Nitrogen overgrowth as a catalytic mechanism during diamond chemical vapour deposition
Lachlan M. Oberg, Marietta Batzer, Alastair Stacey, Marcus W. Doherty
NNitrogen overgrowth as a catalytic mechanism during diamondchemical vapour deposition
Lachlan M. Oberg , Marietta Batzer , Alastair Stacey , and Marcus W. Doherty , Laser Physics Center, Research School of Physics, Australian National University,Australian Capital Territory 2601, Australia Department of Physics, University of Basel, Klingelbergstrasse 82, Switzerland School of Science, RMIT University, Melbourne, Victoria 3001, Australia Quantum Brilliance Pty Ltd, 116 Daley Road, Acton, Australian Capital Territory 2601,AustraliaNovember 11, 2020
Abstract
Nitrogen is frequently included in chemical vapour deposition feed gases to accelerate dia-mond growth. While there is no consensus for an atomistic mechanism of this effect, existingstudies have largely focused on the role of sub-surface nitrogen and nitrogen-based adsorbates.In this work, we demonstrate the catalytic effect of surface-embedded nitrogen in nucleatingnew layers of (100) diamond. To do so we develop a model of nitrogen overgrowth using den-sity functional theory. Nucleation of new layers occurs through C insertion into a C–C surfacedimer. However, we find that C insertion into a C–N dimer has substantially reduced energyrequirements. In particular, the rate of the key dimer ring-opening and closing mechanism isincreased 400-fold in the presence of nitrogen. Full incorporation of the substitutional nitro-gen defect is then facilitated through charge transfer of an electron from the nitrogen lone pairto charge acceptors on the surface. This work provides a compelling mechanism for the roleof surface-embedded nitrogen in enhancing (100) diamond growth through the nucleation ofnew layers. Furthermore, it demonstrates a pathway for substitutional nitrogen formation dur-ing chemical vapour deposition which can be extended to study the creation of technologicallyrelevant nitrogen-based defects.
Introduction
It is well established that trace amounts of nitrogen (N) accelerate the rate of diamond growthduring chemical vapour deposition (CVD) [1–4]. While the magnitude of this enhancement varieswith plasma composition, pressure, temperature [5], and activation technique [6], a ten-fold increasein the growth rate of (100) diamond can be achieved at high plasma intensities [5]. Despite thewidespread adoption of N-enhanced diamond growth for both academic and commercial purposes,there does not exist a consistent and universally accepted mechanism for the effect. Moreover, aprocess detailing overgrowth of N on the (100) surface to form a bulk substitutional defect remainsunknown. This remains a critical pre-cursor for understanding and enhancing the formation oftechnologically relevant N-based defects in diamond during growth. In particular, increasing thelow yield and alignment of nitrogen-vacancy (NV) centers which has applications in quantumcomputing [7], communications [8], and metrology [9, 10].While the catalytic role of N has generated considerable theoretical interest, existing literaturehas failed to identify an atomic mechanism which can demonstrably produce order-of-magnitudeenhancements to diamond growth rates. The majority of studies have focused on the role of1 a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov igure 1: Conventional mechanism for C insertion into a C–C dimer of the H-terminated (100)diamond surface [17, 19, 22], a pivotal stage of CVD diamond growth. For simplicity we depictonly the C–C dimer and neighbouring ions. Full depictions of the surface are presented in thesupplementary material. In step 1 →
2, a terminating H atom is abstracted by an atomic H radicalwithin the plasma. This creates a radical site allowing for the adsorption of a methyl radical in step2 →
3. Further H abstraction of the methyl adsorbate (3 →
4) creates a radical CH adsorbate.This adsorbate incorporates into the diamond surface via a ring opening/closing mechanism inwhich the dimer expands (4 →
5) to allow formation of an energetically favourable six-memberedring (5 → → ab initio works have primarily studied the impactof N on carbon (C) insertion into a C–C dimer of the reconstructed (100) surface [17–21]. Thisprocess has received substantial attention because it is fundamental for all (100) diamond growth byCVD. Most studies follow or slightly develop the widely-accepted model for C insertion presentedby Garrison et al. [22]. This plasma-surface reaction is itself built on extensive earlier work [23–26]and consists of six key steps depicted in Figure 1.Early work by Frauenheim et al. demonstrated that sub-surface N reduced the binding energyof terminating H atoms and therefore enhanced the rate of initial H abstraction [11]. This wasattributed to charge stabilization of the resulting surface radical (structure 2 in Figure 1) due toelectron transfer from the substitutitonal N lone pair [27]. However, this makes methyl adsorption(step 2 →
3) energetically unfavourable. An alternative growth mechanism was therefore proposedinvolving charge transfer to the metastable anti-bonding orbital of C–C surface dimers and subse-quent adsorption of CH radicals. This hypothesis is challenged by CVD plasma experiment andmodeling which demonstrated that methyl radicals (CH ) are the majority C-based radical speciesproximate to the diamond surface during growth and exceed CH concentrations by three ordersof magnitude [28, 29].Alternatively, more recent works have rationalised Frauenheim et al. ’s original findings by as-suming that H abstraction, not methyl adsorption, is the rate-limiting step for diamond growth.These studies identify that near-surface N defects enhance H abstraction from methyl adsorbates(3 →
