Crystallization kinetics during layer exchange of ²⁸Si implanted Al films for fabrication of quantum computers: A theoretical model

Quantum grade Si is a mono-crystalline form of isotopically and elementally pure ²⁸Si absent of any intrinsic bulk defects that could decohere atomic or nuclear spin states.^1–6 When cryogenically cooled and combined with high quality interfaces, quantum grade Si layers could provide ideal “semiconductor vacuums” that could host spin qubits in Si-based quantum computers. In a previous paper,⁷ we demonstrated partial success of a novel enrichment process that could allow the use of industrial CMOS implanters to manufacture quantum grade Si layers. Implanted layer exchange (ILE) is related to conventional, deposition-based layer exchange⁸ but uses ion implantation of ²⁸Si into an aluminum coated Si substrate followed by an anneal to induce layer exchange with epitaxial growth of an enriched, monocrystalline ²⁸Si layer onto the Si substrate. Unlike ILE, enrichment by the direct implantation of ²⁸Si into a Si substrate^7,9,10 produces an enriched layer whose thickness is dependent on the energy (and hence ion range) of the implant. Enrichment is a consequence of dilution of the minor Si isotopes by the implanted ²⁸Si so the enrichment value depends on the implant fluence until a saturation value at very high fluences is reached. The highest enrichment values can only be achieved in regimes where Si self-sputtering yields are unity or smaller. For Si, this means that implantation must proceed either below 3 keV (which is susceptible to layer oxidation) or above 45 keV (which requires high fluences of ∼4 × 10¹⁸ ions cm⁻² to reach the maximum enrichment possible). In contrast, ILE does not implant ²⁸Si into the Si substrate and so avoids oxidation and implant related contamination of the enriched layer.^7,8 As ILE produces an enriched layer thickness that depends on the amount of ²⁸Si introduced into the Al (i.e., implant fluence),⁷ throughput is dictated by the required enriched layer thickness rather than the enrichment value. ILE enrichment depends on the ion beam purity rather than the dilution of diluting minor Si isotopes in the substrate and is independent of implant energy. This gives ILE the significant advantage of requiring lower implant fluences than direct implantation (6.6 × 10¹⁷ ions cm⁻² was used in this study) and implant energy could be chosen to maximize implanter throughput when manufacturing wafers.

Our previous paper⁷ reported that process optimization is required to promote exclusive monocrystalline epitaxial growth on the substrate, rather than poly silicon formation within the Al layer and to eliminate Al contamination in the enriched layer. To understand if there are conditions that might produce mono-crystalline exchanged layers, this paper proposes a naïve theoretical model to predict rates of poly-crystallization and epitaxial growth. Our model builds on approaches used for conventional (i.e., deposition-based) layer exchange but does not involve a barrier layer between Si and Al. The model is used to explain our initial results,⁷ and some further observations of ILE experiments ( Appendix A) that show different fractions of epitaxial and polycrystalline growth after varied anneal conditions. We then predict anneal conditions that have the best chance in the future to produce mono-crystalline quantum grade Si by ILE.

Deposition-based aluminum layer exchange (ALE) has mostly been studied on amorphous substrates because it proceeds below the Al/Si eutectic temperature (577 °C) and so enables the use of low temperature substrates.⁸ The commercialization of ALE was attempted for the manufacture of low-cost solar cells on glass.^8,11,12 The deposition-only process entails depositing an Al layer on a substrate. After the formation of a surface aluminum oxide, amorphous Si is deposited onto Al. An anneal, typically at 500 °C for ∼1 h, dissolves Si into Al which diffuses and then nucleates to form crystals of Si. By the end of the process, the original Al layer is replaced by a layer of poly-crystalline Si with the original Si layer being exchanged with Al. Crucially, the oxide interface between the deposited Si and Al layers limits the speed of Si introduction (and therefore Si concentration) in Al to reduce the Si crystal nucleation rate and produce poly-Si layers consisting of a few large grains (which are favored over layers consisting of many small grains due to their superior electrical conductivity). Majni and Ottaviani¹³ briefly reported that epitaxial layers could be formed when deposition layer exchange was carried out on a crystalline Si wafer as opposed to a glass panel. Later, Civale et al.¹⁴ published a more in-depth study of epitaxial layer exchange of areas on Si.

