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Orientation Sensitive EELS-analysis of Boron Nitride Nanometric Hollow Spheres
Pergamon PII: S0968-4328(98)00030-4
Micron Vol. 29, No. 6, pp. 419–424, 1998 䉷 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0968–4328/99/$ – see front matter
Orientation Sensitive EELS-Analysis of Boron Nitride Nanometric Hollow Spheres C. SOUCHE a,1, B. JOUFFREY a, G. HUG b and M. NELHIEBEL c a
LMSS-Mat URA 850, ECP, Gde Voie des Vignes, 92295 Chaˆtenay-Malabry, France b LEM, ONERA-CNRS, BP 72, 92322 Chaˆtillon, France c Institut fur Angewandte und Technische Physik, Technische Universita¨t Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria (Received 8 January 1997; received in revised form 28 August 1998; accepted 9 September 1998)
Abstract—We present a theory for the inelastic scattering cross-section in hexagonal boron nitride. Explicitly accounting for effects of beam convergence and a finite collection aperture, we present an approach that well describes the form of the observed Boron K-edge. We use this technique to investigate the crystallographic structure of boron nitride nanometric hollow spheres. 䉷 1999 Elsevier Science B.V. All rights reserved. Key words: hexagonal boron nitride, EELS, transmission electron microscopy
INTRODUCTION Boron nitride is a III–V compound which can be found in an hexagonal structure with close resemblance to graphite (Fig. 1). Under particular conditions, one can produce small particles closely related to the fullerene forms of graphite. We studied such particles, synthesized at the Commissariat a` 1’Energie Atomique by a laser driven reaction in BC1 3 – NH 3 gaseous mixture, followed by a subsequent annealing at a temperature between 1400⬚C and 1650⬚C (Baraton et al., 1994). The structure of these particles were investigated in High Resolution Electron Microscopy (HRTEM) (Boulanger, 1994, 1995a, 1995b). These particles are plate-like crystallites or hollow or filled spheres (Fig. 2). The plate-like particles and the smallest spheres (20– 100 nm) are well crystallized and therefore a periodic lattice is observable in HRTEM (Fig. 3). In contrast, there are no visible atomic planes on the larger (100–300 nm) particles which seem to be poorly ordered. Information available from diffraction patterns is difficult to interpret, in particular because of the small size of the particles. Electron Energy Loss Spectroscopy (EELS) is a powerful tool in microanalysis (Egerton, 1986). In order to determine if a general order remains in the organization of the atomic structure in the largest BN particles, we used, similar to O. Stephan in her study of carbon nanotubes (Stephan et al., 1996), Electron Energy Loss Near Edge Structure (ELNES) analysis which is sensitive to local order. In the following we derive an expression for the orientation-dependent K-edge scattering cross-section, following the works of Leapman et al. and Browning et al. (Leapman et al., 1983; Browning et al., 1991, 1993). Based on the result obtained, 1
Corresponding author.
we then interpret the experimental spectra taken on various sites of the BN spheres and confirm that, even when we do not observe atomic planes in HRTEM, there is an hexagonal ordering. The spectra are well interpreted as being the fingerprints of a local hybridized bonding related to the h-BN structure with more or less crystalized planes arranged in concentric spheres where the c axis is parallel to the radius of the spheres (Souche and Jouffrey, 1996).
THEORY Basic approach The differential cross-section expressed for the Generalised Oscillator Strength (GOS) is given for the transition of an atomic inner-shell electron from a state lii to an unoccupied state l f i as (Bethe, 1930): d2 jif (v) 4 ¼ 2 4 lh f lei~q·~r liil2 dQ dE a0 q
(1)
~ ¹ k~ the scattering where a 0 is the Bohr radius and q~ ¼ k⬘ vector. We exclude here the dynamical and interference effects. The fast electron enters the sample with energy E 0 ~ and and wave vector k~ and leaves it with wave vector k⬘ energy E 0-E, where E denotes the energy loss. We define the angle between the two wave vectors as the scattering angle v, and in small-angle approximation we have: q2 ¼ k2 (v2 þ v2E )
(2)
where v E ¼ Emg/~ 2k 2 is the characteristic angle corresponding to the shortening of the fast electron’s wave vector because of the considered energy loss. In the conditions used in our experiments, we can use the dipole approximation: l~q·~r; l p 1. So, developing the ei~q·~r operator up to the 419
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and ~r ¼ r~er : Z hf l~q·~rlii ¼ drr3 Rni li (r)Rnf lf (r) r Z sin bz dbz dF z Yli mi (bz F z )~er ·~qYlf mf (bz , F z ) ⫻ bz , Fz
(4) Crystalline h-BN has three hybridized 2sp xp y (j) bonding orbitals per atom in the layer plane, one p z (p) bonding orbital perpendicular to the layers and the corresponding j* and p* antibonding orbitals. The initial lii state in Eq. (3) is a 1s orbital and the final l f i a p* or a j* one. We can, therefore calculate the two differential cross-sections, corresponding to a 1s → p* and to a 1s → j* transition. The cross-sections have to be calculated separately as they correspond to different energy transitions. The respective cross-sections depend on the angle of the incident beam relative to the basis planes of the sample, and on the scattering angle (Leapman et al., 1983). We are only interested in the angular dependence of the matrix element. The 1s orbital has no angular dependence. The p* angular part is proportional to cos b z, where b z is the radial angle of the spherical coordinates expressed in the (x, y, z) trihedron of the sample corresponding to the p x, p y, p z orbitals (z is parallel to the ~c vector of the crystal), therefore: Zp Z2p dbz dF z ~er ·~q cos bz sin bz (5) hpⴱ l~er ·~ql1si ⬀
Fig. 1. Comparison of the graphite and hexagonal boron nitride structures.