4) through charge transfer [13]. Additionally, near-surface N defects promote the creation ofadsorption sites (1 →
2) through weakening of proximal C–C surface dimer bonds [16]. Further ab initio work has demonstrated that sub-surface N increases activation barriers for CH insertioninto the C–C dimer and CH surface migration [15]. Other studies have found that co-adsorbed Natoms have no bearing on H abstraction [16], reduces the adsorption rate of CH on step edges, butincreases the rate of adsorption (and hypothesized to enhance migration) of CH on step-edges [14].2t has also been proposed that N accelerates (111) growth by nucleating new diamond layers [30].In summary, extensive ab initio modeling has not yet provided a clear consensus on the catalyticrole of N in CVD.As a supplement to ab initio calculations, extensive Monte-Carlo modeling has been performedto understand the complexity of mesoscale diamond growth. These works have highlighted theimportance of surface migration [31] and nucleation of new layers [32] to reproduce experimentalgrowth rates and morphologies. For example, the surface of CVD-grown (100) diamond oftenexhibits a relatively smooth and terraced structure [33]. However, note that this is not ubiquitousand depends on reactor conditions [5, 34]. Such morphologies are indicative of step-flow modes, inwhich migration of hydrocarbon adsorbates and preferential adsorption at step-edges is believed tocontribute largely to layer growth. Recent Monte-Carlo modeling has emphasized the importanceof critical nuclei for propagating layer growth. These are immobile surface features, such as a loneC–C dimer, which act as nucleation points [35, 36]. Furthermore, the inclusion of super-nucleatingspecies in Monte-Carlo models has been found to catalyze diamond growth. These are adsorbates orsurface defects (hypothesised to be N-based) which quickly form critical nuclei following formation.When growth is limited by nucleation of new layers (i.e., growth is dominated by step-flow), super-nucleating species have demonstrated ten-fold enhanced growth rates [35]. The search for a N-basedsuper-nucleating species is therefore well founded and forms the primary aim of this work.We identify a potential super-nucleating species by producing the first atomistic model for sub-surface N formation during CVD. Our calculations demonstrate that surface-embedded N catalyzesthe nucleation of new diamond layers during its overgrowth and subsequent encapsulation intobulk diamond. The atomic structure of surface-embedded N is depicted in Figure 2. It consistsof a substitutional defect which maintains the structure of the dimerised 2x1-(100) surface. Thebonding of the C–N dimer resembles that of a typical C–C surface dimer with a C-H covalent bondsubstituted by a N electron lone pair. Surface-embedded N has previously been identified as themost energetically stable form of substitutional N in the (100) diamond surface [9]. Furthermore, itis reported to be evident in NEXAFS scans of diamond (100) surfaces following N plasma treatment[37]. Consequently, it is likely that surface-embedded N is a common surface defect during CVDgrowth of the (100) surface. It is therefore chosen as the initial point for our overgrowth mechanism.Further motivation for this work includes the refinement of existing ab initio techniques usedin previous CVD-growth literature. Firstly, many works employ cluster models [15, 17, 19–21] torepresent the diamond surface. While cluster models are capable of reproducing results consistentwith periodic slab calculations [38, 39], this is not always the case [40], and careful optimisation ofcluster dimensions is always required. As has been noted in previous works [19], the relaxed stericconstraints associated with some cluster calculations result in an underestimation of structuralstabilities during the ring opening/closing mechanism [20]. Secondly, some diamond studies [16,19]employ the drag method for determining the transition state (or do not explicitly mention themethod used [17, 20, 21]) which can produce inaccurate activation barriers for complex potentialenergy surfaces [41, 42]. Instead, it is preferable to use chain-of-states methods when possible, suchas the nudged-elastic-band (NEB) technique, which provide greater consistency in determiningthe transition state [43]. Consequently, in this work we employ state-of-the-art density functionaltheory, including a fully quantum slab model of the H-terminated diamond (100) surface, the useof hybrid functionals, and the NEB method for determining transition states.Our model for overgrowth begins with surface-embedded N as presented in Figure 2 and isseparated into three distinct stages. Firstly, nucleation of a new diamond layer through C insertioninto the C–N dimer is presented in Section 2. We find that surface-embedded N reduces energyrequirements for nearly every stage of the reaction depicted in Figure 1. In particular, the rateof the ring opening/closing mechanism (4 → →
6) is increased by a factor of approximately3igure 2: Surface-embedded N on the (100) surface forms a dimer unit with a neighbouring C atom.400. This provides a compelling atomistic mechanism for the N catalysis of diamond CVD growththrough enhanced layer nucleation. In Section 3 we present the second growth stage where weconsider layer growth emanating from the nucleation point. We demonstrate the formation of anew C–C dimer and C bridging to form a new dimer row. The key result is that surface-embeddedN does not impede typical diamond layer growth. The final growth stage is considered in Section 4,where we demonstrate that full encapsulation of the N defect into the surface (i.e., the formationof sub-surface N) is mediated by charge transfer.