Our implant-based layer exchange enrichment process (Fig. 1) leverages conventional deposition-based layer exchange approaches but replaces a step of depositing Si above an Al layer with a ²⁸Si implant into the top of an Al layer.

• Al is deposited onto a Si substrate wafer whose surface has been oxide stripped and chemically cleaned. ²⁸Si is implanted into the top of the Al layer at an energy such that ions cannot reach and damage the c-Al/substrate interface.

• An anneal below the eutectic temperature causes diffusion of the implanted ²⁸Si through the Al layer.

• ²⁸Si diffuses to epitaxially grow onto the crystalline Si substrate wafer.

• Ideally, at the end of the anneal, all ²⁸Si has grown epitaxially as a continuous mono-crystalline layer on the substrate and the Al layer has now exchanged position to be above ²⁸Si. In nonideal cases, Si crystal nucleation in the Al layer competes with epitaxial growth and poly-crystalline Si regions can be formed.

The following naïve, qualitative model was developed to estimate characteristic times required to produce ²⁸Si layers through epitaxial growth or nucleation in the untouched Al using simple assumptions and boundary conditions. It is not a quantitative model that follows the changes in the layers as the whole process progresses but is intended to determine which crystallization processes will dominate under the different anneal conditions described in  Appendix A. The framework of the model is discussed broadly in the following sections with more details and equations contained in  Appendix B and full calculations included in Excel spreadsheets in the supplementary material.³⁷ Throughout this description, we combine the common conventions of prefixing crystalline materials with “c-” and enclosing them using chevrons ⟨⟩ and prefixing amorphous materials with “a-” and enclosing them using braces {}.

The naïve ILE crystallization model is summarized in Fig. 2. The implanted layer (blue/green) can be seen above the untouched Al layer (green), which is above the natural Si substrate (gray) onto which an epitaxial ²⁸Si layer (purple) has begun to grow. The implanted ²⁸Si atoms in the implanted layer and those that diffuse into the untouched Al from the implanted layer are considered to be in the amorphous phase before epitaxially growing onto the substrate or crystallizing in Al. The energy released by this Si phase change is the driving force for the ILE process. A polysilicon grain that has nucleated and is growing in the untouched ⟨c-Al⟩ is represented by a (purple) sphere. These layers are generally considered from a macroscopic perspective so that global values for parameters such as dissolved concentrations, Gibbs energy, diffusion coefficient, Miedema and crystal nucleation models are used whereas, in reality, these parameters can vary locally according to the microscopic structure. This macroscopic consideration is valid in the implanted layer but is a simplification in the poly-crystalline untouched Al layer, where values can be very different within the grains compared to around grain boundaries. The thick solid (purple) line in Fig. 2 shows the ²⁸Si concentration throughout these layers soon after ILE initiation (after 0.3 s at 250 °C—see Sec. II B) when the equilibrium Si concentration gradient has become established in the untouched Al layer.

The ²⁸Si atoms that have diffused into the untouched Al out of the implant layer are considered to be in a {a-Si} phase so that the driving force for poly-crystallization or epitaxial growth is largely driven by the Gibbs energy released by the change to the ⟨c-Si⟩ phase.

The diagram also shows the Si diffusion flux (J_diffusion) generated by the Si concentration gradient and velocities of growth of the epitaxial layer (v_epitaxy) and grain (v_grain).

