## Abstract

Analytical models for strained heteroepitaxial quantum dot systems have invariably assumed that the dots have a low-aspect ratio (small slopes) and that the elastic properties of the dot and the substrate are identical. In this paper, a three-dimensional analytical model for the energetics of an array of axisymmetric quantum dots is developed from physical principles. This is valid for high-aspect ratio dots (such as GeSi and InGaAs) and allows the dot and substrate to have different elastic properties. It is shown that these features are very important in determining the strain energy of both isolated dots and arrays of interacting dots. Both the elastic relaxation energy (per unit volume) of a single dot and the elastic interaction energy (per unit volume) between multiple dots are found to be greatest for tall, steep dots and for dots which are stiffer than the substrate. The equilibrium of two-facet dots is investigated and shape transition phase diagrams for small slope monoelastic theory, GeSi and InGaAs are compared. Different features of the bimodal dot size distributions in these systems are explained.

## 1. Introduction

Growth of thin films in heteroepitaxial systems is often preceded by the formation of a wetting layer of the deposited material. The inherent elastic strain energy due to the lattice mismatch between substrate and film causes the film to become unstable as it gets thicker. This can cause the film to break up and grow as a number of discrete islands. If the islands are small enough they can exhibit zero-dimensional electronic confinement and are referred to as quantum dots. These nanoscale particles can be used as building blocks for novel optoelectronic devices if they exhibit a narrow size distribution and a common shape. The elastic strain energy of a single dot and that due to its interactions with other dots is critical to this self-organization process. A number of authors have investigated the rich behaviour of the SiGe system using analytic expressions for the strain energy of the combined multi-dot system (e.g. Floro *et al*. 1998; Chiu 1999; Daruka *et al*. 1999; Gill 2003; Rudd *et al*. 2003; Shchukin *et al*. 2004; Chiu & Poh 2005; among others). These models represent the effect of the dots' presence on the substrate by a distribution of point forces on an elastic half-space. This is a first-order approximation and necessarily only relates to systems in which the dots have small slopes (roughly less than 0.2 (or 11°)). In recent years, a myriad of exotic heteroepitaxial systems have been investigated. The crystallographic planes of the observed island facets are given in table 1 for a number of systems. This shows that small slope theory is adequate for small pyramidal dots in the GeSi system (11°), for instance, but not for large GeSi islands (or domes) which have facet angles up to 33°. Other well-known systems, such as InGaAs, have facet angles of 24 and 54° and are hence completely outside the range of small slope analysis. It is clear that there is a strong motivation for the development of a theory for the elastic strain energy of high-facet angle dot systems. A range of numerical techniques are available for evaluating the energy (e.g. Zhang *et al*. 1999, 2003; Long *et al*. 2001; Zhang & Bower 2001; Eisenberg & Kandel 2005) but these are necessarily complicated, time consuming and less transparent than a simple analytical expression. It is the purpose of §2 of this paper to derive simple approximate expressions for the elastic strain energy associated with isolated, and arrays, of epitaxially strained multi-faceted dots, which are applicable for a wide range of practical aspect ratios. The effect of different elastic properties is naturally accounted for. In §3, equilibrium phase diagrams are constructed to find the minimum energy configurations of dots with two-possible facet orientations. Predictions for the pyramid-to-dome shape transition in the GeSi and InGaAs systems are compared with experimental findings. The state a dot occupies is not solely determined by energetics, however; the kinetics is also very important. This aspect of quantum dot growth is investigated in another paper (Cocks & Gill submitted).

## 2. The strain energy model

We assume that the thermodynamics of quantum dot systems is determined primarily by two contributions: interfacial energy and elastic strain energy. For the purposes of this paper we will ignore the effect of surface stress (Cammarata & Sieradzki 1994), surface reconstructions, any wetting layer thickness dependence (Wang *et al*. 1999), alloying (Koenraad *et al*. 2003) and, in the case of InGaAs, In segregation (Howe *et al*. 2005) and surface termination (Cammarata & Sieradzki 1994). This assumption is consistent with recent models of the equilibrium structure of quantum dots and includes what are regarded as the most important contributions to the free energy of a system of dots. Also, it allows a wide range of different phenomena to be investigated, without the analysis being complicated by the need to identify a wide range of material parameters. Where appropriate, however, we examine how these other contributions influence the predicted response.

The interfacial energy is readily determined from the geometry of the dot. Determination of the elastic strain energy is not such a trivial matter. Small slope solutions for the strain energy of dots on substrates with identical properties have been widely reported in the literature (Floro *et al*. 1998; Daruka *et al*. 1999; Rudd *et al*. 2003; Shchukin *et al*. 2004). However, many novel quantum dot systems are not restricted to small slopes (as illustrated in table 1) and in heteroepitaxial systems the dot and substrate necessarily have different elastic properties. No exact solutions exist for the strain energy of quantum dot systems in these cases. In this section, a simple analytical model for the strain energy of axisymmetric quantum dots is derived using rigorous mechanics principles. The model is physical in the sense that it is not based on empirical evidence and there are no unknown or unphysical fitting parameters. The model is constructed in a number of stages, adding a new physical phenomenon each time. Every stage is validated by comparison with numerical results.

### (a) The problem

In the heteroepitaxial systems identified in table 1 the dots form faceted three-dimensional structures. As a result of the cubic symmetry of the crystal structures, the dots are equiaxed. In order to develop simple analytical expressions which can be employed in large-scale simulations we follow Daruka *et al*. (1999) and Shchukin *et al*. (2004) and idealize these geometries as axisymmetric profiles, such as that shown in figure 1, with the facets smoothed to form the conical sides of the dot. We further assume that both the dot and substrate are isotropic, but they can have different elastic properties. The validity of these assumptions is investigated in appendix A.

We start by determining the strain energy of the simple truncated cone profile of figure 1. This represents an idealized hut profile. Multi-faceted dome profiles can be created by stacking geometries of this type. The analysis presented for this simple axisymmetric profile can therefore be used as a building block for the analysis of multi-faceted dots. This construction is described at the end of this section.

The geometry of figure 1 is defined by three parameters: the base radius, *b*; the height, *h*, and the slope, . In the limit of it becomes a cylinder and when *c*=*h* it reduces to a cone. Polar coordinates are used with the origin at the centre of the dot/substrate interface. The plane is in the plane of the substrate surface and the *z*-axis is normal to this plane, with positive *z* directed out of the substrate. The strain field within the dot is then given bywhich must be compatible with the continuous displacement field where and are the displacements in the *r* and *z*-directions, respectively.

The strain energy in a heteroepitaxial system arises due to a mismatch strain, , between the substrate and deposited material. Assuming the dot material is isotropic and linear elastic, the strain energy due to a single axisymmetric dot is given by,(2.1)where is the volume of the dot or substrate , and is the appropriate material stiffness matrixwhere and are Young's modulus and Poisson's ratio, respectively. In order to determine the strain energy of the system we need to identify a suitable displacement field. A number of possible forms of field are evaluated in the following sub-section.

### (b) Strain energy of a dot on a rigid substrate

In this case, there is no displacement field in the substrate and the only contribution to the strain energy of (2.1) is from the deformation of the dot. The elastic component of displacement along the dot-substrate interface (*z*=0) is necessarily and . The displacement field in the dot is assumed to be the superposition of two orthogonal displacement fields. The first is a field for an incompressible material, such that . The second field is a uniform dilatational field characterized by . Given this restriction, we writewhere and are the magnitudes of the deviatoric and dilatational fields, respectively, and is an arbitrary function of *z*. The constraint at the dot-substrate interface is satisfied by and . Compatibility of strains requires that , and . The strain energy of the system (dot) can therefore be written aswhere we have introduced the dimensionless elastic relaxation factor(2.2)

This is purely a function of the dot shape and can be described in terms of the two ratios:(2.3)

We choose the strain fields which minimize the total strain energy. Then, and(2.4)where is the strain energy density of an unrelaxed dot and provides a measure of the compressibility of the dot. The dimensionless scaling factor, , depends on the dot shape and Poisson's ratio. The dot volume has been written as where . For an incompressible material . Typically, , in which case .