Density functional theory was performed using the VASP plane-wave code [44–47] using PBE [48]and verified using B3LYP [49] functionals. PBE belongs to the generalised-gradient approximation(GGA) class of exchange-correlation functionals. While GGA functionals are computationallyefficient, they are known to over-delocalize electrons and over-stabilize stretched bonds of transitionstates. This can result in underestimations of energy barriers, especially for gas-surface reactions[50]. Hybrid functionals, such as B3LYP, correct the over-delocalisation of GGA functionals byincluding some contribution of exact exchange energy thereby improving estimations of reactionbarriers [51]. However, this increase in accuracy requires significant computational costs and sowe perform the majority of calculations using the PBE functional. To validate the accuracy ofthese PBE calculations we calculate the key ring opening step (4 → × × × < > direction were separated by a distance of 10 ˚A and long-range dipole interactions havebeen corrected for. Γ-point sampling and real-space projection operators have been used for allcalculations. The cut-off energy for the plane-wave basis is 600 eV. The electronic tolerance forsuccessive iterations of the self-consistent field method is taken as 0 . .
05 eV/˚A per ion. Ions composing the top two layersof primitive unit cells (those primarily involved in surface reactions) were allowed to relax in all4BE B3LYP B3LYP B3LYP(slab) (slab) (adamantane) (Cheesman et al. [19])
Step
C–C C–N C–C C–N C–C C–C1 → E -0.026 -0.135 -0.007TS 0.185 0.136 0.2742 → E -4.556 -3.671 -3.868TS 0 0 03 → E -0.228 -0.142 -0.312TS 0.355 0.358 0.3064 → E -0.143 -0.235 0.019 0.290TS 0.057 0.094 0.370 0.4505 → E -0.476 -0.329 -0.540 -0.813TS 0.283 0.571 0.5504 → E -0.350 -0.299 -0.513 ‡ TS 0.959 0.965 † . ‡ → E -4.141 -4.166 -4.573TS 0 0 0Table 1: Reaction (∆ E ) and transition state (TS) energies for C insertion into C–C or C–N dimerson the H-C(100) surface (c.f. Figures 1 and 3). All energies are in eV. † Transition state geometryoptimised using 400 eV cut-off energy. ‡ Effective energy for the direct transition 4 → < > direction. Theeffects of spin polarization on both stable and transition states has been fully accounted for. Theclimbing NEB technique has been used to calculate transition states for all reaction barriers [52].This extension to NEB determines the exact transition state by forcing the highest energy imageto the saddle point. The first stage of growth consists of new layer nucleation through C insertion into the C–N dimer.The results of our ab initio calculations are presented in Table 1, which documents the reactionenergies (∆ E ) and transition state energies (TS) for each of the steps required for C insertion as perFigure 1. This is performed using PBE functionals for both C–C and C–N dimers for the purposesof comparison. We also include several B3LYP calculations performed using both our slab geometryas well as a small adamantane cluster (C H ) as discussed below. Additionally, we compare ourresults to previous values for C–C dimers by Cheesman et al [19]. This is a large-scale hybrid studyin which the reactive C–C dimer and neighbouring atoms were treated using B3LYP functionals,while the remaining ∼ →
6) during C insertion into aC–C dimer. 6 .1 C insertion into C–C
Our results for C insertion into the C–C dimer are largely consistent with existing ab initio works.We identify energy barriers and transition states for initial H abstraction (1 →
2) similar toCheesman et al. and several other studies [17, 18]. However as expected the PBE functionalunderestimates the transition state energy by approximately 100 meV. As with other works, weidentify no barrier to CH adsorption (2 →
3) or H adsorption (6 →
7) [18–21]. While our tran-sition state energy for the H abstraction in step 3 → et al. by49 meV, it underestimates that obtained in the B3LYP cluster studies of Kang and Musgrave byapproximately 100 meV [17].The greatest deviation between our study and existing literature is for the ring opening/closingmechanism, 4 → →
6. We now discuss this discrepancy in-depth due to its relevance for thevalidity of our results and to a general understanding of conventional diamond growth. The majorityof studies find that state 5, the open-ring configuration, is stable with the CH adsorbate realizingan sp bonding configuration (c.f. Figure 1) [17–19, 21]. However, in agreement with the GGAcluster calculations of Oleinik et al. [20], we find that state 5 is not stable and that C insertionoccurs through the direct transition 4 →
6. The transition state for 4 → bonding configuration of state 5. Instead, as depicted in Figure 4, the CH radical inserts into thering structure simultaneously with breakage of the dimer bond.We have confirmed that the instability of state 5 is not a deficiency of the PBE functional.B3LYP functionals in combination with a range of geometry optimisation algorithms (RMM-DIIS[53], conjugate gradient, and velocity damping) also failed to identify a stable sp -like intermediatebetween states 4 and 6. As presented in Table 1, we also perform climbing NEB calculationswith B3LYP for the direct transition 4 → H , resembling the models in Figure 1 if all danglingbonds were hydrogen terminated), the simplest possible cluster model for a surface dimer. Notethat the corresponding C carbon cluster is also the fully quantum region used in several hybridstudies [19–21]. Here we identify that state 5 is stable and that the reaction 4 → → et al. as shown in Table 1.Moreover, we find that the geometry of state 5 in our cluster calculations is considerably relaxedwith respect to our slab calculations. In particular, the separation distance between C atoms ofthe broken dimer of state 5 is large at 2.95 ˚A. As demonstrated in the supplementary, more than1.4 eV is required to achieve a separation distance of this scale within the slab geometry. It isclear that the periodicity enforced by slab models limits the relaxation of the reactive dimer andsurrounding ions. Clusters are therefore inadequate for modeling some aspects of diamond surfacechemistry which are better suited to periodic slabs.Without the erroneous identification of state 5 as stable, cluster models produce results inagreement with our slab model. To see this, consider the effective energies for the direct transition4 → et al . As shown in Table 1, if state 5 is ignored weobtain a transition-state energy of 0.900 eV and a reaction energy of -0.513 eV, similar to respectivevalues of 0.959 eV and -0.350 eV for our work using PBE functionals.7 .2 C insertion into C–N The presence of surface-embedded N greatly enhances C insertion. Firstly, our results in Table 1show that the transition barrier for the initial H abstraction (step 1 →
2) is reduced by 49 meVwhile stability is increased by 109 meV. This is attributable to the greater electronegativity of Nin comparison to C, where we note that the length of the C–N dimer bond is reduced by 5.5%with respect to C–C. A similar reduction in the transition state energy is not observed for theabstraction 3 → adsorbate realizes an sp bonding configuration. Moreover,as per our PBE calculations the transition 4 → → E = − .
