters in Tables and (black, dashed line), the self-limiting width calculated with the model including (001) and (311)A facets (red, solid line, see Sec. 4c) and the experimental data (blue squares) in Ref. 21.

Figure 7

Schematic illustration of the facets near the bottom of a V-groove, as seen in experiments, with the bottom (001) facets bounded by (311)A and (111)A facets. The length of the (311)A facet is denoted by $L_3$. Inset: The self-limiting concentration profiles on the (001) and (311)A oriented facets on the bottom. Segregation in AlGaAs leads to the formation of three narrow Ga enriched branches in the middle of each plane.

Figure 8

Calculated steady state Ga concentration for a model that includes growth on the (011), (311)A, and (111)A facets, which are labeled for the entry for a Ga concentration of 0.8.

Figure 9

Schematic cross section of the bottom of a V-groove sample after the growth of Al${}_x$Ga${}_{1-x}$As. Three layers are sketched. From an initial broad bottom profile (1) the system reaches a self-limited profile (2). Once this profile is reached, for the subsequent layers (3), the bottom width $L_b$ and the angles $\theta$ and $\beta$ remain unchanged for a given Al${}_x$Ga${}_{1-x}$As concentration. The ratio between the growth rates $R_b$ [bottom (100) facet] and $R_s$ [vicinal (111)A facet] is fixed by the geometric constraints.

Figure 10

Variation of the vicinal (111)A sidewall planes angle $\theta$ as a function of growth temperature in Al${}_x$Ga${}_{1-x}$As, with $x=0$ (squares), $x=0.3$ (circles), and $x=0.45$ (triangles). Temperature values have been renormalized from those reported in Ref. 22 to show the estimated real temperature.