- Email: [email protected]
On the measurement of thickness in nanoporous materials by EELS
Ultramicroscopy 111 (2010) 62–65
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
On the measurement of thickness in nanoporous materials by EELS Nan Jiang n, Dong Su 1, John C.H. Spence Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 November 2009 Received in revised form 10 August 2010 Accepted 29 September 2010
This work discusses thickness measurements in nanoporous MgO using the log-ratio method in electron energy-loss spectroscopy (EELS). In heterogeneous nanoporous systems, the method can induce large errors if the strength of excitations at interfaces between pores and the matrix is large. In homogeneous nanoporous systems, on the other hand, the log-ratio method is still valid, but the inelastic scattering mean-free-path is no longer equal to that in the same bulk system. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nanoporous Thickness EELS
1. Introduction Specimen thickness is one of the most crucial parameters for quantitative analysis in transmission electron microscopy (TEM). Several methods have been used to measure relative or absolute specimen thickness. The simplest and the most accurate method involves the use of oxide nanocrystals, which grow naturally as perfect single-crystal cubes, such as MgO, whose orientation can be determined from diffraction data. Other methods include straightforward measurements of planar features intersecting both upper and lower surfaces [1] and contamination spots [2] or lines [3]. More accurate methods, however, are based on convergent beam electron diffraction (CBED), including two-beam CBED [4] and zone-axis-CBED [5] methods. Due to the requirement for strong diffraction, however, these are not applicable to very thin (or porous), fine grained polycrystalline or amorphous specimens. For thin crystals or amorphous materials, the differential characteristic X-ray absorption method has been used, which makes use of the difference in absorption between K and L (or L and M) characteristic lines emitted simultaneously from the same [6] or different [7] elements. This method, unfortunately, is not applicable for specimens containing only light elements. A method that may, in principle, overcome all the above mentioned limitations is based on electron energy-loss spectroscopy (EELS) [8]. The fundamental assumption of this EELS method is that multiple inelastic scatterings consist of a series of independent events that are thickness dependent, and therefore obey Poisson
n
Corresponding author. Tel.: + 1 480 7277169. E-mail address: [email protected] (N. Jiang). 1 Present address: Center for Functional Nanomaterials, Brookhaven National Laboratory, USA. 0304-3991/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.09.011
statistics Pn ¼ ð1=n!Þðt=lÞn expðt=lÞ for n-fold scattering probability [8]. The thickness t can then be determined from the unscattered (n¼0) component (i.e. zero-loss peak): t=l ¼ lnðItot =I0 Þ
ð1Þ
In this log-ratio formula, l is the total inelastic scattering meanfree-path (MFP), and I0 and Itot are integrated intensities of the zeroloss peak and the entire energy-loss spectrum, respectively. The absolute thickness can be obtained if the effective total inelastic MFP is known. In practice, l is sensitively dependent on experimental conditions, such as collection semiangle and beam accelerating voltage [8]. Details of this log-ratio method have been given in many review articles and textbooks [8,9]. Strictly speaking, the total inelastic scattering Itot also includes a contribution from scattering that is independent of specimen thickness, e.g. surface excitations [10]. As a good approximation, intensities due to surface excitations can be ignored if the specimen is not very thin, generally t/l 4 0.1 [9]. One of the advantages of using the log-ratio formula is that the method does not require crystalline perfection. It is therefore widely applied to amorphous materials [11]. In the study of porous materials, the effective thickness of the specimen is crucially important in determining porosity. In specimens that may vary in porosity, thickness is usually measured in terms of mass-thickness (or effective thickness) [12]. Due to high density of nanopores, the effective thickness (or mass thickness) along the path of the electron beam must be shorter than the physical thickness of the specimen. The difference can be used to estimate porosity if the size of the nanopores can be obtained from an image. One may consider the log-ratio EELS technique because the incident electron loses its energy only in the path that contains matter [13,14]. In nanoporous materials, however, there are many
N. Jiang et al. / Ultramicroscopy 111 (2010) 62–65
interfaces between material and nanopores. The interfacial excitations alter the effective dielectric response to the electron beam [15]. Although the physical thickness of a porous specimen may not be small, the large portion of surface area may affect the applicability of the EELS method to porous materials. In this note, we re-evaluate the validity of this method in nanoporous materials.
