Strain Driven Shape Evolution of Stacked (In,Ga)N Quantum Disks Embedded in GaN Nanowires

The fabrication of nanowires with axial multiquantum wells or disks presenting a homogeneous size and shape distribution along the whole stack is still an unresolved challenge, despite being essential for narrowing their light emission bandwidth. In this work we demonstrate that the commonly observed change in the shape of the disks along the stacking direction proceeds in a systematic, predictable way. High- resolution transmission electron microscopy of stacked (In,Ga)­N quantum discs embedded in GaN nanowires with diameters of ∼40 nm and lengths of ∼700 nm and finite element method calculations show that, contrary to what is normally assumed, this change is not related to the radial growth of the nanowires, which is shown to be negligible, but to the strain relaxation of the whole active region. A simple model is proposed to account for the experimental observations. The model assumes that each disk reaches an equilibrium shape that minimizes the overall energy of the system, given by the sum of the surface and strain energies of the disk itself and the barrier below. The strain state of the barrier is affected by the presence of the disk buried directly below in a way that depends on its shape. This gives rise to a cumulative process, which makes the aspect ratio of each quantum disk to be smaller compared to the disk grown just before, in qualitative agreement with the experimental observations. The obtained results imply that strain relaxation is an important factor to bear in mind for the design of multiquantum disks with controlled shape along the stacking direction in any lattice mismatched nanowire system

Summary of boundary conditions.

<p>Summary of boundary conditions.</p

Comparison between results from the PDE model and cells.

<p>Simulated amounts of the different species were generated by running the model for 600 s using constants and expressions in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-t001" target="_blank">Tables 1</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-t002" target="_blank">2</a> and subsequently plotted. Results from cellular experiments show mean SEM, .</p

Extra and intra cellular profiles of BPDE and its metabolites obtained from the PDE model.

<p>The model was run using constants and expressions as described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-t001" target="_blank">Tables 1</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-t002" target="_blank">2</a> and the different species subsequently plotted.</p

Comparison between the compartment model and the PDE model.

<p>The parameters were changed according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-t006" target="_blank">Table 6</a> to describe enhanced diffusion and reactivity of PAH DE (). The panels are ordered in the same way as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-g007" target="_blank">Figure 7</a>.</p

Schematic diagram showing the two step process of iterative homogenization.

<p>The first small scale homogenization assumes ideal layered structures representing the membranes (i.e. periodic homogenization; right cube). The second step assumes that these layered structures are tightly packed, with all orientations equally probable, into a model representative subdomain (left cube). A more detailed view is provided in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-g003" target="_blank">Figure 3</a> Together these steps allow for an efficient and accurate derivation of effective equations governing the diffusion and reactions in the cytoplasm.</p

Definition of the computational domain.

<p>The missing values have been computed as follows. The thickness of the membranes, , has been determined by multiplying by the relative thickness of the nuclear membrane from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-t002" target="_blank">Table 2</a>. Then, and . For , the amount of cell medium per cell has been computed. is the radius of a ball with that volume.</p><p>The nucleaus is described by , the nuclear membrane by , the cytoplasm by , the cell menbrane by , and the extracellular medium by .</p

Sketch of membrane diffusion setting.

<p>A substance with concentrations in compartment I and in compartment II is diffusing through a membrane with thickness .</p

Parameters varied.

a<p>used as baseline values.</p><p>Parameters varied to study the effect on GSH conjugation and DNA-adduct formation.</p

Sketch of compartment system with well-stirred compartments.

<p>In the cytoplasm, the effective quantities are used. Cell and nuclear membrane are handled as sketched in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0023128#pone-0023128-g004" target="_blank">Figure 4</a>.</p