- Email: [email protected]
The structure of nanovolumes of amorphous materials
Journal of Non-Crystalline Solids 318 (2003) 233–238 www.elsevier.com/locate/jnoncrysol
The structure of nanovolumes of amorphous materials W. McBride *, D.J.H. Cockayne Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK Received 8 February 2002; received in revised form 20 September 2002
Abstract In this paper we demonstrate reduced density function (GðrÞ) analysis from amorphous volumes as small as 2 nm in width. Obtaining the GðrÞ from a diffraction pattern is a common starting point in the characterisation of amorphous materials. The use of electrons to form the diffraction pattern allows very small volumes of amorphous material to be investigated. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.43.Dq
1. Introduction There is an increasing need to characterise the structure of small localised volumes of amorphous material, such as are found, for example, in thin amorphous films used as hard coatings [1] in amorphised volumes in semiconductor implantations [2] and in glassy intergranular phases [3]. In many structural studies of amorphous materials, the starting point for characterisation is obtaining the reduced density function, GðrÞ, which can be used to give an accurate measure of nearest neighbour distances, coordination numbers and interatomic bond angles. GðrÞ can be obtained from the X-ray, neutron or electron diffraction
* Corresponding author. Tel.: +44-1865 273 661; fax: +441865 283 329. E-mail address: [email protected] (W. McBride).
pattern of an amorphous material, through Fourier transformation of the scattered intensity IðQÞ (where Q ¼ 2 sinðhÞ=k, h is half the scattering angle and k is the wavelength of the radiation used in the diffraction measurements). When measuring GðrÞ from small volumes of amorphous material it is advantageous to choose electrons because the relatively large scattering cross-section of electrons and the ability to focus electrons with magnetic lenses allows the analysis to be carried out on small volumes selected from within the microstructure. The technique of energy selected GðrÞ analysis using electrons [4,5], developed over the past fifteen years, has been made possible by improvements in CCD cameras, electron energy loss spectrometers, imaging filters and bright field emission sources. The automation of the data collection and subsequent numerical analysis provides routine and rapid access to GðrÞ, with the size of the volume analysed being defined by the
0022-3093/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0022-3093(02)01908-7
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diameter of the incident beam or the size of the selected area aperture, together with the thickness of the sample. This paper reports developments which allow the technique to be applied to volumes as small as 2 nm in width.
hf ðQÞi ¼
U 1 X Ui fi ðQÞ; N i¼1
ð5Þ
where Ui is number of atoms of type i in the specimen, U is the number of PUdifferent types of atom in the sample and N ¼ i¼1 Ui . Once again following [6] 2
IðQÞ N hf ðQÞ i
2. Theory
2
The diffraction pattern of an assembly of atoms can be described mathematically as IðQÞ ¼
N X N X
fm ðQÞfn ðQÞe2piQ:rmn ;
ð1Þ
m¼1 n¼1
where fm ðQÞ is the atomic scattering factor of atom m, rmn ¼ rm rn (rm being the position of the atom m), Q is the scattering vector and N is the number of atoms in the assembly. Following Warren [6] through a series of mathematical operations which include (i) the assumption that the assembly of atoms represents an isotropic amorphous solid containing a single atomic species and (ii) the introduction of a density function pm ðrmn Þ we obtain Z 1 IðQÞ sinðQrÞ 4pr2 dr; ð2Þ ¼ 1 þ ðpðrÞ po Þ Nf 2 ðQÞ Qr 0 where po is the average density of the amorphous material. Rearrangement of Eq. (2) gives the reduced intensity function UðQÞ IðQÞ UðQÞ ¼ Q 1 Nf 2 ðQÞ Z 1 ¼ 4prðpðrÞ po Þ sinðQrÞ dr: ð3Þ 0
From Eq. (3) it is clear that UðQÞ is the Fourier sine transform of GðrÞ and Z 2 1 GðrÞ ¼ UðQÞ sinðQrÞ dQ; ð4Þ p 0 where GðrÞ ¼ 4prðpðrÞ po Þ. For a sample with more than one atomic species it is possible to develop an approximate method for inversion of Eq. (1) to obtain GðrÞ, through the introduction of chemically averaged atomic scattering factors
N hf ðQÞi Z 1 sinðQrÞ 4pr2 dr: ¼ ðpðrÞ po Þ Qr 0
ð6Þ
