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A model accounting for spatial overlaps in 3D atom-probe microscopy
Ultramicroscopy 89 (2001) 145–153
A model accounting for spatial overlaps in 3D atom-probe microscopy D. Blavette*, F. Vurpillot, P. Pareige, A. Menand Group of Material Physics, UMR CNRS 6634FUFR Sciences, Universite de Rouen, 76821 Mont-Saint-Aignan Cedex, France Received 18 July 2000; received in revised form 20 October 2000
Abstract The spatial resolution of three-dimensional atom probe is known to be mainly controlled by the aberrations of ion trajectories near the specimen surface. An analytical model accounting for the spatial overlaps that occur near phase interfaces is described. This model makes it possible to correct the apparent composition of small spherical precipitates in order to determine the true composition. The prediction of the overlap rate as a function of the particle size was found in remarkably good agreement with the simulations of ion trajectories that were made. The thickness of the mixed zone around b precipitates was found to be of 0.3 nm for a normalised evaporation field of b phase of 0.8. Using simulations, the overlap rate could be parameterised as a function of the apparent atomic density observed in particles. This model has been applied to copper precipitation in FeCu. r 2001 Elsevier Science B.V. All rights reserved. PACS: 07.78.+s; 41.20.q; 41.75.i Keywords: Atom probe; Phase composition; Local magnification; Modelling
1. Introduction Three-dimensional Atom Probe (3DAP) is the only nanoanalytical microscope combining atomic-scale resolution with 3D imaging capabilities [1– 3]. Besides the impressive technological improvements that were made during this last decade, many efforts need to be made in the theory of the instrument. One of the major key problems comes from a poor knowledge of the real sample-toimage transfer function. The 3D images produced are reconstructed currently using more or less *Corresponding author. Tel.: 33-2-351466-51; fax: 33-235146652. E-mail address: [email protected] (D. Blavette).
sophisticated procedures, but they have in common that they all rely on the underlying poor assumption that 3DAP is a simple point projection microscope which is evidently not true. The spatial resolution of 3DAP images may be decomposed into two components. The depth resolution, essentially controlled by field screening effects, is excellent, typically one-tenth of a nanometer in metals. In contrast, the lateral resolution, parallel to the emitter surface is far from achieving atomic resolution (a few tenths of a nanometer). The lateral resolution is known not to be limited by the performance of the positionsensitive detector used but instead by the physics of field emission of ions in close vicinity of the tip surface. The simulations of ion trajectories near an
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atomically defined emitter that we have developed recently, well support this idea [4–7]. Most of the aberrations in images could be accounted for. In these simulations, the atomic-scale topology of the field ion emitter and its dynamics of evolution during field evaporation were considered. Crucial problems interfering from the investigation of two-phase samples essentially come from the aberrations in the ion trajectories, notably in the close vicinity of interfaces. These give rise to biased compositions particularly for small particles, which have a smaller evaporation field than the surrounding matrix. A typical system where local magnification effects occur is Fe–Cu. The essential question here is the true composition of copper-enriched particles in the early stages of precipitation or under neutron irradiation. Pure Cu particles are low-field regions (darkly imaged in FIM), compared to the surrounding matrix. High-field iron-rich regions between particles therefore develop a smaller radius of curvature at the tip surface. Defocused iron ions coming from the surrounding matrix are therefore suspected to fall into the precipitate image on the detector. This obviously gives rise to an artificial enrichment in iron within the reconstructed images of precipitates. This local magnification effect is illustrated in the simulations of ion impacts on the detector
(a)
given in Fig. 1. For a low evaporation field of the particle (eB ¼ E b =E a o1), the image of the precipitate (pure in B atoms) is compressed and ions coming from the surrounding matrix (pure in A) fall in the particle image (Fig. 1(a)). In contrast, for eB > 1; precipitate ions fall outside the particle and a depleted zone forms close to the interface (Fig. 1(b)). More details on the procedure and results of simulations can be found in Ref. [7]. Historically, the problem of the true composition of precipitates in Fe–Cu has raised numerous debates. The first investigations of Fe–Cu were made by the group of Brenner in 1973 using 1D atom probe [8]. They found 50–70 at% of Cu in the early stages of phase separation instead of 100 at% as predicted by the classical nucleation theory. The first 2D images were produced with the PoSAP developed by Cerezo et al. in 1988, who claimed to have detected close to 100% of Cu in particles of 5 nm in diameter [1]. The 3D images of neutron-irradiated Fe–Cu were published recently by Pareige et al. using the Tomographic Atom Probe [9]. The physical problem behind is independent of the technological details of the instrument. Composition biases are essentially controlled by the up-to-now unresolved problem of the local magnification effects that occur near phase
(b)
Fig. 1. Simulation of impact images for a binary AB alloy containing a spherical b particle (eight interatomic distances in diameter). b phase is pure in B and the a solid solution is pure in A. These two simulated images are cross sections of reconstructed precipitates with (a) eB ¼ 0:85 and (b) eB ¼ 1:15: B atoms are in black and A are in grey. The (1 0 0) pole is situated close to the left-hand bottom corner of each image.
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interfaces [10]. In this paper, a tentative model accounting for the spatial overlaps which occur near phase interfaces is described. This model makes it possible to correct the apparent composition of small precipitates. Using simulations, the overlap rate could be parameterised as a function of the apparent atomic density in reconstructed particles. In contrast to the evaporation field of precipitates, the apparent density is experimentally available and is a parameter which reflects the importance of local magnification effects. Application to 3D images of neutron irradiated Fe–Cu alloys are discussed.
2. Modelling
A atoms. As a result, the definition of the precipitate region is not ambiguous and the measurement of the precipitate composition poses no problem. Experimentally, the definition of the particle region is not univocal, particularly when the solid solution contains a relatively high content in B. In the latter case, isoconcentration surfaces may be used to define the contour (i.e. interface) of precipitates. A threshold slightly higher than the matrix composition may be used. The sampling statistical fluctuations are, evidently, to be considered in the choice of this threshold. Let us define the overlap rate (Z). This latter quantity is the number of atoms falling in bðdnÞ divided by the initial number of atoms contained in b without overlap: Z ¼ dn=nb ;
Let us consider a binary alloy (AB) containing small spherical particles (b), which have an evaporation field lower than the surrounding a matrix (eB ¼ Eb =Ea o1). This is notably the case of FeCu or of Mg2Zn particles in AlZnMg alloys and this is a more problematic case, compared to systems for which eB > 1 (Fig. 1(b)). In the latter case overlaps also exist, but it is much easier to get a plateau in concentration profiles and then to determine the true composition of the particle in its core. The analytical model developed below can be adapted readily to cases, where eB > 1: Let us consider, a single precipitate embedded in an infinite matrix. The true concentration CB0b is simply: CB0b ¼ nB =nb with nb being the total number of atoms in b that are likely to be detected. The measured concentration (CB ) is biased because some of the atoms located around the particle (dn ¼ dnA þ dnB ) fall inside the image of b (Fig. 1(a)):
ð1Þ
so that CB ¼ ðCB0b þ ZCBa Þ=ð1 þ ZÞ; b 0b a ¼ ðCA þ ZCA Þ=ð1 þ ZÞ: CA
ð2Þ
This result can easily be generalised to systems containing m chemical species: Ci0b ¼ ðCi0b þ ZCia Þ=ð1 þ ZÞ;
i ¼ 1ym:
ð3Þ
(Ci0b )
Conversely, true concentrations can be deduced from measurements through the following expression: Ci0b ¼ Cib þ ZðCib Cia Þ:
ð4Þ
It is worth noting that when the matrix is a dilute solid solution, then Eq. (4) reduces to Ci0b ¼ Cib ð1 þ ZÞ
if Cia 5Ci0b :
CBb ¼ ðnB þ dnB Þ=ðnb þ dnA þ dnB Þ with
Let us consider that the ternary alloy i ¼ 1; 2 are the solutes and i ¼ 3 the major species. It becomes clear that the stoichiometry (K12 ) of a precipitate is not biased by the local magnification effects (for Mg2Zn we have K12 ¼ 2):
a dnA ¼ dnCA
K12 ¼ C1b =C2b ¼ C10 b=C20b :
dnB ¼ dnCBa ;
where C represents the measured atomic concentration, whereas primed superscripts will always hereafter refer to corrected values (C 0 is the true a concentration). CA and CBa are the concentrations away from the particle. They are assumed not to be biased. In Fig. 1(a), the solid solution is pure in
2.1. The apparent phase fraction It is now easy to demonstrate that the molar fraction (f ) of b phase that has been measured
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from the number of atoms in each phase ða; bÞ is also biased: f ¼ ðnb þ dnÞ=N ¼ nb ð1 þ ZÞ=N: Note that N is the total number of atoms collected in the analysis (N) is, in general, not biased as the local magnification effects are conservative in contrast to preferential evaporation effects. The true fraction (f 0 ) is given by f 0 ¼ nb =N so that f 0 ¼ f =ð1 þ ZÞ:
ð5Þ
The latter equation can also be deduced from the lever rule, which expresses the conservation of matter. Writing f 0 ¼ ðCB0 CBa Þ=ðCB0b CBa Þ CBa Þ=ðCBb CBa Þ leads also to f ¼ f ¼ ZÞf 0 with CB0 being the nominal concentration ðCB0
and ð1 þ of B atoms. Note that the apparent fraction of the b phase in reconstructed images can easily be measured by counting the number of atoms (n0b ) inside precipitates f ¼ n0b =N; with N being the total number of atoms in the reconstructed image. Practically, n0b can be estimated using isoconcentration surfaces with the threshold chosen a little bit higher than the matrix level (iso-surfaces should contain the transient zone caused by trajectory aberrations near interfaces). It is worth mentioning that the apparent atomic fraction is not to be mistaken with the apparent volume fraction. Due to the local magnification effects, b particles appear in reconstructed images with a much smaller volume than expected. This problem is discussed later. Eq. (5) indicates that overlaps lead to an overestimation of the atomic fraction of b particles, through a modification of the apparent partitioning ratio ðCib =Cia Þ: As Eq. (2) shows, the concentration in the b particle is simply pushed towards that of the matrix along the tie line linking the composition of the phases. This additional number of atoms in b coming from the close vicinity of interfaces increases the apparent atomic density in reconstructed b zones. But this is not the only factor contributing to this local density effect. The lower evaporation field of b leads to a larger radius of curvature of b zones emerging from the tip
surface and this give rise to a focusing effect. This in turn leads to a compressed image of b on the detector. This can be expressed as a local compression of the reconstructed volume of bðvb Þ: The apparent volume of b can be written as vb ¼ wv0b with w being the compression factor. We can now express the apparent atomic density (r) as r ¼ ðnb þ dnÞ=vb The true density can be written as r0 ¼ nb =v0b so that the normalised density measured in b particles is rr ¼ r=r0 ¼ ð1 þ ZÞ=w;
ð6Þ
with Z being the overlap rate as defined in Eq. (1) and w; the compression factor w ¼ vb =v0b
ðwo1Þ:
ð7Þ
This compression of the particle volume leads to a drastic decrease of the apparent volume fraction F; which can be written as a function of the true volume fraction F 0 in the following form: F ¼ wF 0 : If a and b have the same molar volume then F 0 is equal to the atomic fraction f 0 and the measured volume fraction F is expressed as F ¼ wf 0
with wo1 for eB o1:
ð8Þ
Even though the approach is not quantitative, it is interesting to express w as a function of the local magnifications. As only the lateral dimensions, parallel to the tip surface, are affected by the local magnification effects, only two of the three dimensions of reconstructed precipitates contribute to the compression of the reconstructed volume of b: We assume that the depth scale is not modified which is reasonable as the total number of ions collected in each layer is conserved (the depth is proportional to the cumulated number of collected ions in the reconstruction procedure in 3DAPs [11]; the depth calibration issues are also discussed in [12]). The initial spherical particle therefore transforms into an
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ellipsoid. This is illustrated in the 3DAP image of FeCu in Fig. 4 and confirmed by simulations [13]. The elongation ratio (i.e. the aspect ratio of ellipsoidal precipitates) defined as the ratio (e) of the long axis to the small axis of this ellipsoid (axes parallel to the tip surface) is given by the ratio of local magnifications:
the nanometer scale which is unfortunately the scale at which magnification effects should be taken into account. Note also that the phenomenological parameters ka and kb involved in Eq. (14) are likely to depend on many factors (eB ; particle size r; etc.) and this dependence is unknown.
e ¼ ðGa =Gb Þ
2.2. Theoretical expression of the overlap rate
ð9Þ
and then the compression factor is given by the ratio of the volume of an ellipsoid to that of a sphere; w ¼ 1=e2 ; so that wBðGb =Ga Þ2 :
ð10Þ
Let us introduce a phenomenological expression for the local magnification: G ¼ L=aR
ð11Þ
L is the flight path and a is a parameter related to the projection features. If we now introduce the radii of curvature Ra and Rb of a and b regions at the tip surface, then, we can write w ¼ ðaa Ra =ab Rb Þ2 :
ð12Þ
In this model, the evaporation field of b was chosen smaller than that of aðeB o1Þ so that Ra oRb and wo1: If we now want to express w as a function of eB then, the following phenomenological expression for the electric field generated above a curved surface is needed: E ¼ V=bR:
ð13Þ
V is the applied voltage and b a factor related to the local geometry of the emitter. Writing this latter equation for both a and b phases and introducing the ratio k ¼ b=a; leads to w ¼ ðkb =ka Þ2 e2B :
ð14Þ
For many reasons, one has to be cautious in the use of such an expression. It relies on several assumptions: (i) two distinct radii of curvature and (ii) a unique centre of projection for each phase. This is certainly not true, in particular, near interfaces where the local curvature smoothly changes from that of the a solid solution to that of b zones (there is no negative curvature at the tip surface). Moreover, it is worth mentioning that the notion of radius of curvature looses its meaning in proportion as the size of a precipitate approaches
It is now interesting to express the overlap rate (Z) as a function of both the size and the reduced field ei : We assume that the parasitic atoms falling in the image of bðdnÞ come from a shell around the precipitate that has a volume dv; and a thickness dr: This implies that Z is proportional to dv=v0b (v0b the true volume of the b particle). If we now introduce the probability (p) that atoms of the shell really reach the precipitate region (v0b ), then one can write Z ¼ pdvðdr; rÞ=v0b
and
Z ¼ pðsb =v0b Þdr:
ð15Þ
sb is the area of an interface subjected to overlap. In this first approach, the shell thickness (dr) can be considered as a ‘‘weighted’’ spatial resolution averaged over the contour of particles (only 2 of the 3 dimensions are affected by overlaps). p is a factor accounting for the nonsystematic nature of overlaps, it is discussed further later. We now apply this approach to spherical particles with a radius of curvature r: For dr5r; the surface to volume ratio ðsb =v0b Þ is equal to 3=r so that for small curvatures ð1=rÞ Z ¼ 3pdr=r:
ð16Þ
In a more general way, the volume of a spherical shell can be written as dvðdr; rÞ ¼ ð4=3Þp½ðr þ drÞ3 r3 : Eq. (15) can therefore be written as Z ¼ 3pdr=r þ 3pðdr=rÞ2 þ pðdr=rÞ3 :
ð17Þ
This shows that a non-linearity appears for large values of 1=r (small particles). Note that because of these non-linearities, a small probability (p) is not equivalent to a lower spatial resolution (dr) in Eq. (17).
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2.3. Link with ion trajectory simulations Simulations of ion trajectories were performed to assess this approach. A tip containing a small precipitate (pure in B atoms) embedded in a solid solution (pure in A) was simulated. The evaporation field of B atoms in the b particle had a reduced field eB taken equal to 0.8 or 0.9 (Fig. 2). The overlap rate Z was computed as a function of the curvature 1=r (r ¼ 0:5 nm up to 2.5 nm) for these two reduced fields. As the matrix and particles are pure in A and B atoms, respectively, there is no problem to define the particle volume and the overlap zone. The particle is defined as the smallest region containing all the B atoms and the overlap zone is the outer part of the precipitate containing A atoms. It is then straightforward to compute both the overlap rate and the apparent composition of the particle. Note that Z varies almost linearly with 1=r for small values as predicted (Eq. (16), Fig. 2). This is particularly clear for eB ¼ 0:8: Note also that the influence of eB on Z is very pronounced. The results of simulations were found to fit quite well with the predictions given by Eq. (17). The best fit was obtained for p ¼ 0:3 for both reduced fields
and for an equivalent spatial resolution dr ¼ 0:3 nm for eB ¼ 0:8 and dr ¼ 0:1 nm for eB ¼ 0:9: These are reasonable values. 2.4. The correction of apparent data using this model The previous section has shown that the analytical model proposed is a good approach in spite of the severe assumptions made. The crucial issue is to determine Z; which is the essential parameter involved in the correction of compositions (Eq. (4)). The problem is that the analytical expression (Eq. (17)) requires a knowledge of two parameters (p and dr), which are not directly available from experiments. In addition, as shown in Fig. 2, dr varies with the evaporation field of the b phase (eB ), which is unfortunately not accessible experimentally. In contrast to eB ; the apparent normalised atomic density (rr ) can be estimated from experimental images and it is a parameter that reflects directly the importance of local magnification effects. Using the simulation data, the overlap rate can be plotted as a function of the reduced density (Fig. 3). Data related to both values of eB are
0.7 0.6
η
0.5
ε B = 0.8
0.4 0.3 0.2
ε B = 0.9
0.1 0 0
0.4
0.8
1.2
1.6
2
1/r (nm -1) Fig. 2. Results obtained through the simulation of ion trajectory aberrations near phase interface. The overlap rate Z is plotted as a function of the curvature 1=r of the b particle for two reduced evaporation fields of the b phase (eB ). Best fit lines are superimposed.
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D. Blavette et al. / Ultramicroscopy 89 (2001) 145–153 0.7 0.6 0.5
η
0.4 0.3
ε B = 0.9 ε B = 0.8
0.2 0.1 0 1
1.5
2
2.5
3
3.5
ρr Fig. 3. Simulation of local magnification effects near a spherical particle. The overlap rate Z is plotted against the apparent normalised atomic density in the precipitate region (rr ¼ r=r0 with r the apparent density and r0 the true density). A best fit polynomial has been superimposed in Fig. 3: Z ¼ 0:0808r3r 0:3953r2r þ 0:6994rr 0:4176:
plotted. The largest value of eB (0.9) leads to the smallest effects (small Z). Note that the apparent linearity in the rewritten expression of Eq. (6) (Z ¼ rr w 1) is not real, because w decreases in proportion as rr increases. Clearly, w is an implicit function of rr (Eq. (14)). When no local magnification effects occur (rr ¼ 1), then Z ¼ 0 and the volume of precipitates is not compressed (w ¼ 1). For rr ¼ 2; ZB0:05 and Eq. (6) gives a compression factor w close to 0.5, whereas it is close to 0.4 for rr ¼ 3: It is worth mentioning that the use of Fig. 3 for the apparent concentrations to be corrected relies on the hidden assumption that, whatever eb ; there is a single functional linking Z to the apparent density rr : One argument in favour of this assumption is that no discontinuity exists in Fig. 3 between results related to the two values of eb and it is quite easy to find a single functional fitting with simulations whatever eb : However, in the current state, we have no real proof that there is a univocal relationship between Z and r: It is clear that further simulations are needed to assess this hypothesis.
2.5. Application to the FeCu system The model was applied to neutron irradiated Fe1.4 wt% Cu alloys where Cu enriched precipitates, a few nanometer in size, form. Local magnification effects and distortions of images are produced in this system, because of the lower evaporation field of the Cu precipitates compared to iron. As discussed previously, this leads to an elongation of the reconstructed images of copper enriched particles (Fig. 4). Note that this is not the result of a bad calibration of the depth scale. The investigation was carried out far away from the tip axis, that is why ellipsoids are not oriented parallel to the tip axis (e.g. the largest dimension of the reconstructed volume). The evaporation fields of Fe and Cu are close to 33 and 30 V/nm, respectively [14]. The ratio of evaporation fields, eCu ¼ 0:9 corresponds to one of the values studied in simulations (Fig. 2). For a particle radius close to 1 nm (Fig. 4), simulations predict a moderate overlap rate Z; close to 0.03 (Fig. 2). The order of magnitude of related errors on Cu concentration in precipitates is close to Z (3 at%). The influence
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measurement of composition using 3DAP poses the problem of the definition of the precipitate volume. This is, in general, not a trivial issue. However, in the present case, the concentration of Cu in the solid solution is extremely low as compared to that in particles. As a result, there is no problem to define the contour of Cu enriched particles and then to measure its Cu content. The overlap rate Z was derived from rr using Fig. 3. The corrected composition was derived from the data using Eq. (4). For a dilute solid solution, the corrected concentration is close to ð1 þ ZÞ times the apparent concentration. Note that overlaps and hence corrections do not have the same value for the two precipitates. The largest value of Z for the first precipitate is due to its smallest size. The latter contained 294 atoms compared to 552 for the second. For the first precipitate, Eq. (6) gives a compression factor wB0:5; which corresponds to an elongation factor of the precipitates of O2; which is close to the aspect ratio of ellipsoids in Fig. 4 (close to 1.5 in average). Recall that this ratio gives approximately the ratio of magnifications as wBðGb =Ga Þ2 :
3. Conclusion Fig. 4. 3D reconstruction of a small volume in a neutronirradiated FeCu alloy using the Tomographic Atom Probe. The elongation of Cu particles is thought to be caused by local magnification effects which compress the lateral dimensions at the tip surface. The direction of analysis is close to the largest dimension of the bounding box (B10 10 30 nm3).
of overlaps was studied more precisely for two of the precipitates shown in Fig. 4 (Table 1). Table 1 provides the apparent density and Cu concentration in two of the imaged particles. The
This model relies on several simple assumptions which are of course, not beyond criticism. This analytical model, however, is thought to be the first one accounting for the magnification effects and the related spatial overlaps, which occur near the interfaces between small spherical precipitates and the surrounding matrix. Only the case of low evaporation field precipitates was treated but the extension of this approach to high field precipitates is straightforward. The prediction of the overlap rate (Z) as a function of the precipitate size
Table 1 Experimental data related to two of the precipitates (fB2 nm) shown in Fig. 4 Density (rr Þ
Overlap rate (Z)
Apparent Cu concentration (at%)
Corrected Cu concentration (at%)
Number of atoms within particles
1.92 1.8
0.06 0.03
74 78
78 80
294 552
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was found to be in good agreement with simulations. The thickness of the mixed zone around b precipitates was found to be of 0.3 nm for a normalised evaporation field of b phase of 0.8. This model makes it possible to correct the apparent composition of small precipitates to obtain the true composition. In contrast to the evaporation field of precipitates, the apparent density is a parameter that is experimentally available and that reflects the importance of local magnification effects. The overlap rate was parameterised as a function of the apparent atomic density in reconstructed precipitates through the simulation of ion trajectories near an atomically defined emitter. This model has been applied to true 3DAP experiments which were carried out on neutron-irradiated FeCu alloys. Only spherical precipitates were considered. A similar approach could be applied to non-isotropic systems (platelets or rod-like particles). However, this will require the investigation of the influence of the orientation of the particle with respect to the tip axis. It seems clear that the magnification effects are of much less importance for platelets perpendicular to the tip axis compared to plates which are parallel. The modelling of these two extreme cases is rather straightforward in contrast to intermediate configurations. The influence of the detection efficiency was not discussed. It is clear that for a given precipitate size, lower the
153
efficiency, the more important the sampling errors and statistical fluctuations, and the less reliable the correction of the apparent concentration.