In Fig. 2, ²⁸Si from the implanted amorphous layer has dissolved into the untouched Al layer at their interface (the horizontal dotted line labeled “{a-Si}/⟨c-Al⟩>  interface”). The naïve model initially takes an over-simple assumption that this interface is sharp (“sharp interface model”) to simplify the diffusion calculations (Sec. II B). In reality, the ²⁸Si/30 keV/6.6 × 10¹⁷ cm⁻² implant into Al resulted in a constant ²⁸Si atomic fraction of 80% (a concentration of ∼3 × 10²¹ atoms cm⁻³) over the first 70 nm beneath the surface with an interface that decreased to 0% over the next 100 nm (Fig. 2 of Ref. 7). The model also does not consider the upward movement of the position of the interface as ²⁸Si diffuses out of the implanted layer. The vertical dotted–dashed line labeled “C({a-Si}max)” shows the maximum concentration of {a-²⁸Si} that can be dissolved into Al at the interface. When the dissolved {a-Si} concentration exceeds the equilibrium dissolved ⟨c-Si⟩ concentration, C(⟨Si ⟩ sol), the ratio C({a-Si})/C(⟨Si ⟩ sol) is known as the supersaturation, S. The vertical double dotted–dashed line “C(⟨c-Si⟩ sol)” shows the equilibrium concentration of dissolved ⟨c-Si⟩ in Al, as reported by the equilibrium phase diagram.¹⁵  C({a-Si}) is calculated from the difference of the Gibbs energy between the {a-Si} and ⟨c-Si⟩ phases ( Appendix B). The values of Si phase concentrations and S against temperature are shown in Fig. 3. It should be noted that as the temperature decreases, supersaturation increases, even though the solubilities of both crystalline and amorphous Si in Al decrease.

Calculations ( Appendix B) show how ²⁸Si diffuses through a 150 nm thick untouched Al layer and first reaches the substrate after 0.001 and 0.3 s at 500 and 250 °C, respectively. ILE has no initial delay associated with slow Si diffusion through an interface barrier as in ALE.¹⁶ ILE Si concentrations (∼3 × 10²¹ atoms cm⁻³) at the sharp interface with the untouched Al are consequently higher than those of ALE¹⁶ (∼5 × 10²⁰ atoms cm⁻³) leading to ILE Si fluxes (see spreadsheet “Homo'Nucl in Al” of the supplementary material)³⁷ which are ∼5× higher than those of ALE.¹⁶ 

Consistent with previous epitaxial growth studies,¹⁷ it is assumed that ²⁸Si reaching the substrate is immediately removed by epitaxial growth. The solid (purple) line in Fig. 2 shows the ²⁸Si concentration that decreases linearly at steady state through the depth of the untouched Al for the ideal case of epitaxial growth only. The epitaxial growth rate at the substrate, v_epitaxy, is governed by the diffusion flux, J_diffusion, of ²⁸Si atoms that diffuse out of the implanted region maintained by this Si concentration gradient.

In the model, nucleation rates are estimated using the classical nucleation theory (CNT) with volume and surface energies either extrapolated from experimental values or calculated ( Appendix B). Homogeneous nucleation considers uniform nucleation at nonspecial sites throughout a material. In a material in which there are no pre-existing nuclei, grains must first nucleate before they can grow. After an incubation time, the distribution of existing grain sizes will have built to its steady state distribution and reached sizes that are large enough to be observed. Heterogeneous nucleation can occur if there are special sites at which the rate of nucleation is locally enhanced (for example, on grain boundaries within the untouched Al where the surface energy to produce a nucleus is reduced). Dissolved ²⁸Si can grow onto these two types of nuclei at a rate that, like epitaxial growth, is diffusion-limited.

Poly-crystal nucleation and growth is considered in two regions of the model:

Heterogeneous nucleation can be ignored in the implanted region because we have shown⁷ that the implant destroys all heterogeneous nucleation sites. If Al in the implanted layer is ignored, the layer can be approximated as being pure {a-Si} and the CNT framework of Spinella et al.¹⁸ can be immediately applied in which crystallization proceeds due to jumps of atoms from the {a- Si} onto ⟨c-Si⟩ nuclei. We also consider a more sophisticated approach that uses a Miedema Model¹⁹ (see  Appendix B 5 for more details) to account for the presence of Al in the implant layer when calculating nuclei surface and volume energies before using the same CNT framework.¹⁸ 

ALE models^11,20 considered crystallization in Al during ALE to be caused by nucleation of Si from a supersaturated solid solution of {a-Si} in Al. For ILE, this implies that the nucleation rate varies with depth in Al alongside S and the diffusion driven ²⁸Si concentration gradient. The surface energy changes during homogeneous nucleation are estimated from a Miedema Models ( Appendix B). Heterogeneous nucleation in the untouched Al is considered to be controlled by the density of heterogeneous nucleation sites without a period of incubation.