Eq. (6) is analogous to Eq. (2) and can be converted to GðrÞ in a similar fashion. However in this case GðrÞ is a combination of all the partial radial distribution functions from each of the atomic species and as such, care must be taken when interpreting GðrÞ. Further details are given in [4,7]. Routine GðrÞ analysis with electrons, as described above, and in detail by Cockayne and McKenzie [4], assumes that parallel illumination is used to form the diffraction pattern. However the electron optics used to form the parallel beam restricts the minimum beam diameter that can be produced; for a conventional transmission electron microscope (TEM) this minimum beam diameter is approximately 0.4 lm [8]. Smaller illuminating probes can be formed in a TEM using convergent illumination. However with convergent illumination, a correction must be applied to the diffraction data, to remove the effect of the convergence of the beam on the diffraction pattern [9], before it is used for GðrÞ analysis. Convergent illumination causes the electrons incident upon the sample to encompass a range of different directions. If it is assumed that electrons incident upon the sample in a particular direction are incoherent with electrons from any other direction [10] then the observed diffraction pattern is the superposition of the diffraction patterns formed from each of the different directions of incidence. If it is also assumed that the diffraction patterns formed from each of the different incident directions are the same, then the diffraction pattern formed with convergent illumination, Icon ðQÞ, is the convolution of the diffraction pattern from a single incident beam direction, IðQÞ, with the an-
W. McBride, D.J.H. Cockayne / Journal of Non-Crystalline Solids 318 (2003) 233–238
gular intensity distribution of the convergent beam, J ðQÞ. Z 1 Icon ðQÞ ¼ IðsÞJ ðQ sÞ ds ð7Þ 1
The problem of removing the effect of convergent illumination from the diffraction pattern is then one of deconvolution. There are a number of different algorithms by which deconvolution can be effected [11,12]. One of the more popular algorithms is the Richardson– Lucy algorithm [13], which has been successfully applied to a number of different problems [14,15]. This algorithm is derived by application of Bayes theorem to Eq. (7) and is expressed as Z 1 Icon ðQÞJ ðQ sÞ ds rþ1 r IðQÞ ¼ IðQÞ ð8Þ r H ðsÞ 1 where r
H ðsÞ ¼
Z
1
r
J ðs QÞIðQÞ dQ
ð9Þ
1
Eqs. (8) and (9) show that IðQÞ can be recovered from Icon ðQÞ and J ðQÞ by iteration with IðQÞ0 set as the mean of Icon ðQÞ.
3. Experiment Electron diffraction patterns were formed using a JEOL JEM3000F field emission gun transmission electron microscope operating at 297 kV. Energy filtering of the diffraction patterns was performed using a Gatan imaging filter (GIF) [16] incorporating a 2048 2048 pixel 7941F/20 MegaScan CCD camera. The camera exhibits high linearity over its dynamic range which is approximately 1–16 000 counts per pixel. The linearity and large dynamic range of the CCD are particularly useful when collecting diffraction patterns to high scattering angles, since they have an intrinsically large dynamic range. An energy selecting window of 10 eV, centred on the zero loss peak of the electron energy loss spectrum, was employed. In order to collect data to high scattering angles through the GIF, small camera lengths must be used. This required operating the microscope in
235
Table 1 Convergence angles measured for given probe sizes Spot size (nm) FWHM
Convergence angle (mrad)
1.8 2.0 3.0
1.20 0.85 0.66
free lens mode, allowing the use of camera lengths smaller than the 12 cm available as a preset value. The convergence angle of the electron beam was determined from the diameter of the undiffracted beam as measured in the back focal plane of the objective lens. The conditions for imaging this plane were found by determining the settings of the illumination conditions necessary to obtain parallel illumination (using the method of Christenson and Eades [17]), and then determining the settings of the imaging lenses necessary for imaging the back focal plane of the objective lens (i.e. for imaging the fine spot of the diffraction pattern). Using these same conditions of the imaging lenses for any setting of the illumination forming lenses allowed the convergence of the incident electron beam to be measured. To form the small electron probes used in these studies we used the nanobeam diffraction mode which allows small probes to be formed. Table 1 shows the probe sizes (FWHM) as measured from profiles in the image plane, together with their convergence angles as measured using the method described above. These probes have a high brightness because of the field emission gun, and this is essential for obtaining adequate counting statistics in the diffraction data.
4. Results Electron diffraction patterns were collected from a thin film of amorphous carbon (thickness <20 nm). Fig. 1(a) shows the diffraction pattern formed using parallel illumination, from a surface area of diameter >10 lm. Fig. 1(b) shows a diffraction pattern from the same material collected using a probe of 1.8 nm FWHM, of brightness 1:95 0:03 1013 A m2 sr1 and having a convergence angle of 1.2 mrad. It is apparent that the continuous rings of Fig. 1(a), which are typical for
236
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Fig. 1. Diffraction patterns of amorphous carbon (a) formed using a parallel beam and (b) formed using a probe of 1.8 nm FWHM and a convergence angle of 1.2 mrad.
amorphous materials, are not evident in Fig. 1(b), where the rings are discontinuous or speckled. This speckled ring structure occurs as a result of investigating a very small volume of material. Fig. 2(a) (parallel) and (convergent) are the radial profiles of Fig. 1(a) and (b) respectively and are obtained by azimuthal averaging. To effect the azimuthal average, the location of the centre of the diffraction pattern was determined by fitting circles to intensity contours in the diffraction data, using the method of least squares. Comparison of Fig. 2(a) (parallel) and (convergent) suggests that the convergence angle of the beam (1.2 mrad) has only a modest effect on the diffraction pattern. In order to carry out the deconvolution discussed previously, a profile of the central spot in the diffraction pattern was recorded (Fig. 2(b)), using the same conditions as those used to form the nanobeam diffraction pattern. Deconvolution
of Fig. 2(a) (convergent) using Eq. (8) resulted in Fig. 2(c). When Fig. 2(c) is compared with Fig. 2(a) (parallel) it is seen that deconvolution has removed the effect of beam convergence. Conversion of the diffraction patterns to GðrÞ required calibration of the angular scale against a crystalline silicon standard, and application of Eqs. (3) and (4). An empirical background was substituted for Nf 2 ðQÞ in Eq. (3) as values for f ðQÞ obtained from data tables [18] did not give a good fit to IðQÞ. Conversion of the three diffraction patterns, Fig. 2(a) (parallel), (convergent) and (c), to GðrÞ is shown in Fig. 2(d), where again it is seen that the beam convergence has been effectively removed. Fig. 3(a) shows a cross-section of a thin multilayered specimen, comprised of a crystalline silicon substrate (region E), its native oxide layer which is amorphous (region F) and a layer of crystalline NiFe Permalloy (region G). For the purposes of this experiment it is sufficient to consider the sample as a thin amorphous layer sandwiched between two crystalline regions. The width of the amorphous layer (region F) is 2 nm. There were several reasons for selecting the multilayer as a test specimen: (i) since the crystalline silicon surface bounding the amorphous layer is planar (1 0 0), its diffraction pattern allows the sample to be oriented so that the interfaces are viewed edge on; (ii) the amorphous layer is known to extend through the thickness of the TEM sample; (iii) bounding the amorphous region by two crystalline regions defines the region of interest and allows the dif-
Fig. 2. (a) The azimuthal averages of Fig. 1(a) and (b). (b) Profile of the central spot in the diffraction pattern formed using the same electron optical conditions as in Fig. 1(b). (c) The deconvolution of (a) (convergent) with (b), using the Richardson–Lucy algorithm. (d) GðrÞ from the two profiles in Fig. 2(a) and of (c) show that the effect of beam convergence has been removed.
W. McBride, D.J.H. Cockayne / Journal of Non-Crystalline Solids 318 (2003) 233–238
237
Fig. 3. (a) Image of thin multilayered specimen, E ¼ silicon substrate (crystalline), F ¼ native oxide layer (amorphous), G ¼ NiFe Permalloy (crystalline). (b) Diffraction pattern from region E. (c) Diffraction pattern from region F. (d) Diffraction pattern from region G.
fraction pattern from the amorphous material to be distinguished from the diffraction patterns of the bounding crystals. Fig. 3(b)–(d) show the diffraction patterns collected from regions E, F and G respectively using a probe of diameter 1.8 nm and convergence angle 1.2 mrad. The diffraction patterns taken from regions E and G are both typical crystalline diffraction patterns collected with a convergent beam. Fig. 3(c) contains both crystalline diffraction and the speckled diffraction pattern from the amorphous region. After cutting the crystalline diffraction spots out of Fig. 3(c) using digital imaging software, the diffraction pattern was azimuthally averaged (allowing for the absent data), deconvoluted (using Eq. (8)) and processed through to GðrÞ using Eqs. (3) and (4). The GðrÞ is shown in Fig. 4 along
Table 2 Nearest neighbour distances ) ) SiO2 Exp. (A Known (A a b c
1.64 0.04 2.58 0.04 3.14 0.04
1.60 (Si–O) 2.62 (O–O) 3.13 (Si–Si)
) a-C Known (A 1.51 2.53
with the GðrÞ obtained without deconvolution. Once again an empirical background was substituted for Nf 2 ðQÞ in Eq. (3). The first three peaks in the GðrÞ, marked (a), (b), and (c) are given in Table 2, and compared with known bond lengths for silicon dioxide [19] and carbon [20] (carbon is included, because carbon contamination is a common difficulty with small electron probes). The bond lengths are in excellent agreement with those of silicon dioxide (and not with those of carbon), indicating that the structure of amorphous regions down to 2 nm can be characterised using extended GðrÞ analysis. The straightforward interpretation of the peaks in GðrÞ (Fig. 4) as measurements of the nearest neighbour distances of O–Si, O–O and Si–Si is satisfactory if the individual partial radial distribution functions do not overlap, and neutron diffraction studies of amorphous silicon dioxide (e.g. [21]) show that they do not.
5. Conclusion
Fig. 4. GðrÞs from region F of Fig. 3(a) obtained with and without deconvolution. Deconvolution has changed the heights but not the positions of peaks (a), (b) and (c).
By significantly reducing the minimum volume from which GðrÞ analysis can be performed the applicability of GðrÞ analysis using electrons has been significantly extended to regions 2 nm in width. It would appear that the smallest volume of
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amorphous material that can be investigated using GðrÞ analysis will not be restricted by instrumental limitations but rather by whether or not the ring structure in the diffraction pattern is continuous enough to perform the analysis. Acknowledgements The authors would like to thank Dr Richard Langford for preparing the thin multilayered sample, Mr Ron Doole for expert technical advice and JEOL for financial support.
References [1] P.C. Yashar, W.D. Sproul, Vacuum 55 (1999) 179. [2] B.G. Svensson, C. Jagadish, A. Hallen, et al., Phys. Rev. B 55 (1997) 10498. [3] L.P. Ferroni, G. Pezzotti, T. Isshiki, et al., Acta Mater. 49 (2001) 2109. [4] D.J.H. Cockayne, D.R. McKenzie, Acta Crystallogr. Sec. a 44 (1988) 870. [5] D. Heinemann, W. Mader, Ultramicroscopy 74 (1998) 113. [6] B.E. Warren, X-Ray Diffraction, Dover, New York, 1969.