References [1] A. Cerezo, T.J. Godfrey, G.D.W. Smith, Rev. Sci. Instrum. 59 (1988) 862. [2] D. Blavette, A. Bostel, J.M. Sarrau, B. Deconihout, A. Menand, Nature 363 (1993) 432. [3] D. Blavette, B. Deconihout, S. Chambreland, A. Bostel, Ultramicroscopy 70 (1998) 115. [4] F. Vurpillot, A. Bostel, A. Menand, D. Blavette, Eur. Phys J. AP. 6 (1999) 217. [5] F. Vurpillot, A. Bostel, D. Blavette, J. Microsc. 196 (1999) 332. [6] F. Vurpillot, A. Bostel, E. Cadel, D. Blavette, Ultramicroscopy (2000) 1. [7] F. Vurpillot, A. Bostel, D. Blavette, Appl. Phys. Lett. 7621) (2000) 3127. [8] S.R. Goodman, S.S. Brenner, J.R. Low, Metall. Trans. 4 (1973) 2371. [9] P. Pareige, P. Auger, P. Bas, D. Blavette, Scripta. Metall. 337) (1995) 1033. [10] M.K. Miller, M.G. Hetherington, Surf. Sci. 246 (1991) 442. [11] P. Bas, A. Bostel, B. Deconihout, D. Blavette, Appl. Surf. Sci. 87/88 (1995) 298. [12] J.M. Hyde, A. Cerezo, R.P. Setna, P.J. Waren, G.D.W. Smith, Surf. Sci. 76/77 (1994) 382. [13] F. Vurpillot, A. Bostel, D. Blavette, Ultramicroscopy, these proceedings. [14] P. Pareige, Ph.D. Thesis, University of Rouen, 1994, p. 33.
A model accounting for spatial overlaps in 3D atom-probe microscopy D. Blavette*, F. Vurpillot, P. Pareige, A. Menand Group of Material Physics, UMR CNRS 6634FUFR Sciences, Universite de Rouen, 76821 Mont-Saint-Aignan Cedex, France Received 18 July 2000; received in revised form 20 October 2000
Abstract The spatial resolution of three-dimensional atom probe is known to be mainly controlled by the aberrations of ion trajectories near the specimen surface. An analytical model accounting for the spatial overlaps that occur near phase interfaces is described. This model makes it possible to correct the apparent composition of small spherical precipitates in order to determine the true composition. The prediction of the overlap rate as a function of the particle size was found in remarkably good agreement with the simulations of ion trajectories that were made. The thickness of the mixed zone around b precipitates was found to be of 0.3 nm for a normalised evaporation field of b phase of 0.8. Using simulations, the overlap rate could be parameterised as a function of the apparent atomic density observed in particles. This model has been applied to copper precipitation in FeCu. r 2001 Elsevier Science B.V. All rights reserved. PACS: 07.78.+s; 41.20.q; 41.75.i Keywords: Atom probe; Phase composition; Local magnification; Modelling
1. Introduction Three-dimensional Atom Probe (3DAP) is the only nanoanalytical microscope combining atomic-scale resolution with 3D imaging capabilities [1– 3]. Besides the impressive technological improvements that were made during this last decade, many efforts need to be made in the theory of the instrument. One of the major key problems comes from a poor knowledge of the real sample-toimage transfer function. The 3D images produced are reconstructed currently using more or less *Corresponding author. Tel.: 33-2-351466-51; fax: 33-235146652. E-mail address: [email protected] (D. Blavette).
sophisticated procedures, but they have in common that they all rely on the underlying poor assumption that 3DAP is a simple point projection microscope which is evidently not true. The spatial resolution of 3DAP images may be decomposed into two components. The depth resolution, essentially controlled by field screening effects, is excellent, typically one-tenth of a nanometer in metals. In contrast, the lateral resolution, parallel to the emitter surface is far from achieving atomic resolution (a few tenths of a nanometer). The lateral resolution is known not to be limited by the performance of the positionsensitive detector used but instead by the physics of field emission of ions in close vicinity of the tip surface. The simulations of ion trajectories near an
0304-3991/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 1 2 0 - 6
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atomically defined emitter that we have developed recently, well support this idea [4–7]. Most of the aberrations in images could be accounted for. In these simulations, the atomic-scale topology of the field ion emitter and its dynamics of evolution during field evaporation were considered. Crucial problems interfering from the investigation of two-phase samples essentially come from the aberrations in the ion trajectories, notably in the close vicinity of interfaces. These give rise to biased compositions particularly for small particles, which have a smaller evaporation field than the surrounding matrix. A typical system where local magnification effects occur is Fe–Cu. The essential question here is the true composition of copper-enriched particles in the early stages of precipitation or under neutron irradiation. Pure Cu particles are low-field regions (darkly imaged in FIM), compared to the surrounding matrix. High-field iron-rich regions between particles therefore develop a smaller radius of curvature at the tip surface. Defocused iron ions coming from the surrounding matrix are therefore suspected to fall into the precipitate image on the detector. This obviously gives rise to an artificial enrichment in iron within the reconstructed images of precipitates. This local magnification effect is illustrated in the simulations of ion impacts on the detector
(a)
given in Fig. 1. For a low evaporation field of the particle (eB ¼ E b =E a o1), the image of the precipitate (pure in B atoms) is compressed and ions coming from the surrounding matrix (pure in A) fall in the particle image (Fig. 1(a)). In contrast, for eB > 1; precipitate ions fall outside the particle and a depleted zone forms close to the interface (Fig. 1(b)). More details on the procedure and results of simulations can be found in Ref. [7]. Historically, the problem of the true composition of precipitates in Fe–Cu has raised numerous debates. The first investigations of Fe–Cu were made by the group of Brenner in 1973 using 1D atom probe [8]. They found 50–70 at% of Cu in the early stages of phase separation instead of 100 at% as predicted by the classical nucleation theory. The first 2D images were produced with the PoSAP developed by Cerezo et al. in 1988, who claimed to have detected close to 100% of Cu in particles of 5 nm in diameter [1]. The 3D images of neutron-irradiated Fe–Cu were published recently by Pareige et al. using the Tomographic Atom Probe [9]. The physical problem behind is independent of the technological details of the instrument. Composition biases are essentially controlled by the up-to-now unresolved problem of the local magnification effects that occur near phase
(b)
Fig. 1. Simulation of impact images for a binary AB alloy containing a spherical b particle (eight interatomic distances in diameter). b phase is pure in B and the a solid solution is pure in A. These two simulated images are cross sections of reconstructed precipitates with (a) eB ¼ 0:85 and (b) eB ¼ 1:15: B atoms are in black and A are in grey. The (1 0 0) pole is situated close to the left-hand bottom corner of each image.
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interfaces [10]. In this paper, a tentative model accounting for the spatial overlaps which occur near phase interfaces is described. This model makes it possible to correct the apparent composition of small precipitates. Using simulations, the overlap rate could be parameterised as a function of the apparent atomic density in reconstructed particles. In contrast to the evaporation field of precipitates, the apparent density is experimentally available and is a parameter which reflects the importance of local magnification effects. Application to 3D images of neutron irradiated Fe–Cu alloys are discussed.
2. Modelling
A atoms. As a result, the definition of the precipitate region is not ambiguous and the measurement of the precipitate composition poses no problem. Experimentally, the definition of the particle region is not univocal, particularly when the solid solution contains a relatively high content in B. In the latter case, isoconcentration surfaces may be used to define the contour (i.e. interface) of precipitates. A threshold slightly higher than the matrix composition may be used. The sampling statistical fluctuations are, evidently, to be considered in the choice of this threshold. Let us define the overlap rate (Z). This latter quantity is the number of atoms falling in bðdnÞ divided by the initial number of atoms contained in b without overlap: Z ¼ dn=nb ;
Let us consider a binary alloy (AB) containing small spherical particles (b), which have an evaporation field lower than the surrounding a matrix (eB ¼ Eb =Ea o1). This is notably the case of FeCu or of Mg2Zn particles in AlZnMg alloys and this is a more problematic case, compared to systems for which eB > 1 (Fig. 1(b)). In the latter case overlaps also exist, but it is much easier to get a plateau in concentration profiles and then to determine the true composition of the particle in its core. The analytical model developed below can be adapted readily to cases, where eB > 1: Let us consider, a single precipitate embedded in an infinite matrix. The true concentration CB0b is simply: CB0b ¼ nB =nb with nb being the total number of atoms in b that are likely to be detected. The measured concentration (CB ) is biased because some of the atoms located around the particle (dn ¼ dnA þ dnB ) fall inside the image of b (Fig. 1(a)):
ð1Þ
so that CB ¼ ðCB0b þ ZCBa Þ=ð1 þ ZÞ; b 0b a ¼ ðCA þ ZCA Þ=ð1 þ ZÞ: CA
ð2Þ
This result can easily be generalised to systems containing m chemical species: Ci0b ¼ ðCi0b þ ZCia Þ=ð1 þ ZÞ;
i ¼ 1ym:
ð3Þ
(Ci0b )
Conversely, true concentrations can be deduced from measurements through the following expression: Ci0b ¼ Cib þ ZðCib Cia Þ:
ð4Þ
It is worth noting that when the matrix is a dilute solid solution, then Eq. (4) reduces to Ci0b ¼ Cib ð1 þ ZÞ
if Cia 5Ci0b :
CBb ¼ ðnB þ dnB Þ=ðnb þ dnA þ dnB Þ with
Let us consider that the ternary alloy i ¼ 1; 2 are the solutes and i ¼ 3 the major species. It becomes clear that the stoichiometry (K12 ) of a precipitate is not biased by the local magnification effects (for Mg2Zn we have K12 ¼ 2):
a dnA ¼ dnCA
K12 ¼ C1b =C2b ¼ C10 b=C20b :
dnB ¼ dnCBa ;
where C represents the measured atomic concentration, whereas primed superscripts will always hereafter refer to corrected values (C 0 is the true a concentration). CA and CBa are the concentrations away from the particle. They are assumed not to be biased. In Fig. 1(a), the solid solution is pure in
2.1. The apparent phase fraction It is now easy to demonstrate that the molar fraction (f ) of b phase that has been measured
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from the number of atoms in each phase ða; bÞ is also biased: f ¼ ðnb þ dnÞ=N ¼ nb ð1 þ ZÞ=N: Note that N is the total number of atoms collected in the analysis (N) is, in general, not biased as the local magnification effects are conservative in contrast to preferential evaporation effects. The true fraction (f 0 ) is given by f 0 ¼ nb =N so that f 0 ¼ f =ð1 þ ZÞ:
ð5Þ
The latter equation can also be deduced from the lever rule, which expresses the conservation of matter. Writing f 0 ¼ ðCB0 CBa Þ=ðCB0b CBa Þ CBa Þ=ðCBb CBa Þ leads also to f ¼ f ¼ ZÞf 0 with CB0 being the nominal concentration ðCB0
and ð1 þ of B atoms. Note that the apparent fraction of the b phase in reconstructed images can easily be measured by counting the number of atoms (n0b ) inside precipitates f ¼ n0b =N; with N being the total number of atoms in the reconstructed image. Practically, n0b can be estimated using isoconcentration surfaces with the threshold chosen a little bit higher than the matrix level (iso-surfaces should contain the transient zone caused by trajectory aberrations near interfaces). It is worth mentioning that the apparent atomic fraction is not to be mistaken with the apparent volume fraction. Due to the local magnification effects, b particles appear in reconstructed images with a much smaller volume than expected. This problem is discussed later. Eq. (5) indicates that overlaps lead to an overestimation of the atomic fraction of b particles, through a modification of the apparent partitioning ratio ðCib =Cia Þ: As Eq. (2) shows, the concentration in the b particle is simply pushed towards that of the matrix along the tie line linking the composition of the phases. This additional number of atoms in b coming from the close vicinity of interfaces increases the apparent atomic density in reconstructed b zones. But this is not the only factor contributing to this local density effect. The lower evaporation field of b leads to a larger radius of curvature of b zones emerging from the tip
surface and this give rise to a focusing effect. This in turn leads to a compressed image of b on the detector. This can be expressed as a local compression of the reconstructed volume of bðvb Þ: The apparent volume of b can be written as vb ¼ wv0b with w being the compression factor. We can now express the apparent atomic density (r) as r ¼ ðnb þ dnÞ=vb The true density can be written as r0 ¼ nb =v0b so that the normalised density measured in b particles is rr ¼ r=r0 ¼ ð1 þ ZÞ=w;
ð6Þ
with Z being the overlap rate as defined in Eq. (1) and w; the compression factor w ¼ vb =v0b
ðwo1Þ:
ð7Þ
This compression of the particle volume leads to a drastic decrease of the apparent volume fraction F; which can be written as a function of the true volume fraction F 0 in the following form: F ¼ wF 0 : If a and b have the same molar volume then F 0 is equal to the atomic fraction f 0 and the measured volume fraction F is expressed as F ¼ wf 0
with wo1 for eB o1:
ð8Þ
Even though the approach is not quantitative, it is interesting to express w as a function of the local magnifications. As only the lateral dimensions, parallel to the tip surface, are affected by the local magnification effects, only two of the three dimensions of reconstructed precipitates contribute to the compression of the reconstructed volume of b: We assume that the depth scale is not modified which is reasonable as the total number of ions collected in each layer is conserved (the depth is proportional to the cumulated number of collected ions in the reconstruction procedure in 3DAPs [11]; the depth calibration issues are also discussed in [12]). The initial spherical particle therefore transforms into an
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ellipsoid. This is illustrated in the 3DAP image of FeCu in Fig. 4 and confirmed by simulations [13]. The elongation ratio (i.e. the aspect ratio of ellipsoidal precipitates) defined as the ratio (e) of the long axis to the small axis of this ellipsoid (axes parallel to the tip surface) is given by the ratio of local magnifications:
the nanometer scale which is unfortunately the scale at which magnification effects should be taken into account. Note also that the phenomenological parameters ka and kb involved in Eq. (14) are likely to depend on many factors (eB ; particle size r; etc.) and this dependence is unknown.
e ¼ ðGa =Gb Þ
2.2. Theoretical expression of the overlap rate
ð9Þ
and then the compression factor is given by the ratio of the volume of an ellipsoid to that of a sphere; w ¼ 1=e2 ; so that wBðGb =Ga Þ2 :
ð10Þ
Let us introduce a phenomenological expression for the local magnification: G ¼ L=aR
ð11Þ
L is the flight path and a is a parameter related to the projection features. If we now introduce the radii of curvature Ra and Rb of a and b regions at the tip surface, then, we can write w ¼ ðaa Ra =ab Rb Þ2 :
ð12Þ
In this model, the evaporation field of b was chosen smaller than that of aðeB o1Þ so that Ra oRb and wo1: If we now want to express w as a function of eB then, the following phenomenological expression for the electric field generated above a curved surface is needed: E ¼ V=bR:
ð13Þ
V is the applied voltage and b a factor related to the local geometry of the emitter. Writing this latter equation for both a and b phases and introducing the ratio k ¼ b=a; leads to w ¼ ðkb =ka Þ2 e2B :
ð14Þ
For many reasons, one has to be cautious in the use of such an expression. It relies on several assumptions: (i) two distinct radii of curvature and (ii) a unique centre of projection for each phase. This is certainly not true, in particular, near interfaces where the local curvature smoothly changes from that of the a solid solution to that of b zones (there is no negative curvature at the tip surface). Moreover, it is worth mentioning that the notion of radius of curvature looses its meaning in proportion as the size of a precipitate approaches
It is now interesting to express the overlap rate (Z) as a function of both the size and the reduced field ei : We assume that the parasitic atoms falling in the image of bðdnÞ come from a shell around the precipitate that has a volume dv; and a thickness dr: This implies that Z is proportional to dv=v0b (v0b the true volume of the b particle). If we now introduce the probability (p) that atoms of the shell really reach the precipitate region (v0b ), then one can write Z ¼ pdvðdr; rÞ=v0b
and
Z ¼ pðsb =v0b Þdr:
ð15Þ
sb is the area of an interface subjected to overlap. In this first approach, the shell thickness (dr) can be considered as a ‘‘weighted’’ spatial resolution averaged over the contour of particles (only 2 of the 3 dimensions are affected by overlaps). p is a factor accounting for the nonsystematic nature of overlaps, it is discussed further later. We now apply this approach to spherical particles with a radius of curvature r: For dr5r; the surface to volume ratio ðsb =v0b Þ is equal to 3=r so that for small curvatures ð1=rÞ Z ¼ 3pdr=r:
ð16Þ
In a more general way, the volume of a spherical shell can be written as dvðdr; rÞ ¼ ð4=3Þp½ðr þ drÞ3 r3 : Eq. (15) can therefore be written as Z ¼ 3pdr=r þ 3pðdr=rÞ2 þ pðdr=rÞ3 :
ð17Þ
This shows that a non-linearity appears for large values of 1=r (small particles). Note that because of these non-linearities, a small probability (p) is not equivalent to a lower spatial resolution (dr) in Eq. (17).
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2.3. Link with ion trajectory simulations Simulations of ion trajectories were performed to assess this approach. A tip containing a small precipitate (pure in B atoms) embedded in a solid solution (pure in A) was simulated. The evaporation field of B atoms in the b particle had a reduced field eB taken equal to 0.8 or 0.9 (Fig. 2). The overlap rate Z was computed as a function of the curvature 1=r (r ¼ 0:5 nm up to 2.5 nm) for these two reduced fields. As the matrix and particles are pure in A and B atoms, respectively, there is no problem to define the particle volume and the overlap zone. The particle is defined as the smallest region containing all the B atoms and the overlap zone is the outer part of the precipitate containing A atoms. It is then straightforward to compute both the overlap rate and the apparent composition of the particle. Note that Z varies almost linearly with 1=r for small values as predicted (Eq. (16), Fig. 2). This is particularly clear for eB ¼ 0:8: Note also that the influence of eB on Z is very pronounced. The results of simulations were found to fit quite well with the predictions given by Eq. (17). The best fit was obtained for p ¼ 0:3 for both reduced fields
and for an equivalent spatial resolution dr ¼ 0:3 nm for eB ¼ 0:8 and dr ¼ 0:1 nm for eB ¼ 0:9: These are reasonable values. 2.4. The correction of apparent data using this model The previous section has shown that the analytical model proposed is a good approach in spite of the severe assumptions made. The crucial issue is to determine Z; which is the essential parameter involved in the correction of compositions (Eq. (4)). The problem is that the analytical expression (Eq. (17)) requires a knowledge of two parameters (p and dr), which are not directly available from experiments. In addition, as shown in Fig. 2, dr varies with the evaporation field of the b phase (eB ), which is unfortunately not accessible experimentally. In contrast to eB ; the apparent normalised atomic density (rr ) can be estimated from experimental images and it is a parameter that reflects directly the importance of local magnification effects. Using the simulation data, the overlap rate can be plotted as a function of the reduced density (Fig. 3). Data related to both values of eB are
0.7 0.6
η
0.5
ε B = 0.8
0.4 0.3 0.2
ε B = 0.9
0.1 0 0
0.4
0.8
1.2
1.6
2
1/r (nm -1) Fig. 2. Results obtained through the simulation of ion trajectory aberrations near phase interface. The overlap rate Z is plotted as a function of the curvature 1=r of the b particle for two reduced evaporation fields of the b phase (eB ). Best fit lines are superimposed.