0
0
Expressing ~er in the (x, y, z) trihedron, we get: ~er ·~q ¼ sin bz cos F z qx þ sin bz sin F z qy þ cos bz qz
(6)
whence:
Fig. 2. TEM photography of a group of BN particles whose diameter range from 50 to 500 nm. (Philips CM20-UT, 200 kV).
d2 jpⴱ 1 ⬀ q2 dQ dE q4 z
(7)
Proceeding in the same way for the three sigma orbitals, one obtains: d2 jjⴱ 1 ⬀ (q2 þ q2y ) dQ dE q 4 x
Fig. 3. Filled spherical well crystallized BN particle. (Philips CM20-UT, 200 kV).
first order and remembering that lii and l f i are orthogonal, the different cross-section becomes: d2 jif (v) 4 ¼ 2 4 lhf l~q·~rliil2 dQ dE a0 q
(3)
If we consider the radial and angular parts to be separable,
(8)
where (q2x þ q2y ) and q2z are the square of the projections of the scattering q~ vector on the hexagonal plane and the ~c direction, respectively. From Eqs. (7) and (8) one might conclude that, for incidence parallel to the crystal c-axis, the p*-peak intensity in exact forward scattering is maximal whereas the j* peak should disappear. This is true only if the solid angle Q is very small, this means that the collection aperture is small in comparison with v E. As shown on Fig. 4, for incidence parallel to the c-axis of the crystal, q~ will be in the direction of k~ if we consider forward scattering, at the opposite, the case corresponding to q~B give a (q2z þ q2y ) important as the diffusion angle v becomes larger. This can be very well seen on the results of Leapman et al. (Leapman et al., 1983). The problem is that in most of the EELS experiments the collection angle can easily become much larger than v E (position C on Fig. 4). This means that we have contributions to the spectrum coming from A as well as from B. It is therefore necessary to integrate over the whole collecting angle.
Orientation Sensitive EELS-Analysis of Boron Nitride Nanometric Hollow Spheres
421
microscope system of reference in such a way that Z 1 is perpendicular to the spectrometer aperture and q~ is in the (X 1, Z 1) plane. The first rotation around Z 1 changes the reference trihedral to (X, Y, Z) in which q~ is given by: qX ¼ kv cos J
(10)
qY ¼ kv sin J qZ ¼ kvE
Fig. 4. Illumination parallel to the c-axis of the crystal, A with very small aperture on the beam axis, B with off-axis small aperture, C with big collection aperture.
The second rotation around Y with an angle g changes the reference trihedron to the (x, y, z) coordinate system. The third rotation around z is in fact unnecessary because we are interested only in the projection of q~ onto the (x, y) plane or the z axis, respectively. Thus (X, Y, Z) is related to the (x, y, z) trihedral via the standard matrix describing a rotation around the y axis by an angle g: 0 1 cos g 0 sin g B C R¼B 1 0 C (11) @ 0 A ¹ sin g 0
cos g
As the effects of beam convergence and collection aperture will be essential to our results, we must admit incident waves inclined to the optical axis by the polar angle a and the corresponding azimuthal angle e. So, in the (X, Y, Z) trihedral we have: qX ¼ kv cos J ¹ ka cos e
(12)
qY ¼ kv sin J ¹ ka sin e qZ ¼ kvE and by virtue of q~fxyzg ¼ R~q{XYZ} we obtain: qx ¼ cos g(kv cos J þ ka cos e) þ kvE sin g
(13)
qy ¼ kv sin J þ ka sin e qZ ¼ ¹ sin g(kv cos J þ ka cos e) þ kvE cos g
Fig. 5. Scattering geometry in a case where the incident beam has no divergence. z is parallel to the c axis of the crystal.
Integration over the collecting solid angle We now start from Eqs. (7) and (8) and solve the integrals: Z Z djpⴱ 1 2 djjⴱ 1 2 ⬀ ⬀ q d Q (qx þ q2y ) dQ (9) z Q q4 Q q4 dE dE The components of the scattering vector q x, q y and q z in Eqs. (7) and (8) are taken in the specimens system of reference (x, y, z) (Fig. 5). The experimental setup, however, fixes the scattering geometry in the microscope’s system of reference (X 1, Y 1, Z 1) which is related to the (x, y, z) trihedral of the sample via three rotations. It is possible to fix the
Inserting Eq. (11) into Eqs. (7) and (8), we obtain the crosssections for a particular set of ingoing and outgoing wave vectors, and then integrate the differential cross-sections over the solid angles corresponding to the beam convergence and the collection aperture, i.e. Z2p Zb Za0 Z2p djpⴱ dJ de v dv a da ⬀ 0 0 0 0 dE (14) [ ¹ sin g(v cos J þ a cos e) þ vE cos g]2 ⫻ (v2 þ v2E þ a2 þ 2av cos (J ¹ e))2 djjⴱ ⬀ dE
Z2p
Z2p Zb Za0 dJ de v dv a da 0 0 0 0 (v cos g cos J þ a cos g cos e þ vE sin g)2 ⫻ (v2 þ v2E þ a2 þ 2av cos (J ¹ e))2 [v sin J þ a sin e]2 þ 2 (v þ v2E þ a2 þ 2av cos (J ¹ e))2 (15)
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where a 0 denotes the convergence half-angle and b the collection half-angle. They are small enough to set sin v ⯝ v and sin a ⯝ a. These integrals are valid in the case of a collection aperture centered on the incident electron beam. Parallel illumination with a finite collecting solid angle To get some insight into the physical effects of beam convergence and collection aperture, we first neglect the convergence (a 0 ¼ 0) and only integrate over the collecting aperture to obtain: 2
djpⴱ b b ⬀ (1 þ 3 cos 2g) 2 þ (1 ¹ cos 2g)ln dE b þ v2E
2
þ v2E v2E
We first integrate over e, making use of the periodicity of the sine and cosine functions, and find djpⴱ ⬀p dE
Z2p 0
Zb du
0
Za0 v dv
0
a da
(a2 þ v2 þ 2av cos u) sin2 g þ 2v2E cos2 g ⫻ (v2 þ v2E þ a2 þ 2av cos u)2 djpⴱ ⬀p dE
Z2p
⫻
0
Zb du
0
(19)
Za0 v dv
0
a da
(a2 þ v2 þ 2av cos u)(1 þ cos2 g) þ 2v2E sin2 g (v2 þ v2E þ a2 þ 2av cos u)2 (20)
(16) djjⴱ b2 b2 þ v2E ⬀¹ 2 (1 þ 3 cos 2g) þ (3 þ cos 2g)ln dE b þ v2E v2E (17) We can plot the relative weight of the jp* cross-section, RWp: RWp (g, b) ¼
djpⴱ =dE (djpⴱ =dE) þ (djjⴱ =dE)
(18)
in terms of the incident angle of the electron beam over the sample g and the dimensionless collection angle b/vE (Fig. 6). RWp corresponds to the intensity of the p* peak relative to the j* peak in the EELS spectra. For low collection angles (b/vE ⬍ 3.97) the 1s → p* transition is favoured for g ¼ 0 (incident beam parallel to the c axis vector of the crystal) and for high collection angles (b/v E ⬎ 3.97) the 1s → p* transition is favoured for g ¼ p/2 (incident beam parallel to the atomic layers). For b/v E ¼ 3.97, RWp has no dependence on g. For a small collecting angle, we again find the results of Eqs. (7) and (8). Obviously, the necessary integration of the differential cross-sections may invert the effects predicted by Eqs. (7) and (8).
We rearrange the integrands to get a form G 1/(A þ a cos u) þ G 2/(A þ a cos u) 2, where G 1 and G 2 are functions of g and A ¼ v2E þ v2 þ a2 , a ¼ 2av, and then easily evaluate the integral over u. Solution of the remaining integrals is tedious but straightforward, and at the end we obtain: djpⴱ p2 ⬀ [(1 ¹ cos 2g)(Bb2 þ Aa20 ) þ 2 cos 2g(q2 ¹ W)] dE 4 (21)
Fig. 7. Relative weight of the j p cross-section plotted versus the angle of incidence over the sample, g (g ¼ 0 when the incident beam is parallel to the c axis of the crystal) and the normalized collection angle b/v E for a normalized convergence angle a o/v E ¼ 3.3.
General case (beam convergence) We now evaluate the total integrals Eqs. (14) and (15).
Fig. 6. Relative weight of the jp* cross-section plotted versus the angle of incidence over the sample, g (g ¼ 0 when the incident beam is parallel to the c axis of the crystal) and the normalized collection angle b/v E.
Fig. 8. Relative weight of the p* cross-section as defined in Eq. (18) for collection angle b ¼ 2 mrad for parallel illumination (dashed) and convergence angle a 0 ¼ 1 mrad (solid).
Orientation Sensitive EELS-Analysis of Boron Nitride Nanometric Hollow Spheres
423
djjⴱ dE ⬀
p2 [(3 þ cos 2g)(Bb2 þ Aa20 ) ¹ 2(1 þ cos 2g)(q2 ¹ W)] 4 (22)
with q the q2 ¼ v2E þ a20 þ b2 , abbreviations 2 2 2 4 A ¼ ln[(q ¹ 2a20 þ W)=2v2E ], W ¼ q ¹ 4b a0 , B ¼ ln[(q2 ¹ 2b2 þ W)=2v2E ]. Note, that the final expressions Eqs. (21) and (22) are completely symmetrical in b and a 0, i.e. beam convergence and finite collection aperture have the same consequences. Fig. 7 shows the effect of the convergence angle a 0 on the relative weight of the cross-sections. In Fig. 8, we plot the relative weight of the crosssection RW p both for parallel illumination (a 0 ¼ 0, b ¼ 2 mrad) and also for our experimental setup (v E ¼ 0.3 mrad, a 0 ¼ 1 mrad, b ¼ 2 mrad).
EXPERIMENTAL RESULTS We studied hollow spherical particles such as the one presented in Fig. 9. By tilting the specimen from ¹45⬚ to þ45⬚ we confirmed that the particle is spherical. Operating the microscope in diffraction mode and scanning the probe (diameter 25 nm) over the sphere, we measured the thickness using the area of plasmon peak and confirmed that the particle is a hollow
Fig. 9. Hollow spherical boron nitride particle 220 nm of diameter (Jeol 4000FX, 400 kV). EELS spectra were recorded at position A and B.
Fig. 11. Boron K-edge recorded on the border-position A (solid) and in the middle-position B (dashed) of the particle shown in Fig. 9. The double peak may be approximated (hatched area) by an addition of two Gaussians (dotted).
sphere (Fig. 10). The thickness corresponds rather well to the theoretical thickness of a hollow sphere. Our purpose was to determine the orientation of the microcrystalline particles composing the spheres. We therefore needed a high orientation dependence of the spectra. The best way would have been to use a parallel illumination and a very small collection angle. But as we also wanted to focus precisely the spot on the edge or in the center of the particle and then had a convergence angle of 1 mrad. The only way to increase the orientation dependence of the spectra was, therefore, to increase the collecting angle. We worked in diffraction mode at 400 kV, with a collection angle of 2 mrad, and a convergence angle of 1 mrad. For these conditions and for boron K-edge, v E ¼ 0.3 mrad. We then recorded two spectra from the edge and from the center of the particle (position A and B in Fig. 9), which are presented in Fig. 11. Spectra recorded on the nitrogen K-edge presents exactly the same features. The 1s* → p* transition is favoured at the border of the particle (position A). By fitting a superposition of two Gaussians centered at the p* and j* peaks, respectively, we could measure the areas of the two peaks and thus evaluate the experimental RWp* from Eq. (18) to be 0.27 in the middle and 0.34 at the border of the sphere. Comparing this to the plot of Eq. (18) for our particular set of parameters in Fig. 8, we see that this may only be explained by g → p/2 at the border of the particle and g → 0 in the middle. This indicates that the crystal planes are parallel to the surface of the sphere.
CONCLUSION
Fig. 10. Comparison of the theoretical thickness of a hollow sphere and the thickness of the particle measured using the area of plasmon peak. The plots are in arbitrary units.
This work demonstrates the power inherent in EELS microanalysis: Bonding effects and crystallographic structure drastically affect the shape of ionization edges. Therefore, ELNES techniques may yield results where other methods fail, as was shown for the case of crystalline boron nitride onion-like spheres. However, great care must be taken when interpreting the obtained results as experimental factors like convergent illumination and a finite collection aperture will significantly alter the crosssections. We discussed such effects quantitatively in the
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special case of boron nitride, providing in Eqs. (21) and (22) analytical expressions for the p* and j* cross-sections while accounting for both beam convergence and collection aperture. This result has to be considered also for nanotubes studies in particular. We would like to thank M. Cauchetier and F. Willaime from the CEA for giving us the samples, and the direction of the OM department at the ONERA for allowing us to use the microscope and PEELS. We also want to thank Frank Glas (CNET-Bagneux) for friendly competition in the calculation of integrals in Eqs. (14) and (15).
REFERENCES Baraton, M. I., Boulanger, L., Cauchetier, M., Lorenzelli, V., Luce, M., Merle, T., Quintard, P. and Zhou, Y. H., 1994. Nanometric boron nitride powders: laser synthesis, characterisation and FT-IR surface study. Journal of the European Ceramic Society, 213, 371–378. Bethe, H. A., 1930. Zur Theorie des Durchgangs schneller Korpusku-larstrahlen durch Materie.Ann. Phys., 5, 325–400. Boulanger, L., Willaime, F. and Cauchetier, M., 1994. In: Jouffrey, B., Colliex, C. (Eds.) Nanometric onion-like hollow spheres in laser synthesized boron nitride ultra fine powder. Electron microscopy 1994, vol. 2A. Les Editions de Z Physique, pp. 315–316.
Boulanger, L., Andriot, B., Willaime, F. and Cauchetier, M., 1995. Concentric-shelled and plate-like graphitic boron nitride nanoparticle produced by CO 2 laser pyrolysis. Chemical Physics Letters, 234, 227– 232. Boulanger, L., Willaime, F. and Cauchetier, M., 1995b In: Morphology of nanometric powders produced by laser pyrolysis. Society, Materials Research (ed), Materials research society proceedings, vol. 359, pp. 53–58. Browning, N. D., Yuan, J. and Brown, L. M., 1991. Real-space determination of anisotropic electronic structure by electron energy loss spectroscopy. Ultramicroscopy, 38, 291–298. Browning, N. D., Yuan, J. and Brown, L. M., 1993. Theoretical determination of angulary-integrated energy loss functions for anisotropic materials. Philosophical Magazine, A67 (1), 261–271. Egerton, R. F., 1986. Electron energy loss spectroscopy in the electron microscope. Plenum Press, N.Y. Leapman, R. D., Fejes, P. L. and Silcox, J., 1983 Orientation dependence of core edges from anisotropic materials determined by inelastic scattering of fast electrons. Physical Review, B28 (5), 2361–2373. Souche, C., Jouffrey, B., Marraud, A., Hug, G., Schattschneider, P., Nelhiebel, M., Cauchetier, M. and Willaime, F., 1998. EELS study of nanometric boron-nitride hollow spheres. In: Proceedings of EUREM–II, European Societies of Microscopy (ed.), vol. 2, pp. 397–398. Stephan, O., Ajayan, P. M., Colliex, C., Cyrotlackmann, F. and Sandre, E., 1996. Curvature induced bonding changes in carbon nanotubes investigated by electron energy loss spectroscopy. Physical Review, B53 (20), 13824–13829.
Micron Vol. 29, No. 6, pp. 419–424, 1998 䉷 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0968–4328/99/$ – see front matter
Orientation Sensitive EELS-Analysis of Boron Nitride Nanometric Hollow Spheres C. SOUCHE a,1, B. JOUFFREY a, G. HUG b and M. NELHIEBEL c a
LMSS-Mat URA 850, ECP, Gde Voie des Vignes, 92295 Chaˆtenay-Malabry, France b LEM, ONERA-CNRS, BP 72, 92322 Chaˆtillon, France c Institut fur Angewandte und Technische Physik, Technische Universita¨t Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria (Received 8 January 1997; received in revised form 28 August 1998; accepted 9 September 1998)
Abstract—We present a theory for the inelastic scattering cross-section in hexagonal boron nitride. Explicitly accounting for effects of beam convergence and a finite collection aperture, we present an approach that well describes the form of the observed Boron K-edge. We use this technique to investigate the crystallographic structure of boron nitride nanometric hollow spheres. 䉷 1999 Elsevier Science B.V. All rights reserved. Key words: hexagonal boron nitride, EELS, transmission electron microscopy
INTRODUCTION Boron nitride is a III–V compound which can be found in an hexagonal structure with close resemblance to graphite (Fig. 1). Under particular conditions, one can produce small particles closely related to the fullerene forms of graphite. We studied such particles, synthesized at the Commissariat a` 1’Energie Atomique by a laser driven reaction in BC1 3 – NH 3 gaseous mixture, followed by a subsequent annealing at a temperature between 1400⬚C and 1650⬚C (Baraton et al., 1994). The structure of these particles were investigated in High Resolution Electron Microscopy (HRTEM) (Boulanger, 1994, 1995a, 1995b). These particles are plate-like crystallites or hollow or filled spheres (Fig. 2). The plate-like particles and the smallest spheres (20– 100 nm) are well crystallized and therefore a periodic lattice is observable in HRTEM (Fig. 3). In contrast, there are no visible atomic planes on the larger (100–300 nm) particles which seem to be poorly ordered. Information available from diffraction patterns is difficult to interpret, in particular because of the small size of the particles. Electron Energy Loss Spectroscopy (EELS) is a powerful tool in microanalysis (Egerton, 1986). In order to determine if a general order remains in the organization of the atomic structure in the largest BN particles, we used, similar to O. Stephan in her study of carbon nanotubes (Stephan et al., 1996), Electron Energy Loss Near Edge Structure (ELNES) analysis which is sensitive to local order. In the following we derive an expression for the orientation-dependent K-edge scattering cross-section, following the works of Leapman et al. and Browning et al. (Leapman et al., 1983; Browning et al., 1991, 1993). Based on the result obtained, 1
Corresponding author.
we then interpret the experimental spectra taken on various sites of the BN spheres and confirm that, even when we do not observe atomic planes in HRTEM, there is an hexagonal ordering. The spectra are well interpreted as being the fingerprints of a local hybridized bonding related to the h-BN structure with more or less crystalized planes arranged in concentric spheres where the c axis is parallel to the radius of the spheres (Souche and Jouffrey, 1996).
THEORY Basic approach The differential cross-section expressed for the Generalised Oscillator Strength (GOS) is given for the transition of an atomic inner-shell electron from a state lii to an unoccupied state l f i as (Bethe, 1930): d2 jif (v) 4 ¼ 2 4 lh f lei~q·~r liil2 dQ dE a0 q
(1)
~ ¹ k~ the scattering where a 0 is the Bohr radius and q~ ¼ k⬘ vector. We exclude here the dynamical and interference effects. The fast electron enters the sample with energy E 0 ~ and and wave vector k~ and leaves it with wave vector k⬘ energy E 0-E, where E denotes the energy loss. We define the angle between the two wave vectors as the scattering angle v, and in small-angle approximation we have: q2 ¼ k2 (v2 þ v2E )
(2)
where v E ¼ Emg/~ 2k 2 is the characteristic angle corresponding to the shortening of the fast electron’s wave vector because of the considered energy loss. In the conditions used in our experiments, we can use the dipole approximation: l~q·~r; l p 1. So, developing the ei~q·~r operator up to the 419
420
C. Souche et al.
and ~r ¼ r~er : Z hf l~q·~rlii ¼ drr3 Rni li (r)Rnf lf (r) r Z sin bz dbz dF z Yli mi (bz F z )~er ·~qYlf mf (bz , F z ) ⫻ bz , Fz
(4) Crystalline h-BN has three hybridized 2sp xp y (j) bonding orbitals per atom in the layer plane, one p z (p) bonding orbital perpendicular to the layers and the corresponding j* and p* antibonding orbitals. The initial lii state in Eq. (3) is a 1s orbital and the final l f i a p* or a j* one. We can, therefore calculate the two differential cross-sections, corresponding to a 1s → p* and to a 1s → j* transition. The cross-sections have to be calculated separately as they correspond to different energy transitions. The respective cross-sections depend on the angle of the incident beam relative to the basis planes of the sample, and on the scattering angle (Leapman et al., 1983). We are only interested in the angular dependence of the matrix element. The 1s orbital has no angular dependence. The p* angular part is proportional to cos b z, where b z is the radial angle of the spherical coordinates expressed in the (x, y, z) trihedron of the sample corresponding to the p x, p y, p z orbitals (z is parallel to the ~c vector of the crystal), therefore: Zp Z2p dbz dF z ~er ·~q cos bz sin bz (5) hpⴱ l~er ·~ql1si ⬀
Fig. 1. Comparison of the graphite and hexagonal boron nitride structures.
0
0
Expressing ~er in the (x, y, z) trihedron, we get: ~er ·~q ¼ sin bz cos F z qx þ sin bz sin F z qy þ cos bz qz
(6)
whence:
Fig. 2. TEM photography of a group of BN particles whose diameter range from 50 to 500 nm. (Philips CM20-UT, 200 kV).
d2 jpⴱ 1 ⬀ q2 dQ dE q4 z
(7)
Proceeding in the same way for the three sigma orbitals, one obtains: d2 jjⴱ 1 ⬀ (q2 þ q2y ) dQ dE q 4 x
Fig. 3. Filled spherical well crystallized BN particle. (Philips CM20-UT, 200 kV).
first order and remembering that lii and l f i are orthogonal, the different cross-section becomes: d2 jif (v) 4 ¼ 2 4 lhf l~q·~rliil2 dQ dE a0 q
(3)
If we consider the radial and angular parts to be separable,
(8)
where (q2x þ q2y ) and q2z are the square of the projections of the scattering q~ vector on the hexagonal plane and the ~c direction, respectively. From Eqs. (7) and (8) one might conclude that, for incidence parallel to the crystal c-axis, the p*-peak intensity in exact forward scattering is maximal whereas the j* peak should disappear. This is true only if the solid angle Q is very small, this means that the collection aperture is small in comparison with v E. As shown on Fig. 4, for incidence parallel to the c-axis of the crystal, q~ will be in the direction of k~ if we consider forward scattering, at the opposite, the case corresponding to q~B give a (q2z þ q2y ) important as the diffusion angle v becomes larger. This can be very well seen on the results of Leapman et al. (Leapman et al., 1983). The problem is that in most of the EELS experiments the collection angle can easily become much larger than v E (position C on Fig. 4). This means that we have contributions to the spectrum coming from A as well as from B. It is therefore necessary to integrate over the whole collecting angle.
Orientation Sensitive EELS-Analysis of Boron Nitride Nanometric Hollow Spheres
421
microscope system of reference in such a way that Z 1 is perpendicular to the spectrometer aperture and q~ is in the (X 1, Z 1) plane. The first rotation around Z 1 changes the reference trihedral to (X, Y, Z) in which q~ is given by: qX ¼ kv cos J
(10)
qY ¼ kv sin J qZ ¼ kvE
Fig. 4. Illumination parallel to the c-axis of the crystal, A with very small aperture on the beam axis, B with off-axis small aperture, C with big collection aperture.
The second rotation around Y with an angle g changes the reference trihedron to the (x, y, z) coordinate system. The third rotation around z is in fact unnecessary because we are interested only in the projection of q~ onto the (x, y) plane or the z axis, respectively. Thus (X, Y, Z) is related to the (x, y, z) trihedral via the standard matrix describing a rotation around the y axis by an angle g: 0 1 cos g 0 sin g B C R¼B 1 0 C (11) @ 0 A ¹ sin g 0
cos g
As the effects of beam convergence and collection aperture will be essential to our results, we must admit incident waves inclined to the optical axis by the polar angle a and the corresponding azimuthal angle e. So, in the (X, Y, Z) trihedral we have: qX ¼ kv cos J ¹ ka cos e
(12)
qY ¼ kv sin J ¹ ka sin e qZ ¼ kvE and by virtue of q~fxyzg ¼ R~q{XYZ} we obtain: qx ¼ cos g(kv cos J þ ka cos e) þ kvE sin g
(13)
qy ¼ kv sin J þ ka sin e qZ ¼ ¹ sin g(kv cos J þ ka cos e) þ kvE cos g
Fig. 5. Scattering geometry in a case where the incident beam has no divergence. z is parallel to the c axis of the crystal.
Integration over the collecting solid angle We now start from Eqs. (7) and (8) and solve the integrals: Z Z djpⴱ 1 2 djjⴱ 1 2 ⬀ ⬀ q d Q (qx þ q2y ) dQ (9) z Q q4 Q q4 dE dE The components of the scattering vector q x, q y and q z in Eqs. (7) and (8) are taken in the specimens system of reference (x, y, z) (Fig. 5). The experimental setup, however, fixes the scattering geometry in the microscope’s system of reference (X 1, Y 1, Z 1) which is related to the (x, y, z) trihedral of the sample via three rotations. It is possible to fix the
Inserting Eq. (11) into Eqs. (7) and (8), we obtain the crosssections for a particular set of ingoing and outgoing wave vectors, and then integrate the differential cross-sections over the solid angles corresponding to the beam convergence and the collection aperture, i.e. Z2p Zb Za0 Z2p djpⴱ dJ de v dv a da ⬀ 0 0 0 0 dE (14) [ ¹ sin g(v cos J þ a cos e) þ vE cos g]2 ⫻ (v2 þ v2E þ a2 þ 2av cos (J ¹ e))2 djjⴱ ⬀ dE
Z2p
Z2p Zb Za0 dJ de v dv a da 0 0 0 0 (v cos g cos J þ a cos g cos e þ vE sin g)2 ⫻ (v2 þ v2E þ a2 þ 2av cos (J ¹ e))2 [v sin J þ a sin e]2 þ 2 (v þ v2E þ a2 þ 2av cos (J ¹ e))2 (15)
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where a 0 denotes the convergence half-angle and b the collection half-angle. They are small enough to set sin v ⯝ v and sin a ⯝ a. These integrals are valid in the case of a collection aperture centered on the incident electron beam. Parallel illumination with a finite collecting solid angle To get some insight into the physical effects of beam convergence and collection aperture, we first neglect the convergence (a 0 ¼ 0) and only integrate over the collecting aperture to obtain: 2
djpⴱ b b ⬀ (1 þ 3 cos 2g) 2 þ (1 ¹ cos 2g)ln dE b þ v2E
2
þ v2E v2E
We first integrate over e, making use of the periodicity of the sine and cosine functions, and find djpⴱ ⬀p dE
Z2p 0
Zb du
0
Za0 v dv
0
a da
(a2 þ v2 þ 2av cos u) sin2 g þ 2v2E cos2 g ⫻ (v2 þ v2E þ a2 þ 2av cos u)2 djpⴱ ⬀p dE
Z2p
⫻
0
Zb du
0
(19)
Za0 v dv
0
a da
(a2 þ v2 þ 2av cos u)(1 þ cos2 g) þ 2v2E sin2 g (v2 þ v2E þ a2 þ 2av cos u)2 (20)
(16) djjⴱ b2 b2 þ v2E ⬀¹ 2 (1 þ 3 cos 2g) þ (3 þ cos 2g)ln dE b þ v2E v2E (17) We can plot the relative weight of the jp* cross-section, RWp: RWp (g, b) ¼
djpⴱ =dE (djpⴱ =dE) þ (djjⴱ =dE)
(18)
in terms of the incident angle of the electron beam over the sample g and the dimensionless collection angle b/vE (Fig. 6). RWp corresponds to the intensity of the p* peak relative to the j* peak in the EELS spectra. For low collection angles (b/vE ⬍ 3.97) the 1s → p* transition is favoured for g ¼ 0 (incident beam parallel to the c axis vector of the crystal) and for high collection angles (b/v E ⬎ 3.97) the 1s → p* transition is favoured for g ¼ p/2 (incident beam parallel to the atomic layers). For b/v E ¼ 3.97, RWp has no dependence on g. For a small collecting angle, we again find the results of Eqs. (7) and (8). Obviously, the necessary integration of the differential cross-sections may invert the effects predicted by Eqs. (7) and (8).
We rearrange the integrands to get a form G 1/(A þ a cos u) þ G 2/(A þ a cos u) 2, where G 1 and G 2 are functions of g and A ¼ v2E þ v2 þ a2 , a ¼ 2av, and then easily evaluate the integral over u. Solution of the remaining integrals is tedious but straightforward, and at the end we obtain: djpⴱ p2 ⬀ [(1 ¹ cos 2g)(Bb2 þ Aa20 ) þ 2 cos 2g(q2 ¹ W)] dE 4 (21)
Fig. 7. Relative weight of the j p cross-section plotted versus the angle of incidence over the sample, g (g ¼ 0 when the incident beam is parallel to the c axis of the crystal) and the normalized collection angle b/v E for a normalized convergence angle a o/v E ¼ 3.3.
General case (beam convergence) We now evaluate the total integrals Eqs. (14) and (15).
Fig. 6. Relative weight of the jp* cross-section plotted versus the angle of incidence over the sample, g (g ¼ 0 when the incident beam is parallel to the c axis of the crystal) and the normalized collection angle b/v E.
Fig. 8. Relative weight of the p* cross-section as defined in Eq. (18) for collection angle b ¼ 2 mrad for parallel illumination (dashed) and convergence angle a 0 ¼ 1 mrad (solid).
Orientation Sensitive EELS-Analysis of Boron Nitride Nanometric Hollow Spheres
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djjⴱ dE ⬀
p2 [(3 þ cos 2g)(Bb2 þ Aa20 ) ¹ 2(1 þ cos 2g)(q2 ¹ W)] 4 (22)
with q the q2 ¼ v2E þ a20 þ b2 , abbreviations 2 2 2 4 A ¼ ln[(q ¹ 2a20 þ W)=2v2E ], W ¼ q ¹ 4b a0 , B ¼ ln[(q2 ¹ 2b2 þ W)=2v2E ]. Note, that the final expressions Eqs. (21) and (22) are completely symmetrical in b and a 0, i.e. beam convergence and finite collection aperture have the same consequences. Fig. 7 shows the effect of the convergence angle a 0 on the relative weight of the cross-sections. In Fig. 8, we plot the relative weight of the crosssection RW p both for parallel illumination (a 0 ¼ 0, b ¼ 2 mrad) and also for our experimental setup (v E ¼ 0.3 mrad, a 0 ¼ 1 mrad, b ¼ 2 mrad).
