Kinetic Monte Carlo simulation of growth of Ge quantum dot multilayers with amorphous matrix

 

J. Endres1, S. Daniš1, V. Holý1, M. Mixa1, M. Buljan2

 

1Charles University in Prague, Faculty of Mathematics and Physics, Prague, Czech Republic

2Rudjer Boskovic Institute, Zagreb, Croatia

endres.jan@google.com

 

Quantum dot (QD) multilayers with amorphous matrix are intensively studied because of their physical properties. Due to the size of QDs many quantum effects are observable for these systems, what can be used for many technical applications. Possible applications of QD multilayers are lasers, solar cells, photodetectors or high-speed memories.

Former studies were focused on systems with crystalline matrix [1, 2]. These systems are predominantly prepared by Stranski-Krastanow growth. At first thin strained wetting layer is grown, until it is energetically preferred to relieve elastic energy by creating QDs. The cover layer is then deposited onto the QD layer and the growth can continue with other wetting layer.

QD multilayers with amorphous matrix are mainly prepared by deposition on substrate via magnetron sputtering [3-6]. Samples consist of altering layers of QDs and matrix material. QDs layer is deposited first (material of QDs can be deposited in the same time as matrix material). Then this deposited layer is overlaid by cover layer consisting only matrix material.

Self-ordering of QDs was observed in both types of systems [1-6]. The ordering in multilayer originates in preferential nucleation of QDs during the growth in the minima of the chemical potential (i.e. surface energy) on the surface of the cover or wetting layer. In the case of the systems with crystalline matrix the driving force for self-ordering is elastic strain field originates from mismatch of lattices of materials in wetting and cover layers. However, there is no strain in multilayers with amorphous matrix. In such systems self-ordering is caused only by surface morphology of the cover layers. Two types of stacking of neighbouring layers of QDs (ABC, ABA) were observed in systems of Ge QDs in different amorphous matrix (SiO2, resp. Al2O3), see [4-6]. Finding the conditions which leads to these types of stacking was target of performed simulations.

Chemical potential at point on surface x = (x,y) is given as [7]

                                                          (1)

where μ is a constant reference value of the chemical potential, w is the volume density of elastic energy, γ is the surface tension, κ is the surface curvature and V0 is the atomic volume. Due to the amorphous matrix there is no strain in the multilayer and the second term in equation (1) is thus zero and chemical potential depends only on the curvature (i.e. morphology) of the surface of cover layer.

We consider that the surface shape is originated from deposition of cover layer over the QDs when a hill of material is created above each QD. Multilayer and dots can be described by three indexes j1, j2, j3. Index j3 denotes the position of layer in the z axis direction with zero at substrate and oriented to sample surface. This index is used for both dot and corresponding cover layer. Indexes j1 and j2 then label dots in x and y axes direction in the given layer j3. For the shape of these hills on the surface of cover layer j3 we use function [4]

                                  (2)

where x = (x,y) is point on surface, Xj1,j2,j3-1 and Xj1,j2,j3-2 are positions of QDs in layers j3-1 and j3-2, parameter C is inheritance factor which determines the contribution of height of interlayer surface above QDs layer j3-2 and f(x) is Gauss function

                                                                 (3)

where σ is a parameter determining full width at half maximum. When inheritance factor C is greater than 0, surface height is affected not only by last layer but also by one lower layer, which can cause ordering of QDs in vertical direction.

Simulation of the growth can be performed by Kinetic Monte Carlo (KMC) method. Simulation model was based on several processes which can occur [2]. The adatoms can be deposited onto the surface, adatoms can diffuse on it, adatoms can join into the QD or can escape it. Adatoms are moving by thermally activated hopping between neighboring sites of square discrete simulation lattice. The size of the whole lattice has to be big enough to not be affected by edge effects of finite-size simulation lattice. Periodic boundary conditions are considered during simulation, so if some adatom leaves the simulation area, it reappears on the opposite side. After the completed growth of actual layer the surface of cover layer was computed as hills (defined by analytical functions) over buried dots. This surface shape was used to determine the curvature chemical potential which serves as driven force for simulation of next layer [4].

To analyze the correlation of distances between QDs in lateral direction we used the radial distribution function (RDF). It describes the probability of finding the center of a particle at given lateral distance from the center of another particle. This function can be evaluated for QDs in one layer to determine the lateral ordering in one layer or between chosen layer l and next three layers j + 1,2,3 to determine the ordering of the dots in vertical direction (type of layers stacking, ABC or ABA).

In the case of partially ordered layer of QDs the RDF should have the first highest maximum at the position rm, which correspond with the distance of the first shell of dots around the center dot and can be considered as the most probable distance between all neighbouring dots. In dependence of the degree of ordering, we can seed other lower maximums, which correspond with next shells of dots.

In the case of ABA stacking QDs in layer j+2 is aligned above dots in layer j, for ABC type stacked layers j + 1,2 are offset and layer j + 3 is aligned above dots in layer j. So the ABC stacking correspond with the first maximum of RDF for layer j + 1,2 at about one half of distance rm. The maximums of RDF for layer j + 3 are at 0 and at rm again. In the case of ABA stacking the first maximum of RDF for layer j + 1 is at about one half of rm and maximums for layer j + 2 are at 0 and rm.           

 In performed simulations we observed the short-range lateral self-ordering in particular layers. This ordering (with size of the dots and distance between them) can be influenced by parameters of deposition, like temperature, deposition flux or volume of deposited material. The ordering in vertical direction can be tuned by changing of properties of surface morphology, e.g. the parameters of functions describing the hills over dots like width or inheritance factor.

The different types of stacking order of quantum dot layers (ABC or ABA) were observed, see Fig. 1. RDF corresponding with ABC stacking is in Fig. 1(a). As was mentioned above we see maximum in RDF of positions of dots in layer j = 20 at distance approximately rm = 12 nm. For RDF for dots between layers j and j + 1,2 we see maxim at approximately 7 nm. And for RDF for dots between layers j and j+3 we see maxima at 0 and 12 nm. RDF corresponding with ABA stacking is in Fig. 1(b). Maximum in RDF of positions of dots in layer j = 20 at distance approximately rm = 18 nm. Maximum at the same position is also in part of RDF for layer j + 2 (he we see maximum at 0 too). Maxima at approximately 10 nm are visible at RDF for layers j + 1,2.

rdfABC.gif

rdfAB.gif

Figure 2. Radial distribution functions for dots in layer j = 20 and dots in layers j, j+1, j+2, j+3 for system with (a) inheritance factor C = 0.5 and narrower Gauss functions of surface shape and (b) C = 0.0 and broader Gauss functions for determining the shape of cover layer surface. Value of 1 (value in the infinity distance) is marked by dashed line and positions of maxima determining the distance of nearest neighbor dots by dotted line. Positions of maxima reveal that in case (a) the stacking order of the layers is ABC and in the case (b) it is ABA. For detailed description see the text.

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