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η
0.4 0.3
ε B = 0.9 ε B = 0.8
0.2 0.1 0 1
1.5
2
2.5
3
3.5
ρr Fig. 3. Simulation of local magnification effects near a spherical particle. The overlap rate Z is plotted against the apparent normalised atomic density in the precipitate region (rr ¼ r=r0 with r the apparent density and r0 the true density). A best fit polynomial has been superimposed in Fig. 3: Z ¼ 0:0808r3r 0:3953r2r þ 0:6994rr 0:4176:
plotted. The largest value of eB (0.9) leads to the smallest effects (small Z). Note that the apparent linearity in the rewritten expression of Eq. (6) (Z ¼ rr w 1) is not real, because w decreases in proportion as rr increases. Clearly, w is an implicit function of rr (Eq. (14)). When no local magnification effects occur (rr ¼ 1), then Z ¼ 0 and the volume of precipitates is not compressed (w ¼ 1). For rr ¼ 2; ZB0:05 and Eq. (6) gives a compression factor w close to 0.5, whereas it is close to 0.4 for rr ¼ 3: It is worth mentioning that the use of Fig. 3 for the apparent concentrations to be corrected relies on the hidden assumption that, whatever eb ; there is a single functional linking Z to the apparent density rr : One argument in favour of this assumption is that no discontinuity exists in Fig. 3 between results related to the two values of eb and it is quite easy to find a single functional fitting with simulations whatever eb : However, in the current state, we have no real proof that there is a univocal relationship between Z and r: It is clear that further simulations are needed to assess this hypothesis.
2.5. Application to the FeCu system The model was applied to neutron irradiated Fe1.4 wt% Cu alloys where Cu enriched precipitates, a few nanometer in size, form. Local magnification effects and distortions of images are produced in this system, because of the lower evaporation field of the Cu precipitates compared to iron. As discussed previously, this leads to an elongation of the reconstructed images of copper enriched particles (Fig. 4). Note that this is not the result of a bad calibration of the depth scale. The investigation was carried out far away from the tip axis, that is why ellipsoids are not oriented parallel to the tip axis (e.g. the largest dimension of the reconstructed volume). The evaporation fields of Fe and Cu are close to 33 and 30 V/nm, respectively [14]. The ratio of evaporation fields, eCu ¼ 0:9 corresponds to one of the values studied in simulations (Fig. 2). For a particle radius close to 1 nm (Fig. 4), simulations predict a moderate overlap rate Z; close to 0.03 (Fig. 2). The order of magnitude of related errors on Cu concentration in precipitates is close to Z (3 at%). The influence
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measurement of composition using 3DAP poses the problem of the definition of the precipitate volume. This is, in general, not a trivial issue. However, in the present case, the concentration of Cu in the solid solution is extremely low as compared to that in particles. As a result, there is no problem to define the contour of Cu enriched particles and then to measure its Cu content. The overlap rate Z was derived from rr using Fig. 3. The corrected composition was derived from the data using Eq. (4). For a dilute solid solution, the corrected concentration is close to ð1 þ ZÞ times the apparent concentration. Note that overlaps and hence corrections do not have the same value for the two precipitates. The largest value of Z for the first precipitate is due to its smallest size. The latter contained 294 atoms compared to 552 for the second. For the first precipitate, Eq. (6) gives a compression factor wB0:5; which corresponds to an elongation factor of the precipitates of O2; which is close to the aspect ratio of ellipsoids in Fig. 4 (close to 1.5 in average). Recall that this ratio gives approximately the ratio of magnifications as wBðGb =Ga Þ2 :
3. Conclusion Fig. 4. 3D reconstruction of a small volume in a neutronirradiated FeCu alloy using the Tomographic Atom Probe. The elongation of Cu particles is thought to be caused by local magnification effects which compress the lateral dimensions at the tip surface. The direction of analysis is close to the largest dimension of the bounding box (B10 10 30 nm3).
of overlaps was studied more precisely for two of the precipitates shown in Fig. 4 (Table 1). Table 1 provides the apparent density and Cu concentration in two of the imaged particles. The
This model relies on several simple assumptions which are of course, not beyond criticism. This analytical model, however, is thought to be the first one accounting for the magnification effects and the related spatial overlaps, which occur near the interfaces between small spherical precipitates and the surrounding matrix. Only the case of low evaporation field precipitates was treated but the extension of this approach to high field precipitates is straightforward. The prediction of the overlap rate (Z) as a function of the precipitate size
Table 1 Experimental data related to two of the precipitates (fB2 nm) shown in Fig. 4 Density (rr Þ
Overlap rate (Z)
Apparent Cu concentration (at%)
Corrected Cu concentration (at%)
Number of atoms within particles
1.92 1.8
0.06 0.03
74 78
78 80
294 552
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was found to be in good agreement with simulations. The thickness of the mixed zone around b precipitates was found to be of 0.3 nm for a normalised evaporation field of b phase of 0.8. This model makes it possible to correct the apparent composition of small precipitates to obtain the true composition. In contrast to the evaporation field of precipitates, the apparent density is a parameter that is experimentally available and that reflects the importance of local magnification effects. The overlap rate was parameterised as a function of the apparent atomic density in reconstructed precipitates through the simulation of ion trajectories near an atomically defined emitter. This model has been applied to true 3DAP experiments which were carried out on neutron-irradiated FeCu alloys. Only spherical precipitates were considered. A similar approach could be applied to non-isotropic systems (platelets or rod-like particles). However, this will require the investigation of the influence of the orientation of the particle with respect to the tip axis. It seems clear that the magnification effects are of much less importance for platelets perpendicular to the tip axis compared to plates which are parallel. The modelling of these two extreme cases is rather straightforward in contrast to intermediate configurations. The influence of the detection efficiency was not discussed. It is clear that for a given precipitate size, lower the
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efficiency, the more important the sampling errors and statistical fluctuations, and the less reliable the correction of the apparent concentration.
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