T. Philippe, Université de Rouen, GPM, UMR CNRS 6634 BP 12, Avenue de l’Université 76801 Saint Etienne de Rouvray, Groupe de Physique des Matériaux, France. Tel: +33 (0) 2 35 14 71 84; e-mail: firstname.lastname@example.org
The statistical 1NN method is an elegant way to derive the composition of small B-enriched clusters in a random AB solid solution from 3D atomic fields. An extension of this method is proposed that includes the contribution of interface region and provides an estimate of the core composition of clusters. This model is applied to boron-implanted silicon containing boron-enriched clusters. A comparison with the previous model is performed. This new approach gives relevant information, i.e. the core composition of clusters and the cluster–matrix interface width.
The statistical 1NN method is based on the distribution of the nearest neighbour distances and is a means to get the composition of tiny B-enriched clusters present in a random AB solid solution. The main advantage of this method is that it does not rely on the choice of a composition threshold to identify clusters, a choice that may be sometimes arbitrary. In addition, issues related to sampling errors are avoided as the method does not rely on a 3D composition field but on the raw 3D data set (Danoix et al., 1991). However, one of the main shortcomings of this first model was that the system was considered as the sum of two phases of constant composition with abrupt interfaces.
Because of local magnification effects and trajectory overlaps, cluster–matrix interfaces are often diffuse in 3D reconstructions (Vurpillot et al., 2000). This leads to a transition region at the interface where the composition gradually decreases from the core composition of clusters to the matrix composition (Blavette et al., 2001). Moreover, the real interface can also be diffuse because of kinetic effects during growth. In fact, diffused interfaces are therefore regions where the composition is variable. This was not taken into account in our first model. However, these interface regions may be of crucial importance for tiny clusters (nanometre scale). The width of the transient interface (shell) region may indeed be as large as the core region of cluster (Cojocaru-Mirédin et al., 2009).
In this study, a core-shell model is proposed which takes into consideration the interface between the cluster core and the matrix. This second-approximation model is thought to provide more reliable assessments of the core composition in particular for tiny clusters.
The 1NN method
Let us recall the main principles of the statistical 1NN method (Baddeley, 2004; Philippe et al., 2009). A finely dispersed system composed of B-enriched clusters (β phase) embedded in solute depleted α matrix was considered. The distribution for a two-phase system was computed by considering that both α and β phases are random solid solutions and that both phases have no common regions. Interfaces were not taken into account. The statistical 1NN method relies on the distribution law of the first nearest neighbour distances in a random solid solution. B atoms are randomly distributed among A atoms in both α and β phases.
For n atoms B randomly distributed in a 3D volume V, the probability density to have the first nearest neighbour B at a distance r within dr can be demonstrated as (Philippe et al., 2009):
where Q is the detection efficiency (close to 0.5) and C=n/V. The overall density probability related to the two-phase system P(r) can be written as the weighted sum of probability densities as given by Eq. (1) so that:
Pα(r) and Pβ (r) are the intrinsic distributions of both α and β phases:
Cα and Cβ are the composition of α and β phases and C0 the overall composition. The atomic fraction of 1NN B–B pairs in β phase is expressed as:
The phase composition (Ci) can be derived from the most probable distance in the considered phase i:
A best-fit method is applied to experimental distributions P(r) in order to derive phase composition. It is worth mentioning that the 1NN distribution does not follow the theoretical law as given by Eq. (1), where atomic planes are imaged in atom-probe reconstruction. In this latter case, it is obvious that atomic positions are not randomly distributed any more. However, it was found that the theoretical distribution remains approximately valid for low solute concentrations, let us say less than 5 at.%.
Interfaces are not taken into account in the statistical 1NN method. As a result, this method generally underestimates the composition of clusters Cβ, particularly for tiny clusters where interface contribution may be considerable. In the next section, a core-shell model is implemented to the 1NN method that accounts for the interface region around the cluster core.
A core-shell model accounting for interface regions
The main difficulty in the implementation of interface regions is that the concentration in B atoms gradually increases from the matrix up to the cluster core. The distribution of B atoms is therefore not homogeneous in the interface shell region. The theoretical distribution as given by Eq. (1) does not hold anymore. A new distribution law accounting for this transient region (shell) where the composition is not constant has therefore to be established.
In order to simplify the computation of the weighted probability function P(r) related to the interface shell, we have considered the integral function F(r) of P(r). Compared to P(r), the integral probability F(r) has a simpler form:
F(r) is the probability to get the nearest neighbour distance between B atoms less than r (Philippe et al., 2009). The composition C is no more constant but a function C(x) of the distance. In order to get an analytical solution of the problem, the simplest function for C(x) was chosen:
The origin of x is the shell-core interface so that C(x= 0) =Cβ (core concentration) and C(x=L) =Cα (matrix composition).
The weighted distribution law of the first nearest neighbour distances was obtained by integrating F(C(x)) over the space variable x. For the sake of simplicity, we have chosen in first approximation to make integration in Cartesian coordinates:
where Vi(r) is defined as:
It is worth mentioning that Eq. (8) is nothing but an approximation, as in principle the shape of the real 3D concentration field (shape of clusters) should be taken into account in the integration. For instance, for spherical clusters, integration should be calculated in spherical coordinates. Unfortunately, integration in spherical coordinates revealed very difficult if not unfeasible.
In order to get an analytical solution to the problem, the Cartesian coordinates was chosen (Eq. 8). It is important to keep in mind that most likelihood methods used to fit experiments to a model require the use of an analytical expression of P(r). Numerical integration and solution of F(r) or P(r) would lead to considerable complexity in the algorithm used.
As pointed out above, P(r) is the derivative of F(r). As a consequence, the weighted probability density (in m−1) that the first nearest neighbour distance is r within dr is:
Finally, the probability density <P>x expresses as:
This latter expression gives the probability density PI(r) related to the interface region.
Let us now define the distribution of 1NN distances for a mixture of three regions α, β and the interface region (I). The probability density is written as a new weighted sum of intrinsic distributions:
The variable f is defined by Eq. (4) and the intrinsic distributions Pi(r) related to α and β phases are given by Eq. (3).
The variable k is the weight of the interface shell (I) in the volume of β phase. k represents as the ratio of B–B pairs in the interface shell (NIB−B) over the number of B–B pairs in the β clusters. This is the sum of the interface contribution (NIB−B) with that of cluster core (NβB−B):
Most likelihood methods applied to experiments provide a best-fit set of three parameters (the phase composition Cα, Cβ and k, f being a function of phase composition, Eq. 4). Note that Cβ is now the core composition of clusters and k gives the weight of the interface. The thickness of the interface region can be derived from k considering a given shape of clusters. Let us consider that clusters are cylindrical platelets. Let us define R as the radius of the clusters and h its thickness. WR and Wh are the widths of the diffuse interface regions. Let us define e as the relative width of the lateral interface and e′ as the relative width of the longitudinal interface, i.e. e=WR/R and e′= 2Wh/h. In this case, the number of B–B pairs in the interface shell (NIB−B) is given by:
where CI is the average of the concentration in the interface region (CI= (Cα±Cβ)/2). The numbers of B–B pairs in the clusters (NβB−B) is expressed as:
Considering that Cα≪Cβ and using the definition of k (eq. 13), it is easy to show that:
For spherical clusters (radius R), the relative width e (e=WR/R) with Cα≪Cβ is given by:
Let us take an example. For spherical clusters, for k= 0.5, e= 0.3, the width of the interface is approximately three times smaller than the radius of the clusters.
In this approach, the contributions of the three regions are simply added in an independent way without taking into account boundaries among α, I and β. Reality is however different. This affects in particular the matrix. B atoms of α phase close to the interface region can form pairs with B atoms located in the diffuse interface zone (I). Smaller distances are therefore created in the α phase. The problem of interfaces effects persists but to a much lower level using the second-order model.
Model versus experiment
The core-shell model has been applied to boron implanted silicon samples containing tiny boron-enriched clusters a few nanometres in size (Fig. 1). Clusters situated in the implantation peak were found to have platelet shapes (radius R).
Table 1 gives the best-fit parameters (phase composition, f, k) using both the initial statistical 1NN method and the new approach including interfaces. As in every most-likelihood method, there is no analytical expression of the precision (i.e. standard deviation). One way to access the confidence interval consists in using the jackknife method (Efron, 1982). This method consists in removing a given distance class in histogram (one class corresponds to one angström in Fig. 2) and in computing the new values of compositions and k. This procedure is repeated for each class (1, 2, …, 22 angströms). The model was applied to distances up to the maximum distance observed (the number of class is n= 22) in Fig. 2. A list of 23 values of the different parameters of the model is thus obtained (including the whole data set). Then the standard deviation can be classically computed from the n± 1 estimates of each parameter. For the matrix concentration Cα, the standard deviation σα was calculated as:
where Ci,α are the n± 1 best-fit values of the matrix concentration computing with the jackknife method and C0,α is the average of the concentration:
Table 1. Best data fit obtained for the data set (Fig. 1a) for the initial 1NN method and the new 1NN method with a detection efficiency of Q= 0.54 and X0 the given overall composition of B atoms in the data set (input parameters). Xα and Xβ are, respectively, the atomic fraction of B concentrations in α and β phases. rα, rβ are the most probable distances. f is the fraction of B atoms in β phase and k is the fraction of B atoms of β phase in the interface region.
Initial 1NN method
1.20 ± 0.03
9.26 ± 1.04
1.18 ± 0.04
12.20 ± 0.95
71 ± 0.82
The confidence intervals are given in Table 1. The standard deviations (±2σ) show a reasonable accuracy of the results for each model. As expected, the core composition of cluster is found higher than the ‘mean’ concentration in β phase as obtained with the first-approximation 1NN method (without interface contribution). As a result, the most probable distance rβ is smaller in the second-approximation model.
The best-fit weight of the interface (k∼ 0.7) indicates that most boron atoms of the β phase belong to the interface region. Presence of the interface region clearly appears on the cross-section given in Fig. 1(b). Interface regions (green) prove to dominate compared to the core (red zone). Boron-clusters are platelets perpendicular to the direction of analysis. Neglecting the relative width of the longitudinal interface e′, Eq. (16) becomes:
As a consequence, the best-fit value of k leads to e∼ 0.58 (eq. 19) that is in very good agreement with Fig. 1(b).
Comparison between the initial statistical 1NN method and the new model showed that a better fit with experiments was obtained with the latter. Even if the difference is rather subtle, predicted 1NN distribution is closer to experimental when interfaces are taken into account (Fig. 2). Moreover, it is quite clear that the results obtained using this extended approach are in very good agreement with the 3D concentration map shown in Fig. 1. Indeed, the distance distribution (Fig. 2b) is now very coherent with the information given by the 3D concentration map about the core composition of clusters and the interface width.
The interface preponderance is clearly exhibited in the intrinsic distributions displayed in Fig. 2(b). Interface contribution [fkPI(r)] dominates the core contribution [f(1 –k)Pβ(r)]. This is evident from comparing the relative areas of the intrinsic distributions displayed in Fig. 2(b). Intrinsic peaks of the α and the β phases do not appear in the overall distribution. This indicates that intrinsic most probable distances related to both phases cannot directly be deduced from the distribution, only a best-fit method can give the most likely distance (equivalently, the phase composition, Eq. 5).
The contribution of the interface region was included to the statistical 1NN method. Comparison between the initial statistical 1NN method and the new model on a two-phase system indicates a better fit with experiments. Also, this approach brings new information: the interface width, in good agreement with experimental 3D atomic mapping. Furthermore, this model gives the core composition of clusters. Compositions are close to that expected by 2D concentration maps. The new model that we propose seems to be very close to the reality and keeps the main advantages of the initial statistical 1NN method: no statistical sampling errors, no choice of size of sub-volumes and fast computation. It should be kept in mind that this approach remains valid for low concentrated crystalline systems (<5 at.%) even when atomic planes are observed in atom-probe reconstructions.
This work was supported by French National Agency (ANR) (Project ANATEME n° ANR-08-JCJC-0129-01).