Equations discussed in  Appendix B were used within the model framework described above to predict crystallization times in the implanted region and untouched Al (as shown in the supplementary material).³⁷ 

Figure 4 shows that both implant layer model assumptions (with or without Al) predict Si crystallization times at all temperatures in the implant region to at least four orders of magnitude longer than any of our anneals. The long crystallization time explains that no Si is observed in the surface Al after ILE because the amorphous Si diffuses away and is consumed by epitaxy before it can crystallize in the implanted region. The slow crystallization also accounts for the absence of poly-Si crystallization in our previous experiment (described in Ref. 7) when Si removal by epitaxy from a fully implanted 100 nm thick Al layer (i.e., thinner than the implant range) was stopped by implant damage to the substrate surface.

Compared to the crystallization models of the implanted region, the atomic jump rates in the untouched Al, dictated by the diffusion rate of ²⁸Si through the layer, are much faster than atomic jumps in {a-Si}. Predicted crystallization times in the untouched Al are well within the experimental anneal times. Figure 5 compares crystallization times for all the implanted Si via homogeneously nucleated poly-crystallization and monocrystalline epitaxial growth in the sharp interface model calculated for temperatures in a range up to just below the Si-Al alloy eutectic temperature. Figure 5(a) compares the Si crystallization times for the highest Si supersaturation values present at the sharp interface for full Si dissolution as shown in Fig. 3 and clearly shows that poly-crystallization dominates over epitaxial growth at all temperatures. Figure 5(b) compares the two crystallization mechanisms if the supersaturation were half the maximum values. This corresponds to the situation at the interface if the dissolution of Si at the interface were for some reason halved, or the situation in the middle of the untouched Al layer for full Si dissolution. The most important prediction at this reduced Si supersaturation is that, while the rate of poly-crystallization is higher at low temperatures, epitaxial growth becomes dominant above 325 °C. As a check of the sensitivity of the model, Fig. 6 compare the temperature trends of the epitaxial and poly-crystallization times for different values of the Si solubility. It is evident that the sharp interface model predicts that epitaxy dominates over poly-crystallization in anneals that quickly ramp to 500 °C when the concentration of dissolved {a-Si} is below 0.7 times C({a-Si}max).

The need to restrict the supersaturation of Si in Al is not a surprise because this tactic is used in ALE: the oxide layer reduces Si supersaturation in Al which, in turn, reduces the Si nucleation rate and favors large-grained poly Si growth. A reduction in dissolved Si for ILE in the actual implanted interfaces below that assumed by the sharp interface model is also reasonable. The actual implanted Si concentration at its interface with Al does not rise sharply to its maximum value but has a graded profile (calculated in Ref. 7). He et al.²¹ showed, using thermodynamic considerations, that ALE is initiated by Si dissolving onto the vertical grain boundaries in the untouched Al. The high fluence implant will have damaged the Al grains at the interface and it is reasonable that this will slow down Si dissolution. The upward movement of the interface position into the implant damaged region as Si is consumed is also likely to reduce Si dissolution.

For the assumption that the dissolved Si is indeed lower than the maximum, the experimental observations ( Appendix A) are compared to the model predictions in Fig. 5(b). The short crystallization times predicted by the naïve model when dissolved {a-Si} is less than maximal explain these process outcomes. The 20 s stabilization time in the thermocouple-controlled 500 °C RTA anneals was shorter than the crystallization time at 250 °C. The samples were then quickly heated to 500 °C, epitaxial favoring the crystallization region to form predominantly epitaxial layers that had completed well within a first 30 s dwell time. This explains why a further 30 s cycle anneal did not appear to change the results [ Appendix A, Fig. 8 rows (c) and (d)]. In contrast, crystallization had completed in the low temperature, poly-crystallization favoring regime during the 50 s, 250 °C thermocouple-controlled RTA anneal. Low temperature poly dominated crystallization had also completed during the 120 s 250 °C stabilization step used in the pyrometer-controlled RTA and the slow (hour) temperature ramp of the sample in the edge of the furnace tube. The time subsequently spent at 500 °C for these anneals therefore had no influence on the extent of epitaxial or poly-crystalline growth.