235 eVand a transition state energy of 0 .
094 eV. These are in good agreement with the PBE values of − .
143 eV and 0 .
057 eV respectively. This indicates that the PBE functional possesses similaraccuracy to B3LYP for diamond surface reactions (potentially excluding gas-surface interactionsas discussed above), significantly reducing computational costs.The combination of this low transition state energy and the stability of state 5 drasticallyenhances the rate of the ring opening/closing mechanism. This can be quantified using kinetic rateequations to compare the reaction times between C–C and C–N dimers. Defining the populationof states 4, 5, and 6 as φ , φ , and φ respectively we have that φ ( t ) = − Γ → φ + Γ → φ (1) φ ( t ) = − (Γ → + Γ → ) φ + Γ → φ + Γ → φ φ ( t ) = − Γ → φ + Γ → φ . Here Γ i → j = A i → j e − TS i → j /k B T , where A i → j is the attempt frequency for the transition betweenstates i and j , TS i → j is the transition state energy, and T = 1200 K is the approximate temperatureof the diamond surface during CVD [33]. For C insertion into the C–C dimer, one only needs toconsider the direct transition 4 → φ ( t ) = − Γ → φ + Γ → φ φ ( t ) = − Γ → φ + Γ → φ . A quantitative determination of the pre-exponential factors requires calculating the vibrationalpartition functions of each stable and transition state [54]. To compare the pre-exponential rateof ring opening/closing between C–C and C–N dimers, we may instead approximate the ratio ofthermal occupation between their respective normal modes. We find that this does not differ bymore than a factor of three for vibrational energies in the expected range of 100 −
200 meV. Thereforefor simplicity we assume that A i → j = A is constant for all processes involved in ring opening andclosing. This allows us to solve the above system of equations given the initial conditions φ (0) = 1and φ (0) = φ (0) = 0. Processes interrupting ring opening/closing, such as H adsorption andabstraction, are not included. These are assumed to impact C insertion into C–C and C–N at equalrates and therefore do not influence our relative comparison.In Figure 5 we plot φ as a function of time for ring opening/closing with C–C and C–N dimers.These results indicate that ring closure occurs approximately 400 times faster in the presence of8 - N C - C1 10 100 1000 10 ( units of A - ) O cc upa t i ono f s t a t e6 Figure 5: Occupation of state six as a function of time for the ring opening/closing mechanism ona C–C dimer (orange) and C–N dimer (blue).surface-embedded N. Not only does this drastically enhance the rate of C insertion, it also reducesthe residence time of state 4. This would likely result in less etching of the C adsorbate, believedto occur predominately via β -scission [31, 33, 55].Given the large transition state energies required for ring opening/closing during conventionaldiamond growth, step 4 → ab initio studies of (100) diamond growth, this work demonstrates thegreatest enhancement to any key reaction process. For example, previously suggested catalyticeffects of N include enhanced H-abstraction through weakening of surface bonds proximate to sub-surface N [16]. This increases the rate of H abstraction local to the sub-surface N by 2.4. Thisenhancement is minor compared to the 400-fold increase in the rate of ring opening and closing inthe presence of surface-embedded N. By this metric, our work demonstrates the most compellingatomistic mechanism for N catalysis to date. In this section we demonstrate that layer growth propagating from the nucleation point is nothindered by the presence of surface-embedded N. Due to the complexity of the CVD environmentthere are many possible processes that can contribute to layer growth. Hence, we consider tworepresentative mechanisms; formation of a new surface dimer and C addition between dimer rows.We note that these two mechanisms are not necessarily the atomic processes that produce step-flow growth (which have not yet been conclusively identified in existing literature). Instead, thetwo representative mechanisms involve key elements ubiquitous to most layer growth processes and9igure 6: Mechanism for C insertion into a surface dimer adjacent to an incorporated C. Thisresults in the formation of a new dimer unit on the surface and the beginning of a new layer.Reaction energies (in eV) are presented between each step in the reaction, and where applicable wealso include barrier energies in square brackets.therefore assumed to be highly relevant for step flow. These elements include H abstraction, CH adsorption, ring opening/closing, and migration. We calculate the reaction and transition stateenergies of these two mechanisms in the presence of surface-embedded N and compare these resultsto existing literature of analogous processes without N [19]. Figure 6 presents the reaction mechanism and energy requirements for the insertion of C adjacentto the C already incorporated into the C–N dimer. The result is the formation of a new C–Cdimer on the surface. This mechanism follows that identified by Cheesman et al. for formationof the dimer unit in the absence of N. Our calculations have similar energy requirements with theexception of a secondary dimer/opening closing mechanism as discussed below [19].In step 1 →
2, H abstraction forms a radical site on the previously incorporated C. This Cadopts a bonding geometry similar to a free methyl radical, indicating that the N atom does notdonate an electron and maintains its lone pair. This allows for the stable adsorption of CH in step2 → → → adsorbate to bond at the adjacent surface site. Further H abstraction onthe bridging CH produces a reactive radical. In step 7 →
8, the C–C dimer bond is severed toaccommodate C insertion into the dimer. This is also a ring opening/closing mechanism and has anactivation barrier commensurate with that observed for C insertion on a lone C–C dimer (0.860 eVvs 0.959 eV, c.f. Table 1).In the work of Cheesman et al. , step 7 → → →
8. As10iscussed previously, the deviation between our results and that presented in Cheesman et al. islikely attributable to over-relaxation of the transition state due to their cluster model.The new C–C dimer is highly stable and readily adsorbs a further H radical to satiate the finaldangling bond seen in state 8 of Figure 6 (∆ E = − .