[7] D. Cockayne, D. McKenzie, D. Muller, Microsc. Microanal. Microstruct. 2 (1991) 359. [8] D.B. Williams, C.B. Carter, Transmission Electron Microscopy A Textbook for Materials Science, Plenum, New York, 1996. [9] W.E. McBride, D.J.H. Cockayne, C.M. Goringe, Ultramicroscopy 76 (1991) 115. [10] W. Coene, D. Van Dyck, J. Van Landuyt, Optik 73 (1986) 13. [11] J. Skilling, R.K. Bryan, Monthly Notices Royal Astron. Soc. 211 (1984) 111. [12] N. Tanaka, T. Maki, K. Mihama, et al., J. Electr. Microsc. 38 (1989) 54. [13] W.H. Richardson, J. Opt. Soc. Am. 62 (1972) 55. [14] J. Knodlseder, D. Dixon, K. Bennett, et al., Astron. Astrophys. 345 (1999) 813. [15] G.M.P. Van Kempen, L.J. Van Vliet, P.J. Verveer, et al., J. Microsc. 185 (1997) 354. [16] O.L. Krivanek, A.J. Gubbens, N. Dellby, et al., Microsc. Microanal. Microstruct. 3 (1992) 2. [17] K.K. Christenson, J.A. Eades, Ultramicroscopy 26 (1988) 113. [18] D. Rez, P. Rez, I. Grant, Acta Crystallogr. Sec. a 50 (1994) 481. [19] J.R.G. Da Silva, D.G. Pinatti, C.E. Anderson, et al., Philos. Mag. 31 (1974) 713. [20] D.G. McCulloch, D.R. McKenzie, C.M. Goringe, Phys. Rev. B 61 (2000) 2349. [21] A.C. Wright, J. Non-Cryst. Solids 179 (1994) 84.
The structure of nanovolumes of amorphous materials W. McBride *, D.J.H. Cockayne Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK Received 8 February 2002; received in revised form 20 September 2002
Abstract In this paper we demonstrate reduced density function (GðrÞ) analysis from amorphous volumes as small as 2 nm in width. Obtaining the GðrÞ from a diffraction pattern is a common starting point in the characterisation of amorphous materials. The use of electrons to form the diffraction pattern allows very small volumes of amorphous material to be investigated. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.43.Dq
1. Introduction There is an increasing need to characterise the structure of small localised volumes of amorphous material, such as are found, for example, in thin amorphous films used as hard coatings [1] in amorphised volumes in semiconductor implantations [2] and in glassy intergranular phases [3]. In many structural studies of amorphous materials, the starting point for characterisation is obtaining the reduced density function, GðrÞ, which can be used to give an accurate measure of nearest neighbour distances, coordination numbers and interatomic bond angles. GðrÞ can be obtained from the X-ray, neutron or electron diffraction
* Corresponding author. Tel.: +44-1865 273 661; fax: +441865 283 329. E-mail address: [email protected] (W. McBride).
pattern of an amorphous material, through Fourier transformation of the scattered intensity IðQÞ (where Q ¼ 2 sinðhÞ=k, h is half the scattering angle and k is the wavelength of the radiation used in the diffraction measurements). When measuring GðrÞ from small volumes of amorphous material it is advantageous to choose electrons because the relatively large scattering cross-section of electrons and the ability to focus electrons with magnetic lenses allows the analysis to be carried out on small volumes selected from within the microstructure. The technique of energy selected GðrÞ analysis using electrons [4,5], developed over the past fifteen years, has been made possible by improvements in CCD cameras, electron energy loss spectrometers, imaging filters and bright field emission sources. The automation of the data collection and subsequent numerical analysis provides routine and rapid access to GðrÞ, with the size of the volume analysed being defined by the
0022-3093/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0022-3093(02)01908-7
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W. McBride, D.J.H. Cockayne / Journal of Non-Crystalline Solids 318 (2003) 233–238
diameter of the incident beam or the size of the selected area aperture, together with the thickness of the sample. This paper reports developments which allow the technique to be applied to volumes as small as 2 nm in width.