The model therefore predicts using a 500 °C RTA with no temperature stabilization step promises to improve further the amount of epitaxial growth to poly-crystallization. As ALE anneal times are often hours long,^8,11,12 we had not anticipated before this modeling effort that the crystallization could complete within the anneal thermal ramp times rather than during the high temperature part of the anneal.

Using supersaturation as a proxy for position in Figs. 5 and 6 can also be used to compare the rates of polysilicon crystallization at different depths in the untouched Al. This predicts that poly-crystallization will nucleate near the implant/untouched Al interface where the diffusion profile has a maximum concentration of dissolved {a-Si} and supersaturation. The observations of pockets of Al trapped at the interface between the enriched Si layer and Si substrate [see as examples Figs. 8(e3), 8(f1), and 8(f3)] are consistent with this nucleation at the top of the untouched Al layer followed by downward Si growth that entraps Al upon the substrate. We believe that Al measured in the enriched layer is presently dominated by these Al inclusions.⁷ This Al incorporation mechanism should be eliminated when only epitaxial growth occurs. When this is achieved, sensitive measurements of Al dissolved in the epitaxial Si will become worthwhile.

Surface lumps, which we assign to heterogeneous nucleation and growth, were formed at all anneal conditions and the modeling confirms the idea from ALE that heterogeneous nucleation will be important unless the density of nucleation sites can be made sufficiently small (see below and Fig. 8).

Al poly-crystals in the untouched Al were observed to be approximately 100 nm across [Fig. 8(a)], leading to heterogeneous nucleation site density estimates of 10²¹ m⁻³ (assuming one site per grain boundary, consistent with Ref. 20) to 10²⁴ m⁻³ (if, more pessimistically, each grain boundary were to contain 1000 sites). It is interesting to observe in Fig. 7 that one nucleation site per grain allows epitaxial growth to be faster at higher temperatures, while heterogeneous crystallization is always faster for 1000 nucleation sites/grain boundary. We believe that the number of heterogeneous nucleation sites depends on the quality of the layer deposition and can be made small. As discussed in Secs. II D and III A, the results of implantation into a 100 nm thick Al layer showed that implantation could completely destroy heterogeneous nucleation sites,⁷ but care must be taken not to damage the substrate interface.

Combining the learning from the previous sections suggests the following strategy for the production of mono-crystalline enriched Si using ILE:

Eliminate heterogeneous crystallization with an absence of heterogeneous nucleation sites in Al. This requires development of a high-quality Al deposition technique which could be assisted by an implant step to destroy nucleation sites.

Eliminate homogeneous poly-Si crystallization by using rapid ramp anneals to the highest temperature possible below the eutectic temperature with no temperature stabilization step. The dissolution rate of Si into the Al should be limited. While this appears to be a consequence of the ²⁸Si implant, an additional implant or oxide deposition step may optimize the rate of Si introduction into Al.

Maximize the epitaxial growth rate onto the substrate by ensuring a high-quality substrate surface absent of surface defects, oxide, and other chemical contamination. The maximum epitaxial growth rate will also minimize poly-crystallization by helping to keep Si supersaturation in the Al low.

Although Al contamination is not a consideration of this study, mono-crystalline growth should eliminate Al in Si due to trapped inclusions. Al contamination should then be exclusively due to Al dissolved in Si. While higher temperatures increase equilibrium Al solubility, high sensitivity measurements would be required to assess if the equilibrium concentration of dissolved Al were reached in these fast anneal times.

In this paper, the dissolution, diffusion, and crystallization mechanisms occurring during ILE were investigated theoretically to explain observations of poly- and epitaxial- crystallization rates observed in our annealing experiments. After eliminating heterogenous nucleation sites in the Al, our model suggests that, at high silicon supersaturation in the untouched Al layer, homogeneous nucleation can occur rapidly at low temperatures in timescales similar to the thermal stabilization step of our anneals. At low ultimate anneal temperatures, poly-crystallization in the untouched Al dominates over epitaxial growth onto the substrate. Epitaxial growth can be favored by the use of higher anneal temperatures (>500 °C) with fast ramps if the extent of Si supersaturation in Al is limited, perhaps as a consequence of the interface produced by the implant process that limits the rate of Si dissolution into Al. Further experiments at these conditions will be required to show if exclusively mono-crystalline layers can be produced and high sensitivity measurements can be made to assess the level of Al contamination.