179 eV). Note in particular that N does notpartake in any immediate chemistry throughout dimer formation and maintains its lone pair. Allreactions possess similar energetic requirements for abstraction and adsorption as that for a N-freesurface (c.f. Table 1), and consequently N does not appear to benefit or hinder dimer formation.
Following formation of a new surface dimer, layer growth may propagate between dimer rows (oftendenoted the “trough”). The addition of C across the trough is the beginning of a new surface dimeradjacent to the one previously formed, and therefore represents the first step in establishing a newdimer row. Once again, we identify that surface-embedded N does not hinder or benefit this processas it does not directly participate in bonding.The reaction mechanism for C addition across the trough including reaction and barrier energiesis presented in Figure 7. In step 1 →
2, H is abstracted from a ring-closed C on the surface layer.This permits the adsorption of CH onto the radical site in step 2 →
3. Two further H abstractionsmay then occur on the CH adsorbate (step 3 →
4) and on H from a C on the adjacent dimer row(step 4 → adsorbate. This bridging reaction also represents the first half of CH migration between dimerrows. Alternatively, through a mechanism similar to that presented in Figure 6, a new dimer canbe formed across the trough.The addition of C across the trough appears to be a facile mechanism for propagating dimerrows, requiring only two successive H abstractions following CH adsorption. It is therefore apossible candidate for an atomic process which drives step-flow growth. Regardless, although theenergies we determine in Figure 7 are similar to those identified by Cheesman et al. for the analogousprocess without N [19], they argue that the trough bridging mechanism is not substantially fasterthan C insertion into a C–C dimer.Through the three processes considered above, C insertion into a C–C dimer, formation of anew dimer, and C addition across dimer troughs, layer growth may propagate across the surface.When surface-embedded N is not directly involved in the formation of new bonds and growth, itdoes not appear to hinder or benefit growth processes. Instead, the N maintains its stable electronlone pair rather than donating charge to radical acceptor sites on the surface. The exception tothis typical growth behaviour is C addition in the trough adjacent to N itself, which we treat inthe next section.Our results indicate that surface-embedded N catalyzes new-layer nucleation during step-flowmodes but does not necessarily increase the rate of step-flow growth itself. Consequently, it maybe expected that high N concentrations would result in a growth mode dominated by nucleation.Indeed, this effect is observed in experimental results in which increasing the N concentration leadsto morphology changes indicating a transition from step-flow to nucleation dominated growth [5]. Surface growth may therefore proceed unimpeded until the encapsulation of surface-embedded N bythe new layer. A four-fold N bonding configuration requires C addition across the trough betweenthe N and the C in the adjacent dimer row. However, a trough bridging process analogous to thatdepicted in Figure 7 is not viable. This scenario is depicted in Figure 8. In state 1, a radical CH radical to adsorb directly onto surface-embedded N. Any covalent bonding is prohibited by thestability of the N lone pair.We find that a stable bond between N and the bridging CH adsorbate is only possible followingabstraction of a nearby surface H atom. For example, H abstraction on the newly formed dimerunit in step 1 → →
3, the CH radical is able to bridge the trough and form a stable covalent bondwith surface-embedded N. Encapsulation of surface-embedded N defect therefore requires breakingits lone pair and charge migration to a surface acceptor site. While state 3 is stable, reaction 2 → adsorbate preferentially remains radical. This reflectsthe stability of the N lone pair.Exothermic encapsulation of surface-embedded N can still occur through a different reactionpathway. Consider state 1 depicted in Figure 9. In this scenario, growth of a new layer is completewith the exception of encapsulation of surface-embedded N. Methyl adsorption and subsequent Habstractions result in a CH adsorbate attempting to bridge the final trough between a C and thesurface-embedded N. In step 1 →
2, the bridging C adsorbs to the sub-surface C (analogous to step5 → →
3, H abstraction on the bridging C creates a radical site readyto bond with the surface-embedded N. During this mechanism the remaining H on the bridging Crotates away from the N lone pair, greatly reducing steric interactions. Consequently, step 2 → → −
10% of the surface consists of acceptor sites ratherthan H termination (depending on growth conditions [33]), step 3 → → radical is unable to bond with surface-embedded N. The N lone pairprevents formation of a C–N covalent bond. However, abstraction of a nearby surface H (step1 →
