Computing Equilibrium Shapes of Wurtzite Crystals: The Example of GaN

Abstract

Crystal morphologies are important for the design and functionality of devices based on low-dimensional nanomaterials. The equilibrium crystal shape (ECS) is a key quantity in this context. It is determined by surface energies, which are hard to access experimentally but can generally be well predicted by first-principles methods. Unfortunately, this is not necessarily so for polar and semipolar surfaces of wurtzite crystals. By extending the concept of Wulff construction, we demonstrate that ECSs can nevertheless be obtained for this class of materials. For the example of GaN, we identify different crystal shapes depending on the chemical potential, shedding light on experimentally observed GaN nanostructures. 68.35.Md,68.47.Fg,81.10.Aj Low-dimensional semiconductor nanostructures have attracted a lot of interest in the past decades, largely due to their applications in low-energy consumption and energyharvesting devices [1,2]. Owing to surface effects, the performance of such devices strongly depends on the nanocrystal morphology. To achieve comprehensive understanding and control of the preferred growth morphology, one must know the material's natural shape that results from its crystallographic anisotropy. Ab initio theory can generally provide more insight into this complicated issue through the calculation of surface energies since, according to Wulff's theorem [3], the equilibrium crystal shape (ECS) of a solid can be constructed by the mere knowledge of surface energies of various crystal planes. For a crystalline solid, the surface energy γ is defined as the excess free energy required to create one unit of surface area A [4],G represents the Gibbs free energy of the system that, neglecting temperature and pressure, is replaced by the total energy. The chemical potential µ i is the free energy per atom in the system for species i, and N i denotes the number of atoms of this species. In a bulk material, the total chemical potential is known from the corresponding total energy E bulk = ∑ i n i µ i , where n i is the number of atoms of species i in the bulk. Hence, the surface energy of a nonpolar plane can be extracted from density-functional-theory (DFT) results for a slab that contains two identical surfaces well separated from each other. For some polar and semipolar planes, however, individual surface energies are difficult to access, because different facets may appear at the two surfaces of the slab. To overcome this problem, a method has been proposed [5] involving two surface types on three side faces of a triangular wedge. This approach is, however, not applicable to all surfaces and crystal structures; polar surfaces in wurtzite crystals are one example [6,7]. Consequently, not every individual surface energy of wurtzite crystals can be computed; hence, the construction of the ECS seemed impossible. Recently, neglecting the different layer-stacking sequence, the surface energy of the polar c plane was estimated from the zincblende (111) plane [10].In this Letter, we show that such an approxim...

Equilibrium Morphologies of Faceted GaN under the Metalorganic Vapor‐Phase Epitaxy Condition: Wulff Construction Using Absolute Surface Energies

“…However, the (0001) plateau easily emerges when the difference is lower than 0.06 eV Å −2 , as shown in Figure c under the N‐rich condition. The calculated equilibrium shape under the N‐rich condition is quite different from previous claculations, where the area of side facets is mainly present . This is because the reconstructed polar and semipolar surfaces under the MOVPE condition are not taken into account in the equilibrium crystal shape by Li et al…”

“…Due to crystal symmetry, the SAG on the lateral direction, which consists of (0001), , and planes, can also be considered. Details of the equilibrium Wulff construction are explained elsewhere . Furthermore, it is well known that the MOVPE condition, such as growth temperature, partial pressure, and carrier gas, is crucial for the equilibrium crystal shape in GaN nanostructures.…”

Understanding the advantage of hexagonal WO3as an efficient photoanode for solar water splitting: a first-principles perspective

“…30,35,43 In spite of this, we do not anticipate that the basic ECS morphology to change drastically since the low Miller-index surfaces are thermodynamically the most stable. 30,35,43 In spite of this, we do not anticipate that the basic ECS morphology to change drastically since the low Miller-index surfaces are thermodynamically the most stable.…”

In situ asymmetric island sidewall growth of high-quality semipolar (112̄2) GaN on m-plane sapphire

“…[1][2][3][4][5][6][7] In comparison with nonpolar GaN, semipolar GaN grown on (1122) and (2021) atomic planes has the merits of a high indium incorporation efficiency 8 and a wide growth window. 9,10 (1122) semipolar GaN is of a nearly free electric field and thus preferred as a growth template of long-wavelength GaN-based optoelectronic devices.…”