The authors were supported through the Surrey Ion Beam Centre which is funded by U.K. EPSRC (No. NS/A000059/1) and industrial income. E.S. was a Ph.D. student supported by a Marion Redfearn Scholarship. The Surrey TEM facility was supported by a U.K. EPSRC grant (No. EP/V036327/1). The authors would like to thank Surrey Ion Beam Centre staff Luke Antwis, Alex Royle, Julian Fletcher, and Keith Heasman for running implants, Bob Mirkhaydarov for preparing the Al films using equipment in the Advanced Technology Institute of the University of Surrey, David Cox and Mateus Masteghin of the University of Surrey Advanced Technology Institute for TEM lamella preparation training, Vlad Stolojan of the University of Surrey Advanced Technology Institute for TEM training and Sukanta Biswas, Yifei Meng and Marissa Stevens-Hassell of Eurofins EAG for TEM measurements of some of the annealed samples. We also acknowledge John Watts, University of Surrey School of Mechanical Engineering Sciences, for his helpful discussions during the preparation of the manuscript.

The authors have no conflicts to disclose.

Ella B. Schneider: Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Jonathan England: Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

We have carried out a range of anneals on 150 and 200 nm thick Al layers after high fluence ²⁸Si implants (30 keV/6.6 × 10¹⁷ ions cm⁻²) as an initial survey to determine the best conditions to produce quantum grade Si layers. The substrate preparation, Al layer formation, and implant steps are described in more detail in our previous work.⁷ 

The starting anneal conditions were informed by those used in the conventional ALE^11,22 that had typically used 500 °C for 1 h¹¹ and our earliest unsuccessful attempt²³ that had used 400 °C for 3 h and formed just a few isolated polycrystals.²³ Anneals up to 500 °C were carried out in a Jipilec Jetfirst thermocouple controlled rapid thermal annealer (RTA) and a pyrometer controlled Jipilec Jetstar RTA (which we believe was less accurately temperature controlled due to the low emissivity of Si at such low temperatures). All 500 °C RTA anneals used a standard method of stabilizing the sample temperature at 250 °C during the heat up ramp (for 20 s in the thermocouple- and 120 s in the pyrometer-controlled RTA) to allow the system to stabilize thermally before being heated to the final temperature which was maintained for periods between 30 s to 5 min. Anneals were also carried out in a thermocouple-controlled Carbolite TZF 12/100 tube furnace. The furnace tube took 1 h to heat up to 500 °C during which time the samples were located at the end of the furnace tube outside the temperature-controlled region before being pushed into the full temperature-controlled central region for 1 h. The samples were pulled back into to the end of the furnace tube during cooling, which also took several hours. All anneals were carried out in nitrogen atmospheres.

Top-down optical microscope images and TEM lamellae were made of the as-implanted and post annealed samples. Bright–field (BF) STEM-EDX maps were made of the lamellae to identify the elemental composition of the layers and crystal state was determined by diffraction patterns [by selected area (SAD) measurements or FFT of the image] or inferred by BF image contrast. An earlier paper⁷ describes in more detail the experimental implant and metrology parameters of this study.

Figure 8 shows cross-sectional BF STEM and top-down optical images after the implant and anneals.

Our earlier paper⁷ described that the dark circular and irregular blemishes seen in post anneal, but not post implant, top-down images (columns 2 and 4) of Fig. 8 indicate where large heterogeneous Si grains had grown beyond the thickness of the original deposited Al layer. For the post anneal samples, the lamella shown were made within the lighter-colored planar regions between the blemishes where ILE had progressed. All cross-sectional bright-field (BF) STEM images show the substrate at the bottom with a straight, dark line indicating the position of the original Al/substrate interface. In the future, improved surface cleans and oxide removal before Al deposition could make this interface invisible.