2) creates an acceptor site permitting charge migration of a N electron. This breaks the Nlone pair and allows for covalent bonding with the bridging CH in step 2 →
3. Reaction energies(in eV) are presented between each step in the reaction, and where applicable we also includebarrier energies in square brackets. 13igure 9: Encapsulation of surface-embedded N. Here we consider complete growth of a new surfacelayer with the exception of a final bond between the surface-embedded N and a bridging C. Bondingonly occurs if the N lone pair is broken through donation of an electron to an acceptor site (step3 → Ab initio calculationssupport this hypothesis and predict a similar mechanism for NV formation on the (100) surface [9].This would therefore suggest that the overgrowth mechanism in Figure 9 is a critical pointwhich determines either formation of substitutional N or an NV center. If the CH adsorbate wereto be prevented from bridging the final trough (or CH initially adsorbing to the surface), it isconceivable that a vacancy would form during overgrowth of the next layer (the second layer).While we do not identify any such vacancy formation mechanism in this work, one possibility couldbe if growth of the second layer occurs simultaneously with the first layer. The step-edge of thesecondary layer could then overgrow the local surface depicted in Figure 9 before the reaction canproceed, subsequently forming an NV center. However, given the low energy requirements for Nencapsulation in Figure 9 this is unlikely, thus giving rise to the low NV to substitutional N ratiosobserved experimentally. Three decades of ab initio research has produced no consensus on the atomistic mechanism forN catalysis of CVD diamond growth. In this work we identify a new catalytic effect relevant tostep-flow dominated growth modes on (100) surfaces. Specifically, surface-embedded N drasticallyenhances the rate of new layer nucleation by reducing the energy barrier for C insertion. Thepresence of N increases the rate of the ring opening/closing reaction by a factor of 400, the greatest14nhancement to any key (100) diamond growth process to date. Experimental support for ournucleation model may be obtained through correlation of surface-embedded N and nucleated layersfollowing CVD. One way to achieve this could be use of an NV-based quantum microscope to locateN defects in the surface through their hyperfine structure [58]. Conventional surface imaging tech-niques such as STM could then be used to identify surrounding layer growth which has propagatedfrom the defect [59].This work has also established the first atomic model describing overgrowth of N during (100)diamond CVD. We determine that surface-embedded N does not impede typical layer growthprocesses. N maintains its strong electron lone pair and does not participate in bonding duringformation of the dimer unit or C bridging of non-adjacent troughs. The exception is encapsulationof surface-embedded N to form a sub-surface defect through C bridging on the adjacent trough.This reaction requires donation of a lone pair electron to a surface acceptor site. Moreover, theprocess may be endothermic or exothermic depending on the presence of surrounding new-layergrowth. The resulting sub-surface defect is four-fold coordinated and positively charged.
Acknowledgments
We acknowledge funding from the Australian Research Council (DE170100169). This researchwas undertaken with the assistance of resources and services from the National ComputationalInfrastructure (NCI), which is supported by the Australian Government.
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Lachlan M. Oberg , Marietta Batzer , Alastair Stacey , and Marcus W. Doherty , Laser Physics Center, Research School of Physics, Australian National University,Australian Capital Territory 2601, Australia Department of Physics, University of Basel, Klingelbergstrasse 82, Switzerland School of Science, RMIT University, Melbourne, Victoria 3001, Australia Quantum Brilliance Pty Ltd, 116 Daley Road, Acton, Australian Capital Territory 2601,AustraliaNovember 11, 2020
Introduction
In this supplementary material we demonstrate the accuracy of our ab initio simulations andvalidate our slab model of the H-C(100) diamond surface. All calculations are performed usingversion 5.4.4 of the Vienna
Ab initio
Simulation Package (VASP) using the projector augmentedwave method [1–4] with PBE functionals [5]. In Section 1 we determine the plane wave cut-offenergy required to adequately simulate bulk diamond and optimize the lattice constant for allfurther calculations. In Section 2 we investigate the relationship between slab thickness, the H-C(100) work function, and the energetics of C insertion into the C–N dimer. In Section 3 weinvestigate the relationship between the lateral dimensions of the slab, intra-cellular interactionsbetween N defects, and the energetics of C insertion into the C–N dimer. Using these results weidentify the slab dimensions required to adequately represent the physics and chemistry of thediamond surface. Finally, in Section 4 we further investigate the instability of state 5 during Cinsertion into the C–C dimer.