2. Experimental The samples used in this study were nanoporous and bulk MgO, and were dehydrated from Mg(OH)2 by thermal annealing at 450 and 1000 1C for 2 h, respectively. The samples dehydrated at 450 1C consist of nanopores, but these are sintered to bulk MgO nanoparticles at 1000 1C. The detailed procedure can be found elsewhere [16]. The TEM specimens were prepared by grinding the samples into powders in dry air, and picking them up using a Cu grid covered with a lacy carbon thin film. The specimens were then immediately transferred into and observed in a JEOL 2010F (S)TEM with a fieldemission gun operating at 200 keV and a Gatan Enfina parallel EELS system. The energy resolution is about 1.0 eV measured by the full width at half maximum of the zero-loss peak. The samples were examined using both TEM and STEM modes. The STEM probe was about 2 nm in diameter, and the semiangle of the collection aperture 15 mrad. All spectra were acquired and processed using Digital Micrograph.
3. Results and discussion Fig. 1 compares two spectra acquired in bulk and nanoporous MgO. Both spectra were normalized to their total integrated intensities [0 120 eV]. The relative thicknesses evaluated by the log-ratio methods are about the same, i.e. t=l ¼ 0:80 and 0.78 for bulk and nanoporous MgO, respectively. The ratios of plasmon to zero-loss peak are also about the same; they are 0.055 and 0.052 for bulk and nanoporous MgO, respectively. Following the criterion given by Williams and Carter [9], the errors in composition measurements for these thicknesses by EELS should be less than 7 22.2
23.0
Mg L23-edge
0.03
6 0.02
Normalized Intensity
5 0.01 4
50
60
70
80
lL
90
2
1
20
40
60
80
nanoporous MgO
100 bulk MgO
0 0
20
40
10%. However, the integral of the background-stripped Mg L23-edge in nanoporous MgO is only about half of that in bulk MgO, and the ratio of their integrated intensities is nearly 1:2. Therefore, in apparently equal mass-thickness samples, the amount of Mg in the nanoporous sample is only half of that in the bulk sample. This cannot be true, because both X-ray and electron diffractions indicate that the lattice parameters of MgO in the two samples are not significantly different from each other [15]. The inconsistent results of the method indicate the failure of the logratio method for measuring thickness in nanoporous MgO. This inconsistency can be reconciled by comparing the plasmon excitations in both samples. As shown in the inset of Fig. 1, the loss intensities before the maximum peak in nanoporous MgO are much stronger than in the bulk MgO. In addition, the peak position of the maximum loss shifts from 23.0 eV in bulk MgO to 22.2 eV in nanoporous MgO. All these differences between nanoporous and bulk MgO have been interpreted as the result of existence of nanopores in the MgO matrix [15]. As a simple model, we can consider a fast electron passing through a composite medium in nanoporous MgO, which consists of nanopores and an MgO matrix. It might seem reasonable to determine the energy loss from the sum of pores (vacuum) and MgO according to their volume fractions. However, such a composite system is full of dielectric interfaces between the nanopores and MgO matrix. According to the effective medium dielectric theory [17], the contribution from interfacial excitations must be included. As for surface excitations, the interfacial term does not depend on specimen thickness, and thus can cause error in thickness measurement using the log-ratio method. In a heterogeneous system, the large contribution from the interfaces between pores and matrix may lead to failure of the method. However, if nanopores are homogeneously distributed in the matrix, or if the density of nanopores is constant throughout the specimen, the total contribution from the interfacial excitations may depend on the thickness of the specimen. In this case, the logratio method may work, but in a different manner. In other words, the effective inelastic scattering MFP in nanoporous material will be different from that of the bulk material. To confirm this idea, we acquired spectra from different thicknesses by running a STEM probe across the sample, as shown in Fig. 2a. The ‘‘thickness’’ profile across the sample evaluated using Eq. (1) is plotted in Fig. 2b. For comparison, Fig. 2b also gives the high angle annular darkfield intensity profile across the sample, and the corresponding normalized intensities of the Mg L23-edge, i.e. IMg(D)/IT(D), in which IMg(D) is the integrated intensity of the Mg L23-edge across an energy window of 53–83 eV, and IT(D) the intensity of zero-loss (direct beam) electrons, combined with the low-loss electrons, over an energy window of 30 eV. In the thin specimen, the normalized core-edge intensity is related to the cross-section of the Mg L23-edge sL, the density of bulk MgO rMgO, and the specimen (mass) thickness t, according to Refs. [8,9]: t sL rMgO t IMg ðDÞ=IT ