CHAPTER 1: Distribution, Diffusion and Concentration of Defects in Colloidal Diamond

It is often convenient to think of nanodiamond as pure, and free of defects, but this is not necessarily realistic. Nanodiamonds can (and do) contain a variety of defects, whether we want them there or not. These include intrinsic point defects, such as lattice vacancies, and incidental impurities, such as nitrogen, which are a result of the synthesis and/or purification processes. In general, defects are always thermodynamically unstable, but the relative (in)stability of these defects, and hence the probability that they can be removed from the particle, can depend on the location of the defect within the particle. This is quite different to the case of bulk diamond, where all lattice sites are geometrically (and, therefore, energetically) equivalent.

There are of course, types of defects that are very useful, and are therefore introduced deliberately. Well known examples are the p-type or n-type dopants used in electronic applications, but there are other types of useful point defects that are not dopants. Collectively these are often referred to as “functional defects” (as they provide some functionality), and include the range of optically active defects and colour centres.^1–3  The most simple defect in diamond is a single, neutral lattice vacancy, which is commonly referred to as a GR1 defect (where GR stands for general radiation). Vacancies are omnipresent in diamond, and so this defect has been extensively studied in various states.⁴ 

Since nitrogen is also widespread in diamond, numerous studies have also focused on characterising and understanding the properties of different types of N-related defects, including the single substitutional nitrogen impurity, known as the C-centre. However, arguably, the most widely studied defect in nanocrystalline diamond is the paramagnetic nitrogen–vacancy complex (N–V),^5,6  which forms when a vacancy (GR1) migrates to bind with a C-centre.^7–9  The energy-level structure of the negatively charged N–V defect results in emissions characterised by a narrow zero-phonon line (ZPL) at 637 nm (the neutral N–V centre has a zero-phonon line at 575 nm)^10,11  accompanied by a wide-structured side band of lower energy due to transition from the same excited state, but with formation of phonons localised on the defect. The optical emission from N–V centres in diamond nanocrystals has been shown to strongly depend on the crystal size,^6,7,12–14  and the charge state is related to the temperature.¹⁵ 

Defects, such as GR1 centres and N–V centres, are mobile within diamond, and may migrate if a driving force is sufficient to overcome the kinetic energy barriers associated with diffusion. The diffusion of an N–V centre is vacancy assisted, and the rate-limiting step is the C–N exchange energy. During this migration, if an N–V centre interacts with another single nitrogen atom (or a migrating vacancy interacts with a nitrogen dimer, known as an A-centre), then an H3 centre is formed. The H3 centre consists of two N atoms surrounding a vacancy. It is one of the most studied in diamond,^1,16  and may be formed abundantly by irradiation with 1 to 2 MeV electrons to doses of 10¹⁸–10²⁰ electrons cm⁻², and annealing at 1200 K for 20 h in a vacuum.¹⁷  If we continue this logical progression, then if an N–V centre migrates to an A-centre, or alternatively an H3 centre migrates to a C-centre (which is far less likely), then an N3 centre may be formed. The N3 centre consists of three nitrogen atoms surrounding a vacancy (whereas a vacancy completely surrounded by four N atoms is known as a B-centre). Both the H3 and N3 provide photoluminescence and cathodoluminescence, are known to be thermally stable, and exhibit high quantum efficiency up to temperatures in excess of 500 K.¹⁸ 

A summary of the point group symmetry, zero phonon line (ZPL), central wavelength (λ₀) and photoacoustic imaging quantum yield for the GR1, N–V, H3 and N3 defects is provided in Table 1.1, where we can see that the quantum yield for each of these defects is quite different. The quantum yield is defined as the number of photons emitted via photoluminescence versus the number of photons absorbed during excitation.¹⁹  The N–V and H3 defects are the most efficient, which somewhat explains their popularity in the scientific community. In addition to the quantum yield the wavelength itself is very important as it determines the penetration depth of the irradiation. The penetration depth for 532 nm light is 0.6 mm, for 670 nm is 2.4 mm, and for 750 nm is 2.6 mm. This is insufficient to penetrate human skin, irrespective of the quantum yield.

To be able to effectively exploit any of these functional defects, we need to draw upon a reliable understanding of the thermochemical stability of the defect within the host particle, in addition to the photostability. The optical properties of each of these defects are intrinsically linked to the physical structure of the defect within the lattice as this determines the energy levels of the excited states. In conventional experiments this can be difficult to probe as information on the physical (or structural) stability must be extracted indirectly by measuring the optical spectra, and if the emission decays, blinks or disappears any underlying physiochemical explanations will be obscured. In contrast, by using computer simulations the stability and properties of different configurations can be accessed directly. It is possible to establish the structure of a defect definitively, irrespective of the location within the particle, and the thermochemical stability can be determined unambiguously.

In this chapter a collection of computational studies examining the stability of a range of different defects will be briefly reviewed, with a focus on how the stability of the host nanodiamond affects the stability of the defect.

Before beginning an exploration of defective diamond nanoparticles, it is firstly important to select the right host structures to use, and develop a general understanding of issues related to other “defects” intrinsic to diamond nanoparticles: the surfaces, edges, and corners.

To determine the lowest energy shape for nanodiamonds enclosed by low-index surfaces, Barnard and Sternberg used the density functional-based tight-binding method with self-consistent charges (SCC-DFTB) to simulate a set of nineteen different diamond nanoparticle structures ranging from 1 nm to 3.3 nm in diameter (142 to 1798 atoms).²¹  This method was selected as it had previously been shown to provide good agreement with higher level quantum chemical methods for all-carbon systems, and is capable of accommodating sizes much larger than those accessible to the purely first principles methods (mentioned above). Within this structure set there were four subsets consisting of octahedral, truncated octahedral, cuboctahedral, and cuboid shapes, respectively. The complete octahedral subset contains C₂₈₆, C₄₅₅, C₆₈₀, C₉₆₉, C₁₃₃₀, and C₁₇₇₁ structures enclosed entirely (100%) with {111} surfaces. The truncated octahedral subset contains C₂₆₈, C₅₄₈, C₈₃₇, C₁₁₉₈, and C₁₆₃₉ structures enclosed with ∼76% {111} surfaces and ∼24% {100} surfaces. The cuboctahedral subset contains C₁₄₂, C₃₂₃, C₆₆₀, and C₁₂₇₆ structures enclosed with ∼36% {111} surfaces and ∼64% {100} surfaces. The final subset of cuboid structures contains C₂₅₉, C₇₁₂, C₈₈₁, and C₁₇₉₈ with ∼34% {100} surfaces and ∼66% {110} surfaces. All of the structures were fully relaxed using the conjugate gradient scheme to minimise the total energy.

In this article the authors systematically modelled the evolution of the core–shell structure for octahedral, truncated octahedral, cuboctahedral, and cuboid shapes over this size range, including explicit examination of the fraction of sp³, sp^2+x and sp²-bonded atoms, and their location. This can be seen in the figure plate provided in Figure 1.1, which is based on a combination of visualisation modes, employing a simple ball method for the sp² and sp^2+x hybridised atoms (where 0<x<1), and the polyhedron method for the tetrahedrally coordinated sp³ hybridised atoms (each of which is surrounded by a coordination tetrahedron spanned by the four neighbours of the central atom). In this figure the diamond-like regions appear as collections of interpenetrating tetrahedra, and sp² and sp^2+x atoms participating in the fullerenic (or graphitic) regions appear as simple spheres decorating the outer surface of the diamond-like regions (connecting bonds not shown). This was designed to make the shape and extent of the diamond-like cores easily discernible,²¹  at the expense of detail in the shell region (which is more prevalently displayed in other works).^22–27 

Based on these results it was determined that there is a relationship between the size of the particle and the fraction of diamond-like and/or fullerenic carbon, but that it depended significantly on the overall shape. In shapes when there is greater than 76% {111} surface area, nanodiamonds are likely to prefer a core–shell (bucky-diamond) structure, and the core/shell ratio depends on the overall size. The authors noted a distinct cross-over between predominately sp² and predominantly sp³ structures at ∼1100 atoms. If there is less than 76% {111} surface area, particles are likely to be stable in the diamond structure with a thin (either single or double layer) shell down to approximately 600 atoms (which is discussed below). It was presumed that a type of confinement by multiple layers is responsible for inhibiting relaxation of sp³-bonded atoms into a sp²-bonded shell, and promoting the stability of diamond-like cores at the centre of the structures with a high fraction of {111} surface area.

The reconstruction of the {111} surfaces on smaller (<2.5 nm) nanodiamonds lowers the surface energy (as the anti-bonding electrons become part of the aromatic character of the fullerenic shell), but introduces considerable surface stress. This was explicitly investigated by Barnard et al.²⁸  using a simple thermodynamic theory to compare nanodiamonds and fullerenes directly. By treating only dehydrogenated nanodiamonds (i.e., nanodiamond structures consisting of mostly sp³-bonded atoms as opposed to bucky-diamond), a direct comparison with fullerenes was made.²⁸  The method was based on the enthalpy of formation as a function of size, expressed in terms of the bond energies for diamond-like and fullerenic particles, the surface dangling bond energy, the number of carbon atoms, the number of dangling bonds on the surface of the particle, and the standard heat of formation of carbon at T=298.15 K.

In the case of fullerenes the closed shell eliminates the dependence on the effective surface-to-volume ratio and, therefore, the size dependence. Thus, a term for the strain energy that vanishes in the graphene limit was added by first making the assumption that a fullerene may be approximated as a homogeneous and isotropic elastic sphere. This was derived by considering the bending and stretching of a suitable elastic sheet in terms of the bending energy per unit area, the bending modulus of the sheet, and the mean radius of curvature. A spherical model was assumed and an expression for the strain energy per carbon atom for fullerenes that is proportional to the inverse of the square of the radius of curvature was derived. Using this model the cross-over in the enthalpy of formation of dehydrogenated (stable) nanodiamond crystals and fullerenes was found to be at ∼1100 atoms, which is approximately equivalent to cubic nanodiamond crystals of 1.9 nm in diameter. An important point in this work was the selection of the chemical reservoir and the frame of reference. The model used a reservoir of free (isolated) C atoms, and included the formation enthalpy of a dangling bond so that the nanoparticles were assumed to be in mutual equilibrium with a continuous diamond or graphitic surface, not the bulk.²⁸ 

To investigate the cases where carbon nanoparticles may contain both sp² and sp³ bonding simultaneously, Barnard et al.²⁹  addressed the stability of multi-shell carbon nanoparticles using the same model described above for comparing the stability of nanodiamonds and fullerenes, and applied it to bucky-diamond and carbon onions. The onions were treated as nested fullerenes by adding a term for the van der Waals attraction (0.056 eV) to the expression used to describe fullerenes.³⁰  The bucky-diamonds were treated in the same manner as nanodiamonds, although, obviously, the dangling-bond-to-carbon-atom ratio is different for nanodiamonds and bucky-diamonds (of similar diameter) due to the formation of the graphitised fullerenic outer shells.²⁹ 

The enthalpy of formation (as a function of particle size) for bucky-diamonds and carbon onions was calculated, and extrapolated along with the nanodiamond and fullerene results mentioned above. From this comparison three main conclusions were drawn. First, the sp²-bonded onion and fullerene results were indistinguishable (within uncertainties) below approximately 2000 atoms. Second, the enthalpy of formation of a bucky-diamond is more akin to carbon onions than to nanodiamonds. Finally, in the region from ∼500 to ∼1850 atoms the results predicted that a thermodynamic coexistence region is formed, within which bucky-diamonds coexist (within uncertainties) with the other carbon nanoparticles.²⁹  This region was then further broken into three sub-regions. From ∼1.4 nm to 1.7 nm the enthalpy of formation of bucky-diamonds was found to be indistinguishable from that of fullerenes (within uncertainties), although carbon onions represent the most stable form of nanocarbon. Between ∼1.7 nm and 2.0 nm bucky-diamonds and carbon onions coexist (within uncertainties), and bucky-diamond was found to coexist with nanodiamond (within uncertainties) between ∼2.0 nm and 2.2 nm. Further, the intersection of the bucky-diamonds and carbon onions stability was found to be very close to the intersection for nanodiamonds and fullerenes at ∼1100 atoms, suggesting that at approximately 1100 atoms an sp³-bonded core becomes more favourable than an sp²-bonded core, irrespective of surface structure.²⁹  Once again, the model used in this study assumed a reservoir of free (isolated) C atoms, and the nanoparticles were assumed to be in mutual equilibrium with a continuous diamond or graphitic surface.²⁸ 

As we can see from these examples, advances have been made in understanding the relative stability of sp²- and sp³-bonded particles at the nanoscale, and the basic structure of diamond nanoparticles has been established. These studies have clearly identified the two important size regimes, where (depending upon the phases under consideration) sp²-to-sp³ or sp³-to-sp² phase transitions may be readily expected. In the case of larger particles the cross-over in stability between nanodiamond and nanographite may be expected at around 5 nm to 10 nm in diameter, and for smaller particles, the crossover between nanodiamond and fullerenic particles may be expected at 1.5 nm to 2 nm. They have also identified the lowest energy morphology (the truncated octahedron) in this size regime.

Based on these results, it is acceptable to use a single, model nanodiamond for exploring the stability of different point defects, provided it is a truncated octahedral (and, therefore, provides a range of both diamond-like and graphitised facets, if unpassivated), and sufficiently large to transcend the quantum confinement regime and (at least) occupy the coexistence regime from ∼500 to ∼1850 atoms. It is important that the model structure meets these criteria so as to ensure that all of the possible local bonding environments are included because the stability of a given defect will be sensitive to these issues.

In the following sections the thermodynamic stability (potential energy surface) and kinetic stability (probability of observation) will be reviewed for the GR1, N–V⁰, H3, and N3 defects in model nanodiamonds. The particles used in the studies reviewed in this chapter are a C₈₃₇ truncated octahedral bucky-diamond and a hydrogenated C₈₃₇H₂₅₂ truncated octahedral nanodiamond, each displaying six {100} facets and eight {111} facets.

All of the calculations were originally performed using SCC-DFTB,^31,32  which is a two-centre approach to density functional theory (DFT), where the Kohn–Sham density functional is expanded to second order around a reference electron density. In this approach the reference density is obtained from self-consistent density functional calculations of weakly confined neutral atoms, and the confinement potential is optimised to anticipate the charge density and effective potential in molecules and solids. A minimal valence basis is established and one- and two-centre tight-binding matrix elements are explicitly calculated within DFT. A universal short-range repulsive potential accounts for double counting terms in the Coulomb and exchange-correlation contributions, as well as the internuclear repulsion, and self-consistency is included at the level of Mulliken charges.³²  This method was selected in these studies as it is more computationally efficient than DFT when such a large number of individual calculations are required.

Since there is currently no way of determining experimentally where a defect is likely to be located in a given particle, a study of the stability of defects in diamond nanoparticles must include a range of possible substitution sites in the bucky-diamond and passivated nanodiamond structures in order to develop a reliable statistical description.

In bulk materials many (if not all) lattice sites can be considered as equivalent, so a single site (calculation) is representative. However, this is not the case in nanoparticles because the properties of a defect at one location within the nanoparticle may be very different to the properties at another location. Since the structure is finite, the location of any lattice site may be uniquely defined relative to the position of any collection of surfaces, edges, and corners. Therefore, by definition, all lattice sites in the nanoparticle are unique. To build a robust statistical description of point defects in a nanostructure, individual defects must be introduced at a number of different lattice sites, so as to sample the full range of crystallographically and geometrically unique lattice sites within the particles.

As mentioned above, in the present chapter fully relaxed C₈₃₇ and C₈₃₇H₂₅₂ nanodiamonds were used as initial configurations, and the various point defects were substituted for carbon atoms located along specific (albeit zig-zagged) substitution paths within the lattice. The “paths” extend from the centro-symmetric atom to different points on the surfaces, edges, and corners. The directions of these substitution paths are shown in Figure 1.2, for the substitution paths terminating at the centre of the {100} surface, {111} surface, {100}/{111} edge, {111}/{111} edge, and the {111}/{111}/{100} corner, respectively. If we consider the nanoparticle morphology to be analogous to the shape of the diamond Brillouin zone, the substitution paths begin at the Γ-point and extend along the X, L, U, K, and W directions, respectively. In all, the point defects were introduced at over 50 geometrically unique sites within the diamond nanoparticles to effectively sample configuration space. Following inclusion of the defect, the entire structure was re-relaxed using the same method as described above.

The probability of observation (P_obs(R,E_K)) of a point defect in a diamond nanoparticle of radius (R) is a function of the kinetic energy imparted during probing E_K, the probability that the defect will diffuse to the surface and escape (P_esc(R,E_K)), and the probability that a defect will be initially created during synthesis (P_form(R)), such that:

The probability of the formation of a specific defect will be proportional to the limiting concentration available during synthesis (c), the kinetic energy during growth (E_K,growth), and will be a function of the characteristic energy of the defect E_d at a position r. This may be approximated by a Boltzmann function, so that:

where P(r) is the probability of the defect being at r, when 0<r<R. As briefly outlined above, there are two distinct structural environments that may surround a point defect. It may be in a sp³-bonded environment, in the bulk-like core region, or in a sp²-bonded environment, such as in the shell. We may therefore simplify this to:

where E_d,core is the characteristic energy of the defect in the sp³-bonded core region, E_d,shell is the characteristic energy of the defect in the sp²-bonded shell region, R_core is the radius of the core, and P_core and P_shell are the probability of the defect being located in the core and shell, respectively. The extent of the shell region has been shown to be related to the excitonic radius of the donor or acceptor, and is important in determining the fraction of atoms occupying each region. To define this quantity we may use P_core=N_core/N and P_shell=1 – N_core/N. Shenderova et al.³³  determined that the total number of atoms (N) in a facetted diamond particle with n atoms along the (111)/(111) edge is given by:

As we will see in the subsequent sections, the shell region consists of anywhere from four to eight atomic layers for H-terminated nanodiamond and unpassivated bucky-diamond, respectively, so this formula may be used to determine N_core by simply calculating the number of atoms in a particle that is the size of the core. Note that if the nanoparticle has less than five atoms along the (111)/(111) edge then it is effectively “all shell”.

Similarly, the total probability of escape will be a combination of contributions from the core and shell regions. Each probability of escape will be a function of the input kinetic energy (E_K) and the escape energies. These escape energies are denoted by E_esc,core(E_K) and E_esc,shell(E_K) for the core and shell, respectively, and are once again described using the Boltzmann function. If E_K is significantly lower than the escape energy for each region then P_esc(E_K) will be negligible, whereas, when E_K=E_esc(E_K), the probability for diffusion approaches unity in that region. Hence, the total probability of escape, P_esc(R,E_K), is:

where E_esc,core=|E_diff,core– E_d,core| and E_esc,shell=|E_diff,shell– E_d,shell| are the differences in the kinetic barrier to diffusion E_diff and the energy of the static defect E_d in the core and shell, respectively. Therefore, the calculation of P_obs(R,E_K) requires only c, E_d,core, E_d,shell, E_diff,core, and E_diff,shell.

If we combine this P_obs(R,E_K) with the total number of atoms (which is the total number of possible defect sites) then we may estimate the concentration of defects per particle (C) that may be expected in a nanodiamond of radius R. In the following sections the formation temperature was assumed to be 3000 K, which is consistent with the formation temperatures of detonation nanodiamond,³³  and the kinetic energy is obtained at a temperature of 300 K. In each case the limiting concentration (c) is assumed to be 1%.

In the first of the studies we will review, Barnard and Sternberg used SCC-DFTB computer simulations to investigate the structural and energetic stability of vacancies in the 837 atom model diamond nanoparticles with clean (reconstructed) and hydrogen-passivated surfaces. The concentration of vacancies (or GR1 defects³⁴ ) in bulk synthetic diamond has been estimated to be of the order of ∼26 ppm,^35,36  and the concentration of monovacancies in polycrystalline diamond has been measured at ≤7 ppm.³⁷  There is also evidence to suggest that the majority of these vacancies will be located in the vicinity of the diamond surface.^38,39  Previous studies have also examined the diffusion barrier for individual vacancy defects in bulk diamond, both theoretically^40–42  and experimentally,^43,44  but this was the first study to consider the preferred location, concentration, and stability of these point defects in isolated nanodiamonds.⁴⁵ 

Figure 1.3 (left) reproduces the radial distribution in the vacancy defect energies for the hydrogen passivated nanodiamond (red symbol and line), and bucky-diamond (blue symbol and line). In this study the relative defect energy, E(r)–E(0), was used, defined as the total energy of the nanoparticle with a given vacancy site relative to the energy of the nanoparticle with the vacancy in the centrosymmetric position. The x-axis represents a scaled (dimensionless) nanoparticle radius defined by dividing the distance from the centre to the average vacancy site (r_defect) by the total distance from the centre to the extremum (R_total), and averaging over each path. Hence r_defect/R_total=0 is the centre, and r_defect/R_total=1 is the outermost vacancy site located on a surface, edge, or corner. In Figure 1.3 (left) the uncertainties in the x-axis are related to the geometric differences between paths, and the uncertainties in the y-axis are the statistical variance in the results, which provide a measure of instability. In general the energetic uncertainties are comparable to (or larger than) the diffusion barrier for neutral vacancies in diamond.^39–44 

The results of this study showed that the (in)stability of the GR1 defect does depend on the location of the defect within the nanoparticle, and on the type of surface structure. Even when the surfaces are stabilised by hydrogen, the vacancy is thermodynamically unstable when a substitution site is within six atomic layers from the surface/edge/corner. At this point there is a thermodynamic driving force for diffusion that increases the closer the vacancy is to the surface/edge/corner. The study also showed that the stability of the nanodiamond itself is also affected by the presence of vacancies. This was particularly significant in the case of the bucky-diamond, where dramatic changes in energy were reported, due to sub-surface graphitisation of the “inner surface” of the bucky-diamond core. In these cases the defect did not need to reside in (or near) the “shell” of the bucky-diamond as the sub-surface graphitisation could be activated when the defect was up to as many as eight atomic layers away from the extrema. This resulted in structural asymmetry, where (111) facets in the vicinity of the defect exhibit a dual-shell (onion-like) structure, while the remaining (111) facets retain the single-shell surface structure.⁴⁵ 

These results suggested that diffusion is likely to occur spontaneously at temperatures used during synthesis, or possibly during irradiation, as there is a strong thermodynamic driving force for diffusion of vacancies in diamond nanoparticles towards the surface (escape) even when the surfaces are stabilised with hydrogen. By combining these energetic results with the diffusion barriers in the core and in the shell, we can see from Figure 1.3 (right) that the probability of observing a GR1 defect is different for hydrogenated nanodiamond and a sp²/sp³ bucky-diamond.

The defects are more likely to be observed in a bucky-diamond compared to surface-passivated nanodiamond, which is counterintuitive when we consider that one of the primary benefits of surface passivation with hydrogen is to stabilise the particle and preserve the sp³ hybridisation. This is because, in the case of the H-terminated nanodiamond, while the defect energy is higher in the “core” region (less favourable than in the shell region), the kinetic energy barrier to diffusion is also higher. In the “shell” region the defect energy is lower (more favourable), but the kinetic barrier to diffusion is lower (than in the “core”), making diffusion of the defect out of the particle (escape) more likely: E_d,core>E_d,shell and E_diff,core>E_diff,shell. In the case of the bucky-diamond the defect energy is significantly higher in the “core” region than in the “shell” region, with a kinetic energy barrier to diffusion the same as the core of the nanodiamond. However, in the “shell” region the defect energy is significantly lower (being much more favourable), but the barrier to diffusion is higher (than in the “core”), making diffusion of the defect out of the particle more unlikely: E_d,core>E_d,shell but E_diff,core<E_diff,shell. The graphitised shell effectively traps the defect and prevents escape as the energetic barrier associated with the inter-layer diffusion (moving from one graphitised shell to the next) is too high.

In addition to this (as the probability of escape is always present) when the limiting concentration is low the majority of the defects are likely to escape. Under the conditions used in this example (c=1%) only half of the defects are retained when the particles are ∼50 nm in size.

Nitrogen is ubiquitous in diamond nanoparticles. This is due, in part, to the fact that it is a primary constituent of the source explosive. Detonation-induced transformations of powerful explosives and their mixtures with the composition C_aH_bN_cO_d, with a negative oxygen balance in a non-oxidising medium, yield a number of condensed carbon phases, including diamond nanoparticles.⁴⁶  Therefore, commercial diamond nanoparticles contain a small fraction of nitrogen, usually between 1% and 4%,^46,47  but it can sometimes be as high as 7% to 8%.⁴⁸  In this case a single nitrogen atom substitutes for a carbon lattice atom, and the paramagnetic electron is located in an anti-bonding orbital between the nitrogen and one of the nearest-neighbour carbon atoms. The defect consequently has <111> (trigonal) symmetry, and is known as the C-centre, as mentioned before.

Using the same approach described above for GR1 point defects, and the same model nanodiamond structures, Barnard and Sternberg also investigated the stability of substitutional nitrogen defects using the DFTB method. The study used the same sampling of substitution paths (and sites), and produced results that are directly comparable with the relative stability of vacancies, as shown in Figure 1.4 (left).⁴⁹  In the case of the relaxed hydrogenated nanodiamond the results showed an interesting relation between the relative defect energy and the location of the substitutional site. It was found that, while there is little energetic difference between the substitution paths in the core region of the particle (between 0% to ∼50% of the distance from the centre), beyond this distance the general trend is toward lower energies, indicating that it is energetically preferable for nitrogen to be located near the surface of the nanodiamond.⁴⁹ 

In the case of the relaxed bucky-diamond, with the exception of the interior of the core, the results showed a gradual increase in energy for nitrogen substitution sites approaching the “inner-surface” of the bucky-diamond core, which is also shown in Figure 1.4 (left). This was followed by a sharp decrease in energy for sites located on the “inner-surface” (approximately 75% of the distance from the centre). These sites represent a “trap” if the nitrogen atoms were driven to diffuse. One path did not follow this trend (and did not include a trap), since there is no surface graphitisation (and therefore no inner-surface) in that direction. In addition to this the coordination of N in the bucky-diamond was found to depend upon the distance from the centre. Within the core the nitrogen atoms were found to be four-fold coordinated, and three-fold coordinated in the shell. This correlates with the general “flattening” of the atomic layers on the inner surface of the bucky-diamond core and all subsequent outer layers comprising the shell.⁴⁹ 

More detailed results for a variety of sizes and particles shapes were also reported, and it was confirmed that the stability of the defects were actually independent of particle size, and related only to the location of the defect with respect to the extrema, regardless of shape.⁵⁰  In general these results revealed that nitrogen atoms prefer to reside near the surfaces/edge/corners of diamond nanoparticles, and not within the core; and there was an obvious thermodynamic incentive for diffusion. This fuelled speculation that nitrogen, and hence other more functional nitrogen-complexes (discussed in the following sections), would not be stable with respect to diffusion, particularly in small particles.

However, when we calculate the probability of observation, we find that this is not necessarily the case. By inserting a limiting concentration of 1% for the nitrogen during formation, we can see from Figure 1.4 (right), that a significant proportion of nitrogen is retained under these conditions. Once again, the bucky-diamond is more efficient at retaining nitrogen than hydrogen-passivated nanodiamond due to the energy barrier associated with the diffusion of substitutional nitrogen from the core to the shell, and between layers in the shell. This barrier is moderated by the C–N exchange energy, which is extremely high in the absence of an adjacent vacancy to relieve the strain. Nitrogen is actually more likely to be found in bucky-diamonds than was originally assumed.

Among the various types of optically active defects, the nitrogen–vacancy (N–V) complex is the most widely studied,^51,52  and forms when a GR1 defect migrates to bind with a substitutional N impurity (a combination of the defects described in the last two sections).^40,53  The energy level structure of the negatively charged N–V⁻ defect results in emissions characterised by a narrow zero-phonon line (ZPL) at 637 nm, while the neutral N–V⁰ centre has a zero-phonon line at 575 nm, both accompanied by a wide-structured side band of lower energy due to transition from the same excited state, but with formation of phonons localised on the defect.⁵¹  For many years, all available data pointed to a strong dependence on crystal size and the surface-to-volume ratio, and the optical emission from such defects was rarely seen in small diamond nanoparticles (<40 nm in diameter).^52,54  Photo-physical characteristics for 25 nm particles were later reported,⁵⁴  and most recently N–V emission from 5 nm detonation nanodiamond agglomerates⁵⁵  and isolated 8 nm diamonds⁵⁶  has been shown.

In order to realise any of the diverse applications for N–V centres in nanodiamonds,³ a clearer understanding of this dependence is imperative. To these ends a series of simulations has also been reported on the relationship between the location of the defect, the stability of the nanodiamond, and the probability of observation (akin to those described above). Once again, the computational work used DFTB and employed the 837 atom truncated octahedral model particle, sampling the configuration space of N–V⁰ and N–V⁻ defects by substituting individual defects at the familiar set of 54 geometrically unique sites along specific lattice directions.¹³  The advantage of using the same method, model particle, and configurational sampling is that these results are directly comparable with the results of the intrinsic and incidental defects described above. The relative energy results for N–V⁰ and N–V⁻ centres were found to be thermodynamically degenerate, and will be termed N–V from this point on.

Presented in Figure 1.5 (left) are the site-dependent defect energies for the hydrogen-passivated nanodiamond and the reconstructed bucky-diamond. In the case of the passivated C₈₃₇H₂₅₂ structure the N–V defects were found to be relatively stable within the particle until the substitution site fell within three atomic layers from the surface/edge/corner. Although the defect is thermodynamically unfavourable within the core (with respect to the “shell”), the energetic barrier for a transformation to a lower energy configuration is still high. At r/R>0.7 there is a ∼1.5 eV to 4.5 eV thermodynamic driving force for diffusion that increases the closer the N–V defect is to the surface/edge/corner.¹³ 

In the C₈₃₇ bucky-diamond structure the defect was found to be highly unstable, and with a substantial thermodynamic (up to ∼6 eV to ∼9 eV) driving force for diffusion within the particle core. In bucky-diamond nanoparticles, where the sp² shell exists, the lattice parameter is different from the bulk, and a significant amount of strain already exists in the particle. This means that the energy barrier for distortion of the N–V centres is lowered, and depending on the position of the defect, the structure of the defective region changes to reduce the total stress and the total energy. This was shown to manifest as sub-surface graphitisation since the defect energy for N–V was lower in sp²-bonded regions, and this provides a significant reduction in the site-dependent defect energy. These transformations are common near the surface of nanodiamonds, even when hydrogen terminated, and can be identified in the cores of bucky-diamonds by the large uncertainties in Figure 1.5 (left), which also provide a local minima.¹³ 

Using these radially averaged values of E_d,core and E_d,shell (see Figure 1.5, right), we find that the vacancy assisted diffusion of N–V defects is more likely than a simple substitutional nitrogen defect (C-centre). The diffusion barrier is still dominated by the N–C exchange energy, but this is (once again) higher in the graphitised shell of the bucky-diamond. A defect diffusing out of the stable core region will be required to penetrate the shell in order to escape, and is more likely to be trapped in a bucky-diamond than the H-terminated counterpart. Using a synthesis temperature of 3000 K and nitrogen concentration of c=1%, an estimate of the probabilities of observing an N–V defect in H-terminated nanodiamond and unterminated bucky-diamond are shown in Figure 1.5 (right).

Based on this model and technique, complementary predications for diamond nanoparticles produced with different synthesis techniques have previously been reported, including for detonation or ultra-dispersed diamond (UDD), chemical vapour deposition (CVD) nanodiamond, and high pressure and high temperature (HPHT) diamond nanoparticles.¹³  These results were rigorously validated by explicitly measuring the emission for 3690 individual nanodiamonds, and correlating the size (determined using atomic force microscopy) with single photon fluorescence. By measuring the second order correlation function it was possible to determine if the detected fluorescence was due to single or multiple N–V centres. In this case the probability of detecting two simultaneous photons was normalised by the probability of detecting two photons at once for a random photon source, where an “anti-bunching” dip in the second order correlation function indicated sub-Poissonian statistics of the emitted photons. This revealed the presence of a single quantum system, which cannot simultaneously emit two photons, and was sufficient to confirm the predictions, and provided additional identification of the critical dimension for which the probability of finding a single N–V defect is optimal (under the conditions employed).¹³ 

Subsequent work by the same researchers and their colleagues has further confirmed the existence and behaviour of N–V centres in isolated 5 nm nanodiamond, and reported on the direct, room-temperature observation of profound surface-controllable luminescence intermittency (blinking), which offered a fresh insight into colour centre behaviour in isolated nanodiamonds.⁵⁷  Based on the DFTB simulations (described and shown above) it was determined that the blinking was related to the defects located in or near the shell of the particles, where the structural instability can alter the electronic states (such as the lowest unoccupied molecular orbital, LUMO), and extinguish the luminescence.

Since the diffusion of an N–V centre is vacancy-assisted, then it is reasonable to assume that an N–V centre may migrate and potentially interact with other point defects. If the N–V centre interacts with another single nitrogen atom (or a GR1 migrates and interacts with a nitrogen dimer, known as an A-centre) then an H3 centre is formed. The H3 centre has an N–V–N structure, and can be generated abundantly by the radiation damage of diamond with 1 to 2 MeV electron beams, followed by heat treatment at temperatures above 800 °C in a vacuum.⁵⁸  As indicated in Table 1.1, this defect emits in the green region, and the radiative lifetime is stable up to 500 °C.

When the sampling and modelling procedure describe above is repeated for the H3 defect, the stability of the defect is found to be similar to the N–V defect (comparing Figure 1.5, left, and Figure 1.6, left), with less uncertainty (more like the C-centre). The defect energy of the H3 in the bucky-diamond is not as low as the N–V defect, indicating that the defect is not as easily accommodated by the strained lattice of the bucky-diamond shell. However, the probability of observation of the H3 defect in the two model particles shows an interesting difference. As shown in Figure 1.6 (right), the probability of observation of an H3 centre in the hydrogenated nanodiamond is similar to the case of the N–V, but the probability of observation of an H3 centre in a bucky-diamond is not. The kinetic stability of the H3 centre is independent of the surface chemistry, which indicates that the fullerenic shell does not effectively trap this defect as it did for the GR1 and the N–V.

Returning again to the issue of N–V migration, there are some other defects that can be formed. If the N–V centre migrates to a nitrogen dimer (A-centre), or alternatively an H3 centre migrates to a single nitrogen atom, then an N3 centre is formed. As mentioned above, the N3 centre consists of three nitrogen atoms surrounding a vacancy (whereas a vacancy completely surrounded by four N atoms is known as a B-centre). The N3 centre is also paramagnetic.

When the sampling and modelling procedure described above is repeated for the N3 defect, the stability of the defect is once again found to be similar to the N–V defect (comparing Figure 1.5, left, and Figure 1.7, left), and the kinetic stability (as characterised by the probability of observation) is similar to the N–V defect too. As shown in Figure 1.7 (right), the probability of an N3 centre is lower than N–V (Figure 1.5, right), which is consistent with experimental observations in bulk diamond.¹⁸ 

In each of the cases described above, the N–V centre is assumed to be mobile, and interacting with a (presumably stationary) nitrogen impurity or complex. However, as the diffusion barrier for a GR1 defect is lower than that of an N–V centre, it is also possible that a vacancy may combine with an N–V, and a V–N–V defect may be formed. This is far more likely than the conditions required to form an H3 or an N3, and is significant since a V–N–V defect is not known to be luminescent.