VASP was used to determine the optimal energy cut-off for bulk diamond. The energy cut-offdetermines the size of the plane-wave basis and the minimum value possible must be chosen in orderto maximise computational efficiency. To determine this cut-off, the diamond lattice constant wasfixed to its experimental value of 3.567 ˚A [6] and the k -point sampling was chosen to be a 8 × × ∼ . a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov
00 400 600 800 1000 1200 1400 - - - - - - off energy ( eV ) E ne r g y o f P UC ( e V ) Figure 1: Cut-off energy vs energy of a primitive two-atom diamond unit cell. The energies were de-termined using VASP, assuming the experimental lattice constant of 3.567 ˚A, a 8 × × k -point mesh, and a convergence tolerance of 0.1 meV. The lattice constant resulting in the minimum energy of the bulk diamond PUC was determinedusing VASP. We use an energy cut-off of 600 eV and k -point sampling of 8 × × .
573 ˚A, within 0.25% of the experimental value [6]. This value is used for all future calculations.
The bulk PUC was used to construct slabs of the reconstructed (2x1) H-C(100) surface of varyingdimensions [7]. The computational methodology used is identical to that presented in the maintext.
All surfaces possess a work function, which specifies the energy of the Fermi-level ( E f ) with respectto the vacuum energy ( E vac ) as per W = E vac − E f . (1)For semiconductors, this represents the energy required to relocate an electron from the valenceband maximum (VBM) to the vacuum [8]. The work function therefore provides a reference framefor comparing binding energies and therefore the strength of chemical bonds. The value of W asdetermined by ab initio calculations is a function of slab thickness. Hence, the thickness of the slabmust be chosen to provide a good representation of the surface’s electronic properties.To obtain the vacuum level, VASP was used to produce the xy -planar averaged, local electro-static potential of each slab. The resulting potential is therefore a function of the out-of-planecoordinate (i.e., the < > direction), here defined as z . An example of this potential is shown inFigure 3 for a H-C(100) slab which is four PUCs thick. All energies have been scaled such that thevacuum energy resides at 0 eV. The slab is positioned between 30 and 40 ˚Aand one can observe2 .50 3.52 3.54 3.56 3.58 3.60 3.62 3.64 - - - - - - - ( Å ) E ne r g y o f P UC ( e V ) Figure 2: Lattice constant vs energy of a primitive two-atom diamond unit cell. The energies weredetermined using VASP, with an energy cut-off of 600 eV, an 8 × × − . ≈ .
25 eV for slabs 10 PUCs or greater thick. However, even a thickness of sixPUCs produces a work function within 20 meV of this value.PBE functionals are sufficient to determine the point at which work function convergence occurs.However, due to the band gap problem the converged energies will severely underestimate theexperimental values. This can be resolved through the use of hybrid functionals or GW corrections.Due to the computational demand of the former, the position of the slab VBM and conductionband minimum (CBM) can instead be rescaled according to accepted GW values through the DeltaSCF method. In particular, a recent review by Chen and Pasquarello found that the GW VBMand CBM energies are shifted by − .
66 eV and +0 .
88 eV with respect to the PBE energies [12].3
10 20 30 40 50 60 70 - - - - -
50 xy - averaged electrostatic potential ( eV ) z ( Å ) Figure 3: Averaged xy -planar electrostatic potential vs z -coordinate in a four PUC thick H-C(100)slab supercell. All energies have been scaled such that the vacuum level resides at 0 eV. The slabis positioned between 30 and 40 ˚A into the supercell, indicated by the presence of eight smalloscillations in the potential (the carbon layers) bordered by larger peaks (the H terminations oneither side of the surface). In red, the average potential of the carbon layers has been plotted at − . We also sought to determine whether the trend observed in the work function is reflected in theenergetics for diamond surface reactions. Consequently, we calculated both ∆ E and the barrierenergy (TS) for steps 4 → → Ions in periodic slab models interact with ions in adjacent supercells to effectively reproduce geo-metric properties of the surface. However, this can also result in undesired intra-cellular interactionsbetween single defects or chemical adsorbates which are intended to be studied in isolation. Conse-quently, we now seek to mitigate these interactions by studying the relationship between lateral slabdimensions and the resulting interaction energies. This shall be done in two ways. Firstly, we shalluse the slab band structure to directly calculate interaction energies between surface-embedded Ndefects. Secondly, we will calculate the reaction and barrier energies for steps 4 → →
10 15 20 25 303.163.183.203.223.243.26 Slab thickness ( number of PUCs ) W o r k F un c t i on ( e V ) Figure 4: Work function vs thickness of slab for the H-terminated C(100) surface. - - - -
150 Slab thickness ( PUCs ) Δ E ( → )( m e V ) (a) ∆ E (4 → ( PUCs ) T S ( → )( m e V ) (b) TS(4 → - - - - - -
446 Slab thickness ( PUCs ) Δ E ( → )( m e V ) (c) ∆ E (5 → ( PUCs ) T S ( → )( m e V ) (d) TS(5 → Figure 5: Energetics for steps 4 → → × × × y direction, while adjacent dimers formrows along the x direction as depicted in Figure 6. In these simulations we consider a single N atom embedded in the surface of a H-C(100) slab (e.g.,Figure 6). The interaction energy between N defects in adjacent supercells was determined bycalculating the slab band structure. We sample k -points along the principal axes correspondingto the x and y directions in reciprocal space (i.e., between the Γ and X high symmetries in eitherlateral direction). The resulting wavefunctions are projected onto a Wigner-Seitz sphere centeredon the N defect to determine the corresponding local density of states. This allows us to identifydistinct energy bands which can be attributed to the N defect. As predicted by tight-bindingtheory [13], we find that the dispersion of these bands follows the relationship E ( k ) = α + 2 β (cid:18) πkγ (cid:19) , (2)for some energy off-set α , interaction energy β , and band width γ .In Figure 7 we plot the interaction energy β as a function of the number of primitive unit cellsin the x and y directions. Slabs constructed to demonstrate the dependence on x were two PUCswide in the y direction, while those constructed to demonstrate the dependence of y were two PUCswide in the x direction. This was done to avoid direct bonding between adjacent N defects. Asexpected, the interaction energy decreases with increasing spatial separation. Thinner slabs displayinteraction energies of up to 120 meV, while all interactions become negligible for slabs six PUCswide in either the x or y directions. 6 I n t e r a c t i onene r g y ( m e V ) (a) x direction. I n t e r a c t i onene r g y ( m e V ) (b) y direction. Figure 7: Intra-cellular interaction energy between adjacent N defects as a function of lateral slabdimensions.