hf ðQÞi ¼
U 1 X Ui fi ðQÞ; N i¼1
ð5Þ
where Ui is number of atoms of type i in the specimen, U is the number of PUdifferent types of atom in the sample and N ¼ i¼1 Ui . Once again following [6] 2
IðQÞ N hf ðQÞ i
2. Theory
2
The diffraction pattern of an assembly of atoms can be described mathematically as IðQÞ ¼
N X N X
fm ðQÞfn ðQÞe2piQ:rmn ;
ð1Þ
m¼1 n¼1
where fm ðQÞ is the atomic scattering factor of atom m, rmn ¼ rm rn (rm being the position of the atom m), Q is the scattering vector and N is the number of atoms in the assembly. Following Warren [6] through a series of mathematical operations which include (i) the assumption that the assembly of atoms represents an isotropic amorphous solid containing a single atomic species and (ii) the introduction of a density function pm ðrmn Þ we obtain Z 1 IðQÞ sinðQrÞ 4pr2 dr; ð2Þ ¼ 1 þ ðpðrÞ po Þ Nf 2 ðQÞ Qr 0 where po is the average density of the amorphous material. Rearrangement of Eq. (2) gives the reduced intensity function UðQÞ IðQÞ UðQÞ ¼ Q 1 Nf 2 ðQÞ Z 1 ¼ 4prðpðrÞ po Þ sinðQrÞ dr: ð3Þ 0
From Eq. (3) it is clear that UðQÞ is the Fourier sine transform of GðrÞ and Z 2 1 GðrÞ ¼ UðQÞ sinðQrÞ dQ; ð4Þ p 0 where GðrÞ ¼ 4prðpðrÞ po Þ. For a sample with more than one atomic species it is possible to develop an approximate method for inversion of Eq. (1) to obtain GðrÞ, through the introduction of chemically averaged atomic scattering factors
N hf ðQÞi Z 1 sinðQrÞ 4pr2 dr: ¼ ðpðrÞ po Þ Qr 0
ð6Þ
Eq. (6) is analogous to Eq. (2) and can be converted to GðrÞ in a similar fashion. However in this case GðrÞ is a combination of all the partial radial distribution functions from each of the atomic species and as such, care must be taken when interpreting GðrÞ. Further details are given in [4,7]. Routine GðrÞ analysis with electrons, as described above, and in detail by Cockayne and McKenzie [4], assumes that parallel illumination is used to form the diffraction pattern. However the electron optics used to form the parallel beam restricts the minimum beam diameter that can be produced; for a conventional transmission electron microscope (TEM) this minimum beam diameter is approximately 0.4 lm [8]. Smaller illuminating probes can be formed in a TEM using convergent illumination. However with convergent illumination, a correction must be applied to the diffraction data, to remove the effect of the convergence of the beam on the diffraction pattern [9], before it is used for GðrÞ analysis. Convergent illumination causes the electrons incident upon the sample to encompass a range of different directions. If it is assumed that electrons incident upon the sample in a particular direction are incoherent with electrons from any other direction [10] then the observed diffraction pattern is the superposition of the diffraction patterns formed from each of the different directions of incidence. If it is also assumed that the diffraction patterns formed from each of the different incident directions are the same, then the diffraction pattern formed with convergent illumination, Icon ðQÞ, is the convolution of the diffraction pattern from a single incident beam direction, IðQÞ, with the an-
W. McBride, D.J.H. Cockayne / Journal of Non-Crystalline Solids 318 (2003) 233–238
gular intensity distribution of the convergent beam, J ðQÞ. Z 1 Icon ðQÞ ¼ IðsÞJ ðQ sÞ ds ð7Þ 1
The problem of removing the effect of convergent illumination from the diffraction pattern is then one of deconvolution. There are a number of different algorithms by which deconvolution can be effected [11,12]. One of the more popular algorithms is the Richardson– Lucy algorithm [13], which has been successfully applied to a number of different problems [14,15]. This algorithm is derived by application of Bayes theorem to Eq. (7) and is expressed as Z 1 Icon ðQÞJ ðQ sÞ ds rþ1 r IðQÞ ¼ IðQÞ ð8Þ r H ðsÞ 1 where r
H ðsÞ ¼
Z
1
r
J ðs QÞIðQÞ dQ
ð9Þ
1
Eqs. (8) and (9) show that IðQÞ can be recovered from Icon ðQÞ and J ðQÞ by iteration with IðQÞ0 set as the mean of Icon ðQÞ.
3. Experiment Electron diffraction patterns were formed using a JEOL JEM3000F field emission gun transmission electron microscope operating at 297 kV. Energy filtering of the diffraction patterns was performed using a Gatan imaging filter (GIF) [16] incorporating a 2048 2048 pixel 7941F/20 MegaScan CCD camera. The camera exhibits high linearity over its dynamic range which is approximately 1–16 000 counts per pixel. The linearity and large dynamic range of the CCD are particularly useful when collecting diffraction patterns to high scattering angles, since they have an intrinsically large dynamic range. An energy selecting window of 10 eV, centred on the zero loss peak of the electron energy loss spectrum, was employed. In order to collect data to high scattering angles through the GIF, small camera lengths must be used. This required operating the microscope in
235
Table 1 Convergence angles measured for given probe sizes Spot size (nm) FWHM
Convergence angle (mrad)
1.8 2.0 3.0
1.20 0.85 0.66
free lens mode, allowing the use of camera lengths smaller than the 12 cm available as a preset value. The convergence angle of the electron beam was determined from the diameter of the undiffracted beam as measured in the back focal plane of the objective lens. The conditions for imaging this plane were found by determining the settings of the illumination conditions necessary to obtain parallel illumination (using the method of Christenson and Eades [17]), and then determining the settings of the imaging lenses necessary for imaging the back focal plane of the objective lens (i.e. for imaging the fine spot of the diffraction pattern). Using these same conditions of the imaging lenses for any setting of the illumination forming lenses allowed the convergence of the incident electron beam to be measured. To form the small electron probes used in these studies we used the nanobeam diffraction mode which allows small probes to be formed. Table 1 shows the probe sizes (FWHM) as measured from profiles in the image plane, together with their convergence angles as measured using the method described above. These probes have a high brightness because of the field emission gun, and this is essential for obtaining adequate counting statistics in the diffraction data.
4. Results Electron diffraction patterns were collected from a thin film of amorphous carbon (thickness <20 nm). Fig. 1(a) shows the diffraction pattern formed using parallel illumination, from a surface area of diameter >10 lm. Fig. 1(b) shows a diffraction pattern from the same material collected using a probe of 1.8 nm FWHM, of brightness 1:95 0:03 1013 A m2 sr1 and having a convergence angle of 1.2 mrad. It is apparent that the continuous rings of Fig. 1(a), which are typical for
236
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Fig. 1. Diffraction patterns of amorphous carbon (a) formed using a parallel beam and (b) formed using a probe of 1.8 nm FWHM and a convergence angle of 1.2 mrad.
amorphous materials, are not evident in Fig. 1(b), where the rings are discontinuous or speckled. This speckled ring structure occurs as a result of investigating a very small volume of material. Fig. 2(a) (parallel) and (convergent) are the radial profiles of Fig. 1(a) and (b) respectively and are obtained by azimuthal averaging. To effect the azimuthal average, the location of the centre of the diffraction pattern was determined by fitting circles to intensity contours in the diffraction data, using the method of least squares. Comparison of Fig. 2(a) (parallel) and (convergent) suggests that the convergence angle of the beam (1.2 mrad) has only a modest effect on the diffraction pattern. In order to carry out the deconvolution discussed previously, a profile of the central spot in the diffraction pattern was recorded (Fig. 2(b)), using the same conditions as those used to form the nanobeam diffraction pattern. Deconvolution
of Fig. 2(a) (convergent) using Eq. (8) resulted in Fig. 2(c). When Fig. 2(c) is compared with Fig. 2(a) (parallel) it is seen that deconvolution has removed the effect of beam convergence. Conversion of the diffraction patterns to GðrÞ required calibration of the angular scale against a crystalline silicon standard, and application of Eqs. (3) and (4). An empirical background was substituted for Nf 2 ðQÞ in Eq. (3) as values for f ðQÞ obtained from data tables [18] did not give a good fit to IðQÞ. Conversion of the three diffraction patterns, Fig. 2(a) (parallel), (convergent) and (c), to GðrÞ is shown in Fig. 2(d), where again it is seen that the beam convergence has been effectively removed. Fig. 3(a) shows a cross-section of a thin multilayered specimen, comprised of a crystalline silicon substrate (region E), its native oxide layer which is amorphous (region F) and a layer of crystalline NiFe Permalloy (region G). For the purposes of this experiment it is sufficient to consider the sample as a thin amorphous layer sandwiched between two crystalline regions. The width of the amorphous layer (region F) is 2 nm. There were several reasons for selecting the multilayer as a test specimen: (i) since the crystalline silicon surface bounding the amorphous layer is planar (1 0 0), its diffraction pattern allows the sample to be oriented so that the interfaces are viewed edge on; (ii) the amorphous layer is known to extend through the thickness of the TEM sample; (iii) bounding the amorphous region by two crystalline regions defines the region of interest and allows the dif-
Fig. 2. (a) The azimuthal averages of Fig. 1(a) and (b). (b) Profile of the central spot in the diffraction pattern formed using the same electron optical conditions as in Fig. 1(b). (c) The deconvolution of (a) (convergent) with (b), using the Richardson–Lucy algorithm. (d) GðrÞ from the two profiles in Fig. 2(a) and of (c) show that the effect of beam convergence has been removed.