When the sampling and modelling procedure described above is repeated for a V–N–V defect, we can see that the stability of this defect is different to the other complexes described in this section. As we can see from Figure 1.8 (left), the defect energies for a V–N–V defect within hydrogenated nanodiamond and a bucky-diamond are entirely different. This defect is accommodated much more readily by the bucky-diamond, with up to a 10 meV difference when the defect is located in proximity to a surface, edge, or corner. The defect energies in the bucky-diamond are also much lower than the simple GR1 or N–V, so it is reasonable to assume that if this defect forms it will not separate again into a GR1 and N–V. If this defect were to form, the fluorescence from the participating N–V centre would be permanently quenched.

The kinetic stability of this defect is interesting too (see Figure 1.8, right). The probability of observation for a V–N–V defect is much greater in a bucky-diamond than in an H-terminated nanodiamond, except at small sizes where the situation is reversed. Below ∼12 nm the V–N–V defect is more likely to be found in an H-terminated nanodiamond (unlike all other defects discussed here) as the probability of trapping this defect in a bucky-diamond of this size is negative (due to the exceedingly low defect energy in the shell). Although not shown in Figure 1.8 (right), this disparity increases when we increase the limiting concentrations.

When we compare the stability of all of these defects, some overarching trends become apparent. Figure 1.9 (left) shows the relative stability of all of the N/V defects in the model bucky-diamond, and Figure 1.9 (right) shows the same results for the H-terminated nanodiamond. In the case of the bucky-diamond we can see that, in general, the energy of the defect decreases as the ratio of vacancies to nitrogen increases. When no vacancies are present at all (the nitrogen C-centre) the defect energy is highest. In contrast, when the particle is H-terminated the results for the different defects are practically indistinguishable (within the uncertainties associated with the different possible distributions), irrespective of the composition of the defect.

The relationship between the defect energy and the ratio of nitrogen and vacancies in the defect does not extend to the relative probabilities of observation (see Figure 1.10). As the probabilities are governed by the diffusion barriers, the defects with the greatest C–N exchange energies are the ones that are most likely to be trapped. For this reason the probability of observing the nitrogen C-centre is far greater than the other defects discussed here. The mobility of this defect is effectively zero under ambient conditions, and the observed concentration of substitutional nitrogen impurities is ultimately determined by the formation conditions.

By altering the synthesis temperature, E_K,growth, and c, the probability of observing all of these defects may be improved. In general, increasing the synthesis temperature and quantity of N in the precursors increases the probability that luminescent N–V and H3 defects will be present in the lattice, assuming that diffusion occurs on the same timescale as observed in bulk diamond and graphite. We can also see that the probability of observation is size-dependent, and so by increasing the concentration of N in larger HPHT nanodiamonds a reasonable concentration of functional defects is virtually assured. This is summaries in Table 1.2, which provides a comparison of the luminescent defects present in diamond particles of different characteristic sizes.

The study of other impurities and defects within isolated diamond nanomaterials^46,59  and nanocrystalline diamond films^60–63  with grain sizes in the order of ∼5 nm to 100 nm is also receiving some attention. Some results have also been reported on other types of optically active defects, such as the analogous silicon–vacancy complex.⁶⁴  This defect has a much higher probability of observation,⁶⁵  but due to difficulties associated with Si introduction during synthesis, the N–V defect remains the firm favourite for the majority of relevant applications.

Diamond nanoparticles are never perfect. Given that nitrogen is ubiquitous during the synthesis of diamond nanoparticles (irrespective of the synthesis method), the inclusion of N in the nanodiamond lattice is practically guaranteed. Although it is preferable for N to reside near the surfaces, edges, or corners, diffusion of a C-centre is highly unlikely (due to the barrier associated with the C–N exchange energy), and so under normal conditions it will be trapped at or near the site of its inclusion. On the other hand, GR1 centres are highly mobile, and by introducing lattice vacancies that can migrate and bind with a C-centre, it is possible to generate a measurable concentration of “functional” defects. Depending on the relative concentration of vacancies and N impurities (and the different N configurations), these defects emit stable fluorescence in the visible spectrum, provided the structural integrity of the host particle and the defect can be preserved.

In the case of the host particle the crystallinity is sensitive to the shape of the particle (which determines the crystallographic orientation of the surface facets), and the types of surface reconstructions. In the absence of stable surface passivation, the low energy {111} facets of nanodiamond reconstruct to form a fullerenic shell around the diamond-like core. The resultant “bucky-diamond” has reduced crystallinity, but the formation of this sp²/sp³ core–shell structure has its advantages. The stability of defects in colloidal diamond particles is sensitive to the structure of the exterior shell, and deliberate removal of surface groups to encourage the bucky-diamond reconstructions can be useful to trap defects and prevent diffusion.

Another way of increasing the probability of observation of stable functional defects in diamond nanoparticles is to simply increase the concentration of nitrogen and/or vacancies either during formation or during post-synthesis treatment. This will be challenging in smaller nanodiamonds, less than ∼40 nm, where the expected concentration of functional defects (per particle) is less than one, even with 4% nitrogen in the precursors. In cases such as these a better strategy may be to simply add more nanodiamonds to the sample as then we can be confident that at least some of them will be sufficiently defective.

This project was supported by the Australian Research Council under grant number DP0986752.