Shapes of GexSi1xSi001 QDs

The most frequently reported QD shapes in the GexSi1-x/Si system are pyramid (with square bases) bounded by {501} facets, hut (with rectangular bases) bounded also by {501} facets [25, 89-92], multifaceted dome (including {311}, {111}, and other {h11} facets) [88, 93], and dome without facets [24]. Examples of hut, pyramid, and dome QDs are shown in Figure 2.

Pyramids with {501} facets are seen at the early stage of Gex Si1-x/Si QD growth. As the QD growth continues, pyramids evolve into either huts or domes, depending on the

Figure 2. (a) and (b) scanning tunneling microscopy (STM) images of a hut-shaped Ge/Si(001) island. Reprinted with permission from [25], Y.-W. Mo et al., Phys. Rev. Lett. 65, 1020 (1990). © 1990, American Physical Society; (c) STM images of Ge/Si(001) islands with (left) a dome-shaped and (right) an island at the transition stage and a small pyramid-shaped. Reprinted with permission from [88], G. Medeiros-Ribeiro et al., Science 279, 353 (1998). © 1998, American Association for the Advancement of Science.

Figure 2. (a) and (b) scanning tunneling microscopy (STM) images of a hut-shaped Ge/Si(001) island. Reprinted with permission from [25], Y.-W. Mo et al., Phys. Rev. Lett. 65, 1020 (1990). © 1990, American Physical Society; (c) STM images of Ge/Si(001) islands with (left) a dome-shaped and (right) an island at the transition stage and a small pyramid-shaped. Reprinted with permission from [88], G. Medeiros-Ribeiro et al., Science 279, 353 (1998). © 1998, American Association for the Advancement of Science.

growth conditions. At lower growth temperatures, transformation from pyramids into elongated huts with {501} facets [25, 94] occurs; at higher temperatures, equiaxial islands (i.e., pyramids and domes) are more stable than huts [95], and the pyramids evolve into domes. The lower-temperature transformation from pyramid into {501}-faceted huts has been explained by Tersoff and Tromp [96]. If we consider the case of a high density of adatoms being deposited, diffusion is much more rapid on the substrate than on the QD, so that atoms tend to stick to the QD side-wall and not diffuse to the top facet. As a result, the QD height remains roughly constant. Under these circumstances, for QDs with size smaller than a specific value, the minimum energy, which includes the surface energy and the elastic energy, corresponds to islands of the square shape. Above this value, it is energetically favorable for the QDs to be elongated in one direction, through a second-order shape transition (the second derivative of the energy, with respect to QD size, is discontinuous), to maintain a smaller lattice strain along the shorter axis of the QDs. Conversely, for high-temperature growth, diffusion is rapid, and so QDs can grow in height. Assuming that the stable configurations are a square-based pyramid (at small QD size) and a dome (at large QD size), as a QD grows, it should adopt increasingly steeper facets (the facets of a dome are steeper than those of a pyramid) [95].

Pyramids and domes can coexist in a system, giving the so-called bimodal size distribution. Many groups have observed the transformation from pyramid to dome. However, interpretation of the experimental evidence, in terms of the nature of the transformation process, remains controversial and, as a result, two different explanations for the transformation have been given.

Medeiros-Ribeiro et al. [88] suggested that pyramid and dome shapes represent two volume-dependant energy minima, separated by a large energy barrier resulting from the interplay between strain relaxation at the facets and stress concentration at the edges. Therefore, the bimodal size distribution reflects two equilibrium states with different energies. From an in-situ, scanning tunnelling microscopy (STM) investigation, they observed an abrupt shape change from pyramid to dome, which involves overcoming the energy barrier and a rapid increase in volume. In an annealing experiment involving alloying between a Ge island and the Si substrate, Kamins et al. [97] found that a dome-shaped island can change back to a pyramid, even though the island volume continues to increase substantially. This supports the conclusion that both pyramid and dome are equilibrium shapes.

On the other hand, Ross et al. [98] argued that the pyramid is not a stable shape. In their model, mean field theory is used to give an expression for the island energy E, which depends upon the contact angle and the total volume of the island V. In the case of both pyramids and domes, E decreases monotonically with V, with E(dome) > E(pyramid) for any V < V1, and E(dome) < E(pyramid) for any V > V1. Consequently, for a given V, pyramids are predicted to be stable for V < V1 and unstable for V > V1. At the same time, the chemical potential for both pyramids and domes decreases monotonically with V, and for any V > V1 is much lower for domes than for pyramids. Consequently, at V = V1, a pyramid transforms into a dome, and the system chemical potential drops discontinu-ously causing these islands to grow preferentially, consuming the smaller islands [99]. Using in-situ low-energy electron microscopy, they [98] found that the transformation from a pyramid to a dome is a slow process passing through a series of transition shapes. Pyramidal islands have two fates: either transforming to domes or disappearing through a form of Ostwald ripening [100].

Rastelli et al. [101, 102] investigated the evolution process of Ge/Si(001) islands during exposure to a Si flux using STM. They found gradual morphological changes from dome to pyramid, which involve intermediate shapes, caused by the incorporation of Si into the Ge islands—a phenomenon similar to that reported by Kamins et al. [97]. They concluded that the shape of an island is determined by its volume and average composition and that an island is stable if material exchange with its environment is kinetically suppressed. This conclusion is different from that obtained by Williams' group [88, 97] who suggest that pyramids and domes are both thermodynamically stable (energy minima).

According to Ross et al. [98, 99], as island growth continues, pyramidal islands will not survive and all islands will end up as domes. Indeed, single-modal, dome-shaped Ge/Si(001) islands with very uniform size distribution have been successfully grown by Wang et al. [103] using molecular beam epitaxy (MBE) at the high growth temperature of 700 ° C. The islands are of a square base with edges parallel to (100) and rounded corners. Extensive investigation showed that most of the islands are uniform in size with a base edge of approximately 95 nm. The reason for the size uniformity in strained island growth has been discussed [99, 104-106] and has been attributed to a self-limiting growth mechanism [107]; the adatom attachment to a large island is slowed down by the build-up of elastic strain around the island perimeter, which allows the size of small islands to catch up with that of the large islands.

Zhang [108] proposed that the above experimental inconsistencies in the stability of GeSi/Si(001) island shapes could be explained in terms of island surface energy anisotropy through its dependence on temperature and material composition [109]. It has been reported that (001), (105), and (103) facets are thermodynamically stable and, therefore, their surface energy densities are local minima [110]. Zhang [108] called these minima the "first, second, and third" minima. Through three-dimensional computer simulations, which took into account the strain energy density, surface energy density, and anisotropy, Zhang concluded that

(1) in the case where surface energy anisotropy is weak or where there is no anisotropy (which is true for very high temperatures), ripening occurs and islands with different shapes and sizes can coexist. This is consistent with the results of 850 °C controlled annealing experiments of GeSi/Si(001) islands carried out by Ozkan et al. [111];

(2) when the second minimum is shallow and the energy barrier between the second and the third minima is high, islands are almost uniform in size and self-organize into a regular square array, similar to the experimental results of Floro et al. [112] at 755 °C;

(3) in the situation where there is a deep second minimum and a high barrier between the second and third minima, islands evolve and coalesce until they reach a state that does not undergo ripening. Large islands are hut-shaped, while small islands are pyramidal-shaped with a square base. This result resembles those observed at temperatures below 330 °C [25, 91, 92];

(4) if the barrier between the second and third minima is intermediate in height, pyramids or huts appear first, followed by a bimodal size/shape distribution in which both shapes are stable, which is consistent with the results of Ge/Si(001) grown at 550 ° C reported by Medeiros-Ribeiro et al. [88];

(5) with a low barrier between the second and third minima, square base pyramids form first, and then all islands transform into dome-shaped islands, as observed by Ross et al. at 650 °C [99].

Although many other shapes for Gex Si1-x/Si(001) QDs have been reported, including a truncated pyramid with {111} facets [113-116], {311} faceted [117-119], cone-shaped circular island with {711} facets [120], all can be regarded as dome-shaped.

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