EXPERIMENTAL RESULTS We studied hollow spherical particles such as the one presented in Fig. 9. By tilting the specimen from ¹45⬚ to þ45⬚ we confirmed that the particle is spherical. Operating the microscope in diffraction mode and scanning the probe (diameter 25 nm) over the sphere, we measured the thickness using the area of plasmon peak and confirmed that the particle is a hollow
Fig. 9. Hollow spherical boron nitride particle 220 nm of diameter (Jeol 4000FX, 400 kV). EELS spectra were recorded at position A and B.
Fig. 11. Boron K-edge recorded on the border-position A (solid) and in the middle-position B (dashed) of the particle shown in Fig. 9. The double peak may be approximated (hatched area) by an addition of two Gaussians (dotted).
sphere (Fig. 10). The thickness corresponds rather well to the theoretical thickness of a hollow sphere. Our purpose was to determine the orientation of the microcrystalline particles composing the spheres. We therefore needed a high orientation dependence of the spectra. The best way would have been to use a parallel illumination and a very small collection angle. But as we also wanted to focus precisely the spot on the edge or in the center of the particle and then had a convergence angle of 1 mrad. The only way to increase the orientation dependence of the spectra was, therefore, to increase the collecting angle. We worked in diffraction mode at 400 kV, with a collection angle of 2 mrad, and a convergence angle of 1 mrad. For these conditions and for boron K-edge, v E ¼ 0.3 mrad. We then recorded two spectra from the edge and from the center of the particle (position A and B in Fig. 9), which are presented in Fig. 11. Spectra recorded on the nitrogen K-edge presents exactly the same features. The 1s* → p* transition is favoured at the border of the particle (position A). By fitting a superposition of two Gaussians centered at the p* and j* peaks, respectively, we could measure the areas of the two peaks and thus evaluate the experimental RWp* from Eq. (18) to be 0.27 in the middle and 0.34 at the border of the sphere. Comparing this to the plot of Eq. (18) for our particular set of parameters in Fig. 8, we see that this may only be explained by g → p/2 at the border of the particle and g → 0 in the middle. This indicates that the crystal planes are parallel to the surface of the sphere.
CONCLUSION
Fig. 10. Comparison of the theoretical thickness of a hollow sphere and the thickness of the particle measured using the area of plasmon peak. The plots are in arbitrary units.
This work demonstrates the power inherent in EELS microanalysis: Bonding effects and crystallographic structure drastically affect the shape of ionization edges. Therefore, ELNES techniques may yield results where other methods fail, as was shown for the case of crystalline boron nitride onion-like spheres. However, great care must be taken when interpreting the obtained results as experimental factors like convergent illumination and a finite collection aperture will significantly alter the crosssections. We discussed such effects quantitatively in the
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special case of boron nitride, providing in Eqs. (21) and (22) analytical expressions for the p* and j* cross-sections while accounting for both beam convergence and collection aperture. This result has to be considered also for nanotubes studies in particular. We would like to thank M. Cauchetier and F. Willaime from the CEA for giving us the samples, and the direction of the OM department at the ONERA for allowing us to use the microscope and PEELS. We also want to thank Frank Glas (CNET-Bagneux) for friendly competition in the calculation of integrals in Eqs. (14) and (15).
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Boulanger, L., Andriot, B., Willaime, F. and Cauchetier, M., 1995. Concentric-shelled and plate-like graphitic boron nitride nanoparticle produced by CO 2 laser pyrolysis. Chemical Physics Letters, 234, 227– 232. Boulanger, L., Willaime, F. and Cauchetier, M., 1995b In: Morphology of nanometric powders produced by laser pyrolysis. Society, Materials Research (ed), Materials research society proceedings, vol. 359, pp. 53–58. Browning, N. D., Yuan, J. and Brown, L. M., 1991. Real-space determination of anisotropic electronic structure by electron energy loss spectroscopy. Ultramicroscopy, 38, 291–298. Browning, N. D., Yuan, J. and Brown, L. M., 1993. Theoretical determination of angulary-integrated energy loss functions for anisotropic materials. Philosophical Magazine, A67 (1), 261–271. Egerton, R. F., 1986. Electron energy loss spectroscopy in the electron microscope. Plenum Press, N.Y. Leapman, R. D., Fejes, P. L. and Silcox, J., 1983 Orientation dependence of core edges from anisotropic materials determined by inelastic scattering of fast electrons. Physical Review, B28 (5), 2361–2373. Souche, C., Jouffrey, B., Marraud, A., Hug, G., Schattschneider, P., Nelhiebel, M., Cauchetier, M. and Willaime, F., 1998. EELS study of nanometric boron-nitride hollow spheres. In: Proceedings of EUREM–II, European Societies of Microscopy (ed.), vol. 2, pp. 397–398. Stephan, O., Ajayan, P. M., Colliex, C., Cyrotlackmann, F. and Sandre, E., 1996. Curvature induced bonding changes in carbon nanotubes investigated by electron energy loss spectroscopy. Physical Review, B53 (20), 13824–13829.