The top row of Fig. 8 [(a1)–(a4)] shows the post implanted layers and indicates the crystal poly-crystalline structure of the untouched ⟨c-Al⟩ layer that sits upon the Si substrate. The following rows [(c)–(f)] of Fig. 8 show post anneal results. Rows (b)–(d) show anneals carried out in the thermocouple-controlled (RTA), row (e) used the pyrometer-controlled RTA and row (f) shows furnace anneal results. The exchanged layers immediately above the original Al/substrate interface are Si rich (as confirmed in many by the false color STEM-EDX maps). Poly-crystallinity in rows (b), (e), and (f) is indicated by regions of different contrasts (caused by different electron transmission through crystals of different orientations) and confirmed in (f1) by a SAD diffraction pattern that shows characteristic poly-crystal rings rather than the spots from the mono- crystalline substrate. Epitaxial growth is indicated in rows (c) and (d) by the exchanged Si having the same image contrast as the substrate. The mono-crystallinity and identical orientation of the exchanged layer and substrate is confirmed in (c1) and (d3) by FFT and SAD diffraction patterns, respectively.

It should be noted that the exchanged layer (e1) appears to contain both poly-crystals and a region of epitaxial growth. Layers (e3) and (f1) and (f3) include Al trapped on the original Si substrate interface.

Al displaced during the ILE anneal is visible as the topmost layers above the exchanged Si, confirmed in the false color EDX maps and by the characteristic dark stripe as described by us previously.⁷ Si is not detected by EDX in the exchanged Al after any anneal.

The ultimate anneal temperature and stabilization time were important for influencing the crystallinity of the Si layer. The TEM results are summarized in Table I. Epitaxial growth dominated for a high ultimate temperature (500 °C) reached after the shortest (20 s) stabilization step (rows c and d). There appears to be little difference between 1 and 2 cycles in the thermocouple-controlled RTA, indicating that ILE had completed during the first 30 s. The low temperature [250 °C, row (b)] or slow to reach 500 °C anneals [rows (e) and (f)] produced predominantly poly-Si layers. Large poly-grains are observed after the furnace anneal where Ostwald ripening^11,24 had time to occur.

This appendix uses a slight change of notation from that used in the body of the paper to describe parameters. Notations such as C({a-Si}max) are clearer in diagrams and the main text, these are replaced using sub- and super-scripts, such as $C { a - Si } max$⁠, which are clearer in equations.

The diffusion coefficient can vary over many orders of magnitude depending on the microscopic characteristics of the Al layer.^17,25,28,29 In this paper, we assume a pre-exponent value D_Si,0 = 2.9 × 10⁻⁷ m² s⁻¹ and activation energy E_a = 0.79 eV/atom.³⁰ These values predict times for ²⁸Si to diffuse through a 150 nm thick untouched Al layer and first reach the substrate after 0.001 and 0.3 s at 500 and 250 °C, respectively. This rapid diffusion through the Al is consistent with the macroscopic models developed by Nast¹¹ and Sarikov and Schneider.¹⁶ Once the diffusing ²⁸Si reaches the substrate, excess ²⁸Si is removed from the solid solution by epitaxial growth and a linear ²⁸Si concentration gradient (shown in Fig. 2) is established in the untouched Al layer with a constant ²⁸Si flux at all depths. Unlike ALE, the ILE model does not include an oxide barrier between the amorphous (implanted) Si and polycrystalline Al.

Classical nucleation theory (CNT) has been usefully described by Spinella et al.¹⁸ in the context of crystallization anneals of Si amorphized by implantation. Our ILE model uses both their theoretical framework and parameters that they extracted from experiments.

Homogeneous nucleation considers uniform nucleation at nonspecial sites throughout a system. In a system in which there are no pre-existing nuclei, grains must first nucleate before they can grow. After an incubation time, t_inc, the distribution of existing grain sizes will have built to its steady state distribution and reached sizes that are large enough to be observed. This model considers an observable radius, using a TEM, to be 25 nm¹⁸ describes how the incubation time follows two stages in Si. Initially, in the single kinetic phase, atoms attach themselves onto the lattice orientation of the pre-existing nuclei. Above a threshold radius, R_T, it becomes faster for particles to attach via a twin assisted mechanism.

Although the Miedema model in the untouched Al predicts slightly higher critical nucleus energy barriers (Table II), the effect of higher barriers on retarding crystallization times is more than compensated by the higher transition rate.