In Figures 8 and 9 we present the relationship between x and y slab dimensions and the energeticsof steps 4 → → x were two PUCs wide in the y direction, while those constructedto demonstrate the dependence of y were two PUCs wide in the x direction.Figure 8 reveals that reaction energies have the greatest dependence on number of PUCs in the x direction, differing by up to 40 meV for ∆ E (4 →
5) and 25 meV for ∆ E (5 →
6) between cells threeand six PUCs wide. The dependence of slab width on transition energy is much less pronounced,differing by a maximum of 7 meV. Similar trends are observed in Figure 9 for the dependence onPUCs in the y direction. Note that these increase in multiples of two due to the diamond surfacereconstruction. We find that ∆ E (4 →
5) and ∆ E (5 →
6) differ by up to 12 meV for slabs betweenfour and eight PUCs thick, with the corresponding transition state energies differing by only 7 meV.Despite our limited data for dependence on PUCs in the y dimension, it appears that all energiesconverge beyond eight PUCs in agreement with the results seen for the x dependence.The energy difference observed in the reaction energetics as a function of slab dimensions isconsiderably lower than that observed in the interaction energy of N defects. This is attributableto the fact that N is embedded within the lattice and is therefore capable of long-range interactionsthrough bonding with three adjacent C ions. This is not the case for C insertion, in which themethyl group is adsorbed to the surface and so has a less pronounced effect on overall surfacegeometry.Based on these results we choose for our final slab geometry five PUCs in the x direction andfour in the y direction, as depicted in Figure 6. Five PUCs in the x direction result in reactionbarriers and energies within 1–2 meV of the converged result, and so we sacrifice no major losses inaccuracy with this choice. Based on the results in Figure 9 it would be desirable to choose a slabat least six PUC wide in the y direction. However, this is beyond our computational capabilitiesfor the B3LYP functional and so we have compromised an accuracy of at most 12 meV by selectingthe four PUC wide slab. 7 - - - - -
140 PUCs in x direction Δ E ( → )( m e V ) (a) ∆ E (4 → T S ( → )( m e V ) (b) TS(4 → - - - - - -
450 PUCs in x direction Δ E ( → )( m e V ) (c) ∆ E (5 → T S ( → )( m e V ) (d) TS(5 → Figure 8: Energetics of steps 4 → → x direction. - - - - - - -
180 PUCs in y direction Δ E ( → )( m e V ) (a) ∆ E (4 → T S ( → )( m e V ) (b) TS(4 → - - - - - - -
436 PUCs in y direction Δ E ( → )( m e V ) (c) ∆ E (5 → T S ( → )( m e V ) (d) TS(5 → Figure 9: Energetics of steps 4 → → y direction. 8 .6 1.8 2.0 2.2 2.4 2.6 2.80.00.20.40.60.81.01.21.4 Dimer width ( Å ) E ne r g y ( e V ) Figure 10: Slab supercell energy as a function of C–C dimer width using PBE functionals. Thisrepresents the change in energy to the system during the ring opening mechanism (4 → ≈ .
17 ˚A corresponds to state 4 in Figure 1 of the main text. No minimum isobserved, indicating that state 5 of the C insertion mechanism is a transition state instead of astable geometry. The dotted line is a guide for the eye.
The instability of state 5 during C insertion into the C–C dimer was further investigated by de-termining the relationship between slab energy and dimer width using the PBE functional. Todo so we fix the lateral position of the C which bonds with the CH adsorbate while allowing allother ions to relax. If state 5 is stable, we should expect to see a local energy minimum when thedimer bond is dissociated, corresponding to a C–C bond length of ≈ → et al. [14] and Kangand Musgrave [15] considered this transition and both identified activation energies in excess of2 eV. Our considerably lower estimates of 0.96 eV presented in the main text may be attributed tothe climbing NEB technique, which performs a rigorous search of the potential energy surface.9 eferences [1] G. Kresse and J. Furthm¨uller. Efficiency of ab-initio total energy calculations for metals andsemiconductors using a plane-wave basis set. Computational Materials Science , 6(1):15–50,1996.[2] G. Kresse and J. Hafner. Ab initio molecular-dynamics simulation of the liquid-metalamorphous- semiconductor transition in germanium.
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