Ion bombardment induced buried lateral growth: the key mechanism for the synthesis of single crystal diamond wafers

A detailed mechanism for heteroepitaxial diamond nucleation under ion bombardment in a microwave plasma enhanced chemical vapour deposition setup on the single crystal surface of iridium is presented. The novel mechanism of Ion Bombardment Induced Buried Lateral Growth (IBI-BLG) is based on the ion bombardment induced formation and lateral spread of epitaxial diamond within a ~1 nm thick carbon layer. Starting from one single primary nucleation event the buried epitaxial island can expand laterally over distances of several microns. During this epitaxial lateral growth typically thousands of isolated secondary nuclei are generated continuously. The unique process is so far only observed on iridium surfaces. It is shown that a diamond single crystal with a diameter of ~90 mm and a weight of 155 carat can be grown from such a carbon film which initially consisted of 2 · 1013 individual grains.


Monte Carlo simulations of the domain shapes
A crucial additional check for the validity of our model consists in its ability to describe the formation of the different shapes of the domains as we find them in our experiments. They comprise compact circular or nearly square-shaped patterns with smooth edges, corral like structures with the central part nearly free of epitaxial diamond up to highly branched fractal shaped structures (see Fig. 1). Figure 1 also shows crystal islands that evolve in a short CVD growth step from the different types of domains. Similar variations in shape of islands ranging from regular polygons to dendritic fractal patterns have been observed in experiments with a variety of thin layer systems like sub-monolayer growth of metals 1,2 , or the synthesis and etching of intrinsically 2D material systems, e.g. graphene 3 or h-BN 4 . The highly branched shape is typically interpreted as a nonequilibrium state which is caused by kinetic constraints (see for example the diffusionlimited-aggregation (DLA) growth mechanism 1 ).
For a better understanding of the mechanisms which generate the large variety of selforganized domain patterns during BEN on Ir, a second type of MC simulations was performed. It is based on two fundamental observations: 1) Domains can increase in size and shrink on different spots of the same sample only due to slight differences in the local ion bombardment conditions 5 . Thus, there is a kinetic competition between lateral spreading of the crystalline diamond regions along the Ir interface (transformation amorphous-to-crystalline) and the local destruction of crystallinity (back-transformation crystalline-to-amorphous). 2) A continuous 2D layer is a rare metastable situation. Normally, isolated grains with a distance of 15 -20 nm are found after BEN and a short growth step (see Fig. 2(a) and schema in Fig. 3(e) in the main article). This distance corresponds to a nucleation density of 310 11 cm -3 in good agreement with reports from other groups 6,7,8 . We start with a two-dimensional matrix (201x201) of square cells (Fig. 2). The grid spacing of the matrix corresponds to a distance of 18 nm between nuclei in the real experiment. Thus, the simulated region is equivalent to an area of 3.6 x 3.6 µm 2 similar to the regions shown in Fig. 1. Each cell can have two possible states N = 0 (no nucleus) and N =1 (nucleus). The transition "01" is a nucleation event, the transition "10" is a destruction event. In the subsequent simulation the cells of the matrix are selected in a stochastic process. In case the state of the chosen cell is "0", it can be changed to "1" with a certain nucleation probability p N using a random number generator. Switching from "0" to "1" of an initially empty cell can only occur when one of the neighbouring cells is in state "1", i.e. lateral spread only works via direct neighbours. The probability for a "01" transition of a cell is given by the contributions of all neighbouring cells with state "1" (see Eq. (1)). The control parameter "S" is called the nucleation strength. The physical background for the last assumption in the real IBI-BLG process is that carbon atoms in the amorphous region are the easier transformed to crystalline diamond the more direct neighbours are already located in the ordered lattice structure. Diagonal and off-diagonal neighbours are distinguished to account for crystal anisotropy with the parameter "A".
When an occupied cell in the state "1" is selected in the stochastic process it can be destroyed ("10") with a certain probability p D (destruction strength "D"). The probability for this transition is minimal (p D,0 ) when all neighbouring cells are in the state "1" and it increases with every neighbour in the state "0" (see Eq. (2)). Isolated nuclei are most unstable.
Finally, in order to reproduce the observed annular patterns the destruction probability is designed as a dynamic parameter which increases with the number "N n/d " of nucleation/destruction events that have occurred in a given cell (x,y) with progressing biasing time (see Eq. (3).). The potential physical background is a change in the conditions of the Ir surface for prolonged biasing treatment.
with v,w  {-1,0,1} or v,w  {-2,-1,0,1,2} when only first or up to second nearest neighbors are taken into account, respectively (with v,w ≠ 0,0). A schema of the different steps of the MC simulation is shown in Fig. 2. The results are plotted in Fig. 3. In the first row of Fig. 3, homogeneous patterns with smooth edges are shown. They are obtained for a low and constant value of "D". In the framework of our simple model, anisotropy could only be obtained when second nearest neighbours were taken into account.
In the second row of Fig.3 ((e) -(h)), patterns are shown for increasing destruction parameter "D". It is remarkable to see that a rather limited variation of "D" between 0.04 and 0.054 causes a dramatic change from homogeneous to highly branched fractal like patterns. Above 0.054 destruction completely dominates and all nuclei disappear.
In the lower row of Fig. 3 ((i) -(l)) the corral shape patterns could be reproduced. In conclusion, our simple MC simulation which only assumed a dynamic competition between nucleation and destruction events, could reproduce a large variety of features of the domains observed by us and other groups in BEN experiments on Ir.

Transition from highly oriented heteroepitaxial grains to a single crystal
The wafer shown in the main article was synthesized by first nucleating heteroepitaxial diamond on Ir/YSZ/Si(001) followed by growth over several days in a high power 915 MHz microwave setup. During growth, the heteroepitaxial diamond layer passes through different stages as schematically shown in Fig. 4: First isolated crystallites grow and merge to form a closed layer. This layer forms a mosaic crystal in which single crystals are separated by a polygonized network of grain boundaries. During growth of the first microns, the grains coarsen but the network persists. Before reaching a thickness of 30 µm, the network dissolves into isolated bands of dislocations. Growth proceeds as a single crystal accompanied by a continuous reduction of the dislocation density.

Structural properties of the single crystal diamond wafer shown in the main article
In order to characterize the structural quality and homogeneity of the wafer shown in Fig. 5 of the main article, rocking curves have been measured along two perpendicular lines across the wafer (see Fig 5 (a) below). The average value and the standard deviation provide quantitative measures for the structural quality and the homogeneity. In Fig. 5 (b) these values are plotted in a diagram taken from Ref. 10. According to the nitrogen content (< 10 ppm) the wafer is classified as type IIa 11 . The comparison with the rocking curves of natural IIa crystals clearly shows that the wafer is structurally in the range typical for single crystals of this type.