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237
Fig. 3. (a) Image of thin multilayered specimen, E ¼ silicon substrate (crystalline), F ¼ native oxide layer (amorphous), G ¼ NiFe Permalloy (crystalline). (b) Diffraction pattern from region E. (c) Diffraction pattern from region F. (d) Diffraction pattern from region G.
fraction pattern from the amorphous material to be distinguished from the diffraction patterns of the bounding crystals. Fig. 3(b)–(d) show the diffraction patterns collected from regions E, F and G respectively using a probe of diameter 1.8 nm and convergence angle 1.2 mrad. The diffraction patterns taken from regions E and G are both typical crystalline diffraction patterns collected with a convergent beam. Fig. 3(c) contains both crystalline diffraction and the speckled diffraction pattern from the amorphous region. After cutting the crystalline diffraction spots out of Fig. 3(c) using digital imaging software, the diffraction pattern was azimuthally averaged (allowing for the absent data), deconvoluted (using Eq. (8)) and processed through to GðrÞ using Eqs. (3) and (4). The GðrÞ is shown in Fig. 4 along
Table 2 Nearest neighbour distances ) ) SiO2 Exp. (A Known (A a b c
1.64 0.04 2.58 0.04 3.14 0.04
1.60 (Si–O) 2.62 (O–O) 3.13 (Si–Si)
) a-C Known (A 1.51 2.53
with the GðrÞ obtained without deconvolution. Once again an empirical background was substituted for Nf 2 ðQÞ in Eq. (3). The first three peaks in the GðrÞ, marked (a), (b), and (c) are given in Table 2, and compared with known bond lengths for silicon dioxide [19] and carbon [20] (carbon is included, because carbon contamination is a common difficulty with small electron probes). The bond lengths are in excellent agreement with those of silicon dioxide (and not with those of carbon), indicating that the structure of amorphous regions down to 2 nm can be characterised using extended GðrÞ analysis. The straightforward interpretation of the peaks in GðrÞ (Fig. 4) as measurements of the nearest neighbour distances of O–Si, O–O and Si–Si is satisfactory if the individual partial radial distribution functions do not overlap, and neutron diffraction studies of amorphous silicon dioxide (e.g. [21]) show that they do not.
5. Conclusion
Fig. 4. GðrÞs from region F of Fig. 3(a) obtained with and without deconvolution. Deconvolution has changed the heights but not the positions of peaks (a), (b) and (c).
By significantly reducing the minimum volume from which GðrÞ analysis can be performed the applicability of GðrÞ analysis using electrons has been significantly extended to regions 2 nm in width. It would appear that the smallest volume of
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amorphous material that can be investigated using GðrÞ analysis will not be restricted by instrumental limitations but rather by whether or not the ring structure in the diffraction pattern is continuous enough to perform the analysis. Acknowledgements The authors would like to thank Dr Richard Langford for preparing the thin multilayered sample, Mr Ron Doole for expert technical advice and JEOL for financial support.
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