The increase of Si supersaturation at lower temperatures (Fig. 3) causes important temperature trends in the volume crystallization values which ultimately affect the trends in the steady state critically sized nuclei production rates, R. For the highest Si supersaturation values (Fig. 3), R increases with temperature. If S were limited to half of those values, R decreases with increasing temperature (Table II).

Sarikov et al.²⁰ quote a value of Z = 0.01 at 500 °C for heterogeneous nucleation, a value they justify fits experimentally observed nucleation rates. We calculate the Zeldovich factor for homogeneous nucleation using the first term in our Eq. (12). For homogeneous nucleation, and consistent with our approach in the implanted region, we again take the number of nucleation sites to be related to the local concentration of a-Si, C_{a-Si}. The transition frequency has been estimated from the diffusion flux hitting the cross-sectional area of the critical sized nucleus. These values can then all be used to calculate the incubation and crystallization times (see the supplementary material,³⁷ spreadsheet “Homo'Nucl in Al”).

It is tempting to apply the identical approach to estimate characteristic times for heterogeneous nucleation in the untouched Al. After all, this is the approach developed by Ref. 20 to describe heterogeneous nucleation in ALE. For nucleation sites, the number of dissolved a-Si atoms is now replaced by the number of heterogeneous sites on grain boundaries. With typical grains observed to be approximately 100 nm across, this leads to heterogeneous nucleation site densities of 10²¹ m⁻³ (for one site/grain boundary) to 10²⁴ m⁻³ (if each grain boundary were to contain 1000 sites). The calculation of the nucleation volume term, $Δ G v o l n u c l$ stays the same as for homogeneous nucleation but Sarikov et al.²⁰ assumed that $γ i n t e r f a c e n u c l$ was reduced to 0.01 J m⁻² at 500 °C on heterogeneous nucleation sites, a value justified, alongside the choice of Z, that allowed their ALE model to reproduce experimental data. However, applying CNT to heterogeneous nucleation for ILE in its regime of transition rates driven by unrestricted diffusion does not make sense for two reasons. First, reducing the surface energy leads to a prediction that the radius of the critical nucleus would be much smaller than a single atom. Secondly, the steady state critical nuclei creation rate would exceed the number of heterogeneous nucleation sites available at all temperatures considered. An alternative estimation has been taken in which the nucleation rate has been limited to the density of heterogeneous nucleation sites. It is apparent that a period of incubation need not be considered before growth commences.

The crystallization times predicted using CNT with a limitation of the steady state nucleation rate to the density of pre-existing nucleation sites leads to the interesting hint that heterogeneous growth in the Al layer could become slower relative to epitaxial growth onto the substrate as the temperature is increased.

Benedictus et al.³³ described a thermodynamic model for solid-station amorphization at interfaces and grain boundaries in binary systems using methodology originally developed by Miedema to calculate values of volume and surface energies.¹⁹ Miedema's model is a semiempirical, thermodynamic approach for estimating the formation enthalpy of alloys.¹⁹ An alloy is considered to be composed of cellular atoms akin to macroscopic pieces of the constituent elemental material. The alloy formation energy consists of a “melt” energy and “mixing” energy. The “melt” energy is related to the fusion enthalpies of Si and Al (energy absorbed when a substance changes from solid to liquid) as the atomic structure of a solid amorphous substance is assumed to be similar to that of a liquid. The “mixing” energy is the energy required to mix the two “molten” phases, which depends on the number of interfaces or “interactions” between dissimilar Si and Al atom-cells. The mix energy also contains a term describing the entropy change associated with the atomic arrangement of atoms: mixing increases configurational disorder which is energetically favorable.

Zhao et al. applied the Miedema approach to investigate the driving forces behind ALE.^21,34–36 We have also used this approach to calculate volume and surface energies for our ILE system using values from, and comparing our calculated values against, these publications to check the validity of our calculations (see the supplementary material,³⁷ spreadsheet “Miedema”). As some values reported in these papers sometimes differ and due to uncertainties in the material properties, we have changed some of calculated volume and surface energies by ±10% to check that the model conclusions are not overly sensitive to the input parameters (see the supplementary material,³⁷ spreadsheet “Homo'Nucl in Al”).

A selection of calculated values from the model is compared in Table II for two temperatures.