The strain energy is now completely defined for a dot on a rigid substrate given a function, , for the variation of strain vertically within the dot. Two functions are now proposed and evaluated. The first is a simple linear function(2.5)where *d* is an unknown degree of freedom to be determined; requires *a*=1 (the reason for introducing *a* will become evident in §2*e*). This model allows the top of a dot to be completely unstrained elastically if it is very tall. The cut-off at *z*=*d* introduces a discontinuity in the shear strain, however, which results in a discontinuity in the rate of change of strain energy with respect to the geometric parameters. This is not the case for the second function, which is a quadratic and chosen to have continuous first derivatives(2.6)

Determination of the relaxation function (2.2) is straightforward for each of the above functions. In the case of the linear function, , for ,(2.7)where

The degree of freedom, *d*, is always chosen such that the total strain energy (2.4) is minimized, i.e. the relaxation factor is maximized such that . Given *a*=1, this yields and the optimal relaxation factor for the linear strain variation is simply(2.8)

For a conical dot , the relaxation factor is a simple function of the dot aspect ratio, *η*,

This result is consistent with the small slope analysis of Shchukin *et al*. (2004) which predicts that , i.e. the strain energy (for a dot on a rigid substrate) simply scales with the volume to first order.

The other cases described in (2.5) and (2.6) can also be easily evaluated, but are too unwieldy to be reproduced here. However, the same trend in relaxation factor with dot shape is observed for all choices of displacement field. The strain energy predictions for the linear and quadratic strain variations are compared in figure 2 with the small slope predictions and numerical results generated using the finite element method. The finite element results were generated using FEMLAB with 1000 6-noded axisymmetric isoparametric triangular elements. The results for the total stored energy are, however, insensitive to the number of elements used. This is not surprising since the analyses presented here essentially model the dot as a single finite element.

Figure 2 illustrates that the small slope prediction is poor for taller dots in all cases and does not reproduce the essential features of the numerical results. As seen in figure 2*a,b*, the predictions for the linear and quadratic functions are excellent for low-aspect ratios, , over the entire range of heights. Introducing a cut-off at *z*=*d* only reduces the strain energy for very tall cones (see figure 2*c*). The cut-off transition is smooth for the quadratic model, but the slope is discontinuous for the linear model, as expected. Both models predict that the strain energy asymptotes to a constant value as the height is increased. The asymptote predicted by the quadratic model is slightly lower than that obtained from the linear model and closer to the numerical simulations. The prediction of the asymptote becomes less accurate as the slope of the facet increases. Given that this paper aims to provide a simple expression for the strain energy of a quantum dot, we will simply use the linear strain variation function without a cut-off, i.e. for all *z*>0 for the majority of the situations we examine. The relaxation factor is then given by (2.7). For tall dots, we truncate the strain energy at the asymptotic value. This can be determined from a single computation of a very tall dot, with the appropriate slope. In the remainder of this section we concentrate on the situation where (2.7) applies, although the results can be readily extended to tall dots. We examine the effect of the asymptotic energy in §3, where we consider the behaviour of practical systems.

The effect of changing Poisson's ratio of the dot is illustrated in figure 3. The general trends of the numerical results are reproduced and the results for small slopes (less than 30°) are excellent. Conventional small slope theory predicts no dependence on Poisson's ratio.

### (c) Strain energy of a dot on an elastic substrate

The model developed in §2*b* has been shown to give a good approximation for the strain energy of an elastic dot on a rigid substrate. In this subsection, this model is extended to consider an elastic dot on an elastic substrate. We assume that the strains are constant (but unknown) at the dot-substrate interface as for the previous rigid substrate case. In order to include the effect of the elastic substrate, an expression for the strain energy of the substrate as a function of the strain (or surface tractions) at the dot/substrate interface is required. Shchukin *et al*. (2004) have determined the elastic strain energy for small sloped conical dots on an elastically identical substrate. This analysis is equivalent to the calculation of the strain energy due to a constant radial surface traction distributed over a circular patch on a strained elastic half space. For a constant radial force density, *f*, the change in the strain energy of the substrate due to the applied tractions is(2.9)where *s* denotes the properties of the substrate and *J*=1.059. The force density is simply related to the dot slope and is equivalent to in the small slope analysis of Shchukin *et al*. (2004). This is not the case for the general model proposed here but our model should converge to this result in the small slope limit, i.e. .

By combining the analysis of §2*b* with equation (2.9) we can obtain the elastic strain energy for any dot profile. First consider the substrate, which experiences a constant tangential radial force density *f* over a circular patch of radius *b*. We assume that the radial displacement, *u*, at the surface varies linearly and can be characterized by a strain , such that . The strain energy is then given by

Equating this with (2.9) provides a linear relationship between and *f*, which allows the strain energy to be expressed as a function of ,(2.10)

If the substrate immediately below the dot strains radially by an amount , continuity of displacement at the interface requires the elastic component of the radial displacement in the dot to satisfy the relationship . The strain energy in the dot is now given by (2.4) but with replaced by .

The total energy of the system (substrate and dot) is therefore given by (2.4), and (2.10), i.e.(2.11)

This is minimized when(2.12)where is a dimensionless constant incorporating the elastic properties of the substrate and dot and *g* is defined in (2.4). The total change in energy of the system due to the addition of a dot is therefore obtained by substituting (2.12) into (2.11), to give(2.13)where is given by (2.4). Hence, the energy of a system with an elastic substrate can be easily determined from the energy of the equivalent system with a rigid substrate. Note that, in the limit of , one finds and the rigid substrate result is recovered. Also note that for small sloped cones, , and setting and we recover the small slope result of Shchukin *et al*. (2004).

The predictions of (2.13) using the relaxation factor (2.7) are compared with numerical finite element results in figures 4 and 5. Figure 4 considers dots and substrates with identical elastic properties and shows the dependence on their mutual Poisson ratio. The agreement between the numerical results and the model are good and are an improvement on the equivalent rigid substrate model, particularly for incompressible materials, where the result is within 10% of the finite element prediction over the full range of aspect ratios considered. Figure 5 shows the strain energy for two cases: a stiff dot on a compliant substrate and a compliant dot on a stiff substrate. The agreement is again good, especially for stiff dots on a compliant substrate. The results also show that a mismatch in the elastic properties can significantly alter the strain energy curves. To the authors' knowledge, all other models for the strain energy of quantum dot systems assume the elastic properties of the dot and substrate are identical. Also note that, in each of the set of conditions considered in figures 4 and 5, the conventional small slope model is shown to be invalid above a slope of about 0.2.

### (d) Strain energy of a regular array of dots

A model for the strain energy of a random array of interacting dots is derived in appendix B. For a hexagonal array of identical dots a simple closed form relationship can be obtained for the elastic strain energy per dot,(2.14)where *I* is an interaction term which is a function of the base radius of the dot, *b*, and centre-to-centre spacing, *L*, which is well approximated by equation (B 7).

Equation (2.14) is a significant result. It captures the effects of dot shape, elastic property mismatch and interaction effects in a form that readily allows the effects of changes in geometric and physical properties on the behaviour of a system to be evaluated and understood. It forms the basis of the analyses presented in the remainder of this paper. Note that this relationship reduces to the strain energy for a single dot on an elastic substrate (2.13) when the interaction term *I* becomes small . When the dot has a low-aspect ratio or is very compliant the interaction affect is a second-order term. This is still of importance, as the change in strain energy with shape is also second order for a dot of constant volume. However, the interaction term is first order for stiff, high-aspect ratio dots indicating that elastic interactions are more important in this case.

The effect of the interaction term is explored in figure 6, in which the strain energy of a single dot on an elastic substrate (2.13) is compared with that for a similar dot in a regular array of identical dots (2.14) for a cone and a 45° truncated cone. The solutions are the same when as expected, i.e. *h*=*c*=*b*. The dots are taken to be touching to maximize the interaction. It is clear that the interaction term always increases the energy of the system and therefore dots will always repel each other. The interaction is stronger for a shallow 45° truncated cone than for a shallow cone of the same volume. It is weak for compliant dots on a stiff substrate and most pronounced for stiff high-aspect ratio dots on a compliant substrate, i.e. when is large.

The elastic properties of some typical heteroepitaxial quantum dot systems are given in table 2. For a system in which the dots and the substrate have identical elastic properties , which is equal to 1.38 for . The widely used SiGe and InGaAs systems tend to have a stiffer substrate than dot, with and , respectively. This corresponds most closely to figure 6*a*. Their elastic properties are such that the interaction is overestimated by the monoelastic models. The third heteroepitaxial system, PbSe/PbEuTe, has a stiffer dot than substrate and a higher value of . The dot–dot interactions in this system will be the strongest of the three, as they have a high-aspect ratio and the largest elastic *β* parameter. This is observed in practice as these dots have an unusually narrow size distribution of ±10% (Raab & Springholz 2000). Strong spatial ordering is also observed in stacked systems with a transition from vertical stacking to fcc-like stacking as the capping layer thickness is increased beyond 380 Å (Raab *et al*. 2002).

### (e) Energetics of multi-faceted dots

At small volumes, quantum dots are highly facetted structures, indicating that they have a finite number of stable minimum energy surface orientations. As the volume of the dots increases, the surface energy becomes less dominant and the strain energy drives the dot shape towards a higher aspect ratio (the strain energy per unit volume is minimized by an infinitely tall, infinitely narrow column). This type of shape transition has been observed in a number of systems (the first four in table 1) but the best known is the SiGe system (Floro *et al*. 1998). The dots initially form low-aspect ratio pyramids with {105} facets. As the volume increases, steeper {113} facets appear and the dot undergoes what is known as a hut-to-dome transition. Small slope (Daruka *et al*. 1999; Chiu 2004; Chiu & Poh 2005) and numerical (Zhang & Bower 2001) analyses have been used to examine this transition. This process was analysed by Daruka *et al*. (1999) for an isolated dot within the confines of two-dimensional, small slope, monoelastic theory. These calculations are now revisited in the context of the axisymmetric, large slope, bielastic model proposed here.

We adopt the two-facet dot configuration model of Daruka *et al*. (1999) but now for axisymmetric dots. The dot is constructed from a stack of two truncated cones, denoted 1 and 2 in figure 7. The slopes of the upper and lower cones are fixed and are defined to be and , respectively, subject to . The lower cone 2 has base radius *b* and its shape is defined by and . The upper cone 1 has base radius and its shape is defined by and such that it has a top radius of . The dot is assumed to sit on a wetting layer with surface energy per unit area. We also assume that the top surface of the dot (0) has the same surface energy. These assumptions are made to reduce the number of unknown parameters. The surface energy densities of the two sloped facets are and . The change in surface free energy due to the introduction of the dot is therefore(2.15)where is the additional energy per projected area for facet *n* (=1 or 2), and .

The strain energy can be calculated from the strain energy model of §2*a* and is the summation of the strain energies of two stacked truncated cones. For continuity of strain, the strain in the upper dot is where is the height of the lower dot section and . The strain in the lower dot section is the same as the single section dot of §2*a* with . The linear strain gradient, , is taken to be the same in both sections (for continuity of strain) and is chosen such that it minimizes the combined strain energy of both sections, using the notation of equation (2.4). This is a simple quadratic in so the minimization is straightforward and an analytical expression is readily derived. The total change in energy of the system per dot is therefore given by (2.14), with (2.4), plus (2.15),(2.16)and is a function of the dot shape defined by the variables and and various prescribed geometric and material parameters. We introduce a reference length, , such that and . For a given dot volume, , the equilibrium shape is that which minimizes the energy of the system, which is when(2.17)where and . The resulting dot shape is a function of six-dimensionless parameters: three geometric parameters , two elastic material parameters (*β* and *k*) and one surface material parameter (*α*), plus whether or . Note that Daruka *et al*. (1999) introduced a reference length of which rendered the change in free energy invariant to changes in the slope given a fixed ratio. This is only valid for small slopes. This linear scaling does not exist for larger aspect ratio dots and is not introduced here.

## 3. Equilibrium configurations

The absolute values of the surface parameters in different heteroepitaxial systems are not well known, so it is the aim of this section to explore the relative differences between systems. The most common experimental systems are the GeSi and InGaAs systems and these will be investigated further here. Their behaviour has been observed under consistent experimental conditions by Constantini *et al*. (2005), who demonstrate that they exhibit broadly similar behaviour. It is observed that both systems undergo a Stranski–Krastanow transition from two-dimensional to three-dimensional growth at 1.6 ML (InAs) and 4.0 ML (Ge) of deposited material, where 1 ML corresponds to an atomistic monolayer of material. Pyramidal dots (P) then form, which grow in size and eventually undergo a shape transition to form domes (D) via a transition dome (TD) stage, i.e. a truncated two-facet dome. This is a first-order transition (i.e. there is a transition energy barrier) so some dots transform and some do not. At a particular instant, this generates a bimodal distribution of dot types and sizes as shown in figure 8. There are, however, a number of distinct differences between the two systems. We concentrate on two of these differences here.

First, the aspect ratio for the InAs dots (figure 8*b*) is much higher than for Ge dots (figure 8*a*). The pyramids have aspect ratios of about 0.45 (InAs) and 0.2 (Ge), and the domes have ratios of about 0.9 (InAs) and 0.4 (Ge). As shown in table 1, the InAs facets are {137}, {110} and {111} which have slopes of 0.45, 1 and 1.41. The Ge facets are {105}, {113} and {15 2 23} which have slopes of 0.2, 0.47 and 0.66. The pyramids in each case are formed with the shallowest facet. The domes are necessarily a mixture of facet types, consisting of a mixture of all three facets. They are effectively double rather than triple facet dots, with the lower (steep) section of the dot constructed from the two steepest facets on different sides of the dot.

Second, there is a large difference between the volumes of the InAs domes and pyramids (figure 8*b*), which is not observed with Ge dots (figure 8*a*). Dashed lines have been added to figure 8 to make this clearer. The smallest InAs dome (about 4000 nm^{3}) has a volume 10 times that of the largest InAs pyramid (about 400 nm^{3}) with no structures observed in-between. The largest Ge pyramid, however, is almost the same size as the smallest dome (about 12 000 nm^{3}). There is one transition dome in the GeSi system, indicating that the transition from pyramid to dome is quite rapid. The aspect ratio of the tallest pyramid (0.25) is much lower than the aspect ratio of the shortest dome (0.4). By contrast, the number of transition domes in the InGaAs system is equal to the number of domes, indicating that the transition is slower. Also, the aspect ratio of the tallest pyramid (0.53) is almost the same as that of the shortest transition dome. It appears, therefore, that Ge pyramids undergo a rapid transition to a higher aspect ratio dome without a significant increase in volume and that InAs pyramids undergo a rapid transition to a much larger volume without a significant change in aspect ratio.

These differences are now explored from an energetics perspective. This is expected to provide a partial explanation for these features, but as we have already stated, the structures observed in these systems exist due to a combination of kinetic and thermodynamic effects. Kinetic processes are investigated in another paper (Cocks & Gill submitted). The strain energy model of (2.17) is now used to explore the equilibrium configurations of these heteroepitaxial quantum dot systems. Daruka *et al*. (1999) observed that there are seven possible classes of dot shape for the two-facet dot configuration. These are listed in table 3 and illustrated in figure 9. Three systems are investigated here using the material parameters from table 2: the monoelastic small slope system; Ge dots on Si(100); and InAs dots on GaAs(100). For the small slope case, we take , and *ν*=0.3. For InGaAs we take and *s*_{1}=0.45 from table 1. For GeSi we take and *s*_{1}=0.2. The strain energy model of §2 has already been shown to be valid for a moderate range of slopes.

Before proceeding further, we now compare the strain energy predictions of the two-facet stacked dot model of §2*e* with numerical calculations. In these calculations we completely prescribe the geometry of a dot and substitute for *g* in (2.13) to find the strain energy of a two-facet dot on an elastic substrate. The dot has a complete top cone , i.e. it is of type 2 in figure 9. The strain energy is shown in figure 10 as the dot changes from a low-facet cone to a high-facet cone . The maximum experimentally observed aspect ratios are about 0.4 (GeSi) and 0.9 (InGaAs). This corresponds to (GeSi) and (InGaAs). The linear model without a cut-off is accurate for aspect ratios up to 0.6 (30°). This covers the evolution of GeSi over the whole range of shapes and the low-facet states of InGaAs . Introducing a quadratic model (see §2*b*) for the strain variation improves the approximation for InGaAs for , but the quadratic solution does not have a simple analytic form and the improvement is not significant enough to justify using this more complex model. Introducing the theoretical cut-off in both the linear and quadratic models (see §2*b*) improves the prediction for but the error is still of the order of 10%. These models correctly predict an asymptotic value of the elastic stored energy. Rather than construct more complex displacement fields to try and improve the prediction of the asymptote it is relatively straightforward to calibrate this feature of the response directly using the finite element results. The numerical simulations of figure 10 suggest an asymptotic value of the normalized strain energy of 0.547 for InGaAs. This corresponds to an upper limit on of 0.484.

Figures 11 and 12 show the transition phase diagrams predicted by (2.17) for the three systems under consideration for and , respectively. For a given system, the geometric parameters and the two elastic material parameters (*β* and *k*) are prescribed. For a given dot volume, and surface energy ratio *α*, (2.17) provides the minimum energy and the shape associated with this energy. Shapes corresponding to different regions of the phase diagrams are illustrated on the figures using the notation of figure 9. The values of the reference length, *l*, are selected to rescale the volume so that a similar region of the phase diagram is shown in each case. For small slopes, the strain energy density scales with *β*, so the reference length scales as , and the volume plotted on the *x*-axis of the phase diagram, , scales as . This has a significant effect on the region of a phase diagram that is applicable to a particular material system and is an effect that is usually ignored. For the systems in table 1, the ratio of the volume predicted by a monoelastic analysis to that predicted by the bi-elastic (small slope) analysis, , is 0.48 for SiGe, 0.26 for InGaAs and about 2 for PbSe/PdEuTe. These rescaling factors are all quite considerable, especially in the case of InGaAs, even without taking into account the large slope response. For high-aspect ratio dots, there is a nonlinear rescaling of the phase diagram, although the topology remains unchanged. The topology of the transition boundaries is identical to that seen in figure 3 of Daruka *et al*. (1999), who used a two-dimensional small slope, monoelastic model, although the predicted dot shapes are very different, as can be seen from the aspect ratio contours superimposed on the phase diagrams. The significance of this topology is discussed in detail in their paper.

To make further progress, it is helpful to identify regions of the phase diagrams which might relate to the GeSi and InGaAs systems. In a system which undergoes a pyramid to dome shape transition the dots change from type 1 (or 1′) to type 2 (or 2′) as their volume increases.1 Assuming that the surface energy ratio *α* remains constant during dot growth2 then three regions present themselves as candidates. Taking the GeSi diagrams in figures 11*b* and 12*b* as a reference, we can see that these are:

and . This corresponds to a 1′–2′ transition for and a 0–1′–2′ transition for .

and . This is a large region of the phase diagram and corresponds to a 1–2 transition.

and . This also corresponds to a 1–2 transition but is a small area of the phase diagram.

The model of (2.17) assumes that an initial wetting layer exists, which is the case for both the material systems considered. InGaAs undergoes a transition from two-dimensional to three-dimensional growth at 1.6 ML, which is small enough to be indistinguishable from the initial wetting layer (1.0 ML). GeSi undergoes this transition at 4.0 ML, which is large enough to suggest that a stable flat film (0) must exist below a certain volume. This would confine the GeSi system to a narrow region of the phase diagram, i.e. to case A above, with the restriction that . However, as Wang *et al*. (1999) have noted, the surface properties in such systems depend on the thickness of the wetting layer (i.e. *α* can change with thickness for small volumes) and hence one cannot strictly rule out other areas of the phase diagram for the evolution of Ge dots on a Si substrate.

We can use the results employed to construct the phase diagram to provide information about the driving forces for the growth process. As a dot grows, the state moves horizontally across the phase diagram at a height corresponding to the value of *α* for the system. At a given point on the diagram the dot chemical potential, defined as the change in energy with volume, i.e. , can be determined. Figure 13 shows the phase diagrams of figures 11*b* and 12*b* with contours of constant chemical potential. Dots with low-chemical potentials will grow faster than dots with higher chemical potentials. A simple generalized model can be constructed by assuming that dots with an above average chemical potential shrink, while those with a below average chemical potential grow at their expense (Ross *et al*. 1998). One can see that transitions from a flat film (0) or a pyramid (1 or 1′) to a dome (2 or 2′) are always first order. There is always a discontinuous drop in the chemical potential (and a discontinuous increase in the height of the dot) implying that the growth rate of a pyramid which has just undergone a transition to a dome will be increased.

To gain further insight into these phase diagrams it is useful to look at the evolution of the dot shape along a line of constant surface energy ratio in the three shape transition regimes identified above. The first contour is taken at *α*=0.15 for which corresponds to case A. This is illustrated in figure 14 for all three systems. It is clear that, although the topologies of the phase diagrams in figure 11 are similar, the predictions of the dot shape and growth rate (as indicated by the chemical potential) are not. It is not worth comparing specific values for the dot aspect ratio with experiment as *α* is chosen arbitrarily and growth kinetics are not accounted for. However, the general features of each case are the same, whatever value of *α* is chosen in a particular region. Figure 14*b* shows that there is a large sudden increase in height when Ge pyramids make the transition to domes, whereas figure 14*c* shows that this is not the case for InAs dots, where there is only a small increase in height associated with the transition. Figures 15 and 16 show profiles for the Ge and InAs dots along the contour *α*=−0.15 of figure 11 (case B) and the *α*=1.1 contour of figure 12 (case C) respectively. In all three cases one sees that there is a large increase in height for a small increase in volume for Ge dots and that there is only a large increase in height for a large increase in volume for InAs dots. This matches well with the overall features of these systems, as illustrated by figure 8. In all cases InGaAs dots need to increase their volume by an order of magnitude to reach the expected dome aspect ratio of about 0.9, which is what is observed experimentally. It does not, however, explain why there are only small and large dots in the InGaAs system and no dots of an intermediate size. There are a number of explanations for this.

First, at the equilibrium transition point there is an energy barrier to the growth of a new lower facet of finite length (Lin *et al*. 2005). Further evaluation of this requires the development of a kinetic model for the transition from one shape to another and we do not explore this further here. Another explanation follows from an examination of the nature of the chemical potential evolution. In each of the three cases, the chemical potential evolves in a different way, apart from the fact that the chemical potential always undergoes a discontinuous drop at the pyramid to dome transition. Figure 14 demonstrates that the chemical potential in case A monotonically decreases with volume in the pyramid and dome stages. Therefore, the larger the dot the faster it grows. This picture is consistent with the observed distribution in the GeSi system in figure 8*a*. For case B, figure 15 illustrates that the chemical potential increases with volume in the pyramid stage and decreases in the dome stage. This implies that small pyramids grow faster than large ones, but small domes grow slower than large domes. For the InGaAs example in figure 15*b* there is a region where small pyramids grow faster than large pyramids or small domes. For case C, figure 16 shows that the chemical potential increases continuously in both the pyramid and dome stages and only drops at the transition. This suggests that small dots grow faster than large dots, except near the transition where there is a region in which small domes grow faster than large pyramids. Domes grow by consuming dots with a higher chemical potential. A gap in the InGaAs size distribution could appear if large domes rapidly consumed large pyramids and/or small domes, but not small pyramids. Both cases B and C are consistent with this picture. The stability of these three different cases is of great interest and will be the subject of a further paper which combines the energetics presented here with the kinetics of dot shape transitions and dot coarsening (Cocks & Gill submitted).

## 4. Conclusions

A simple analytical model for the strain energy of a system of steep axisymmetric quantum dots has been formulated in terms of the dot shapes and the relative dot positions. The different elastic properties of the dot and substrate have been accounted for and the effect of the elastic interaction of the dots has been taken into account. All these effects have been captured in the form of a simple analytical expression (equation (2.14)). The model has been verified with numerical calculations. Multi-facetted dot shapes have been considered and used to investigate the pyramid-to-dome shape transitions observed in many heteroepitaxial systems. Equilibrium transition phase diagrams have been constructed for the GeSi and InGaAs systems based on the observed facet angles and their respective elastic properties. The phase diagrams are found to have the same topology, but the predictions of the nature of the shape transitions is generally explained by simple energetic arguments. It is found that Ge/Si(100) dots undergo a large increase in aspect ratio with a small increase in volume, whereas InAs/GaAs(100) dots only undergo a large increase in aspect ratio with an order of magnitude increase in volume. This is consistent with experimental observations.

## Footnotes

↵Although pyramids are typically of type 1, one often finds that the type 1′ dots have such a small top facet that they are nearly type 1. Daruka

*et al*. (1999) also observed pyramids with a very small top facet in the GeSi system. Given that a number of thermodynamic contributions have been neglected here, we will assume that type 1′ dots with a small top facet (say ) can be classed as complete (type 1) pyramids.↵This assumes that the surface energies are independent of strain—this is equivalent to a zero surface stress model.

- Received December 13, 2005.
- Accepted May 11, 2006.

- © 2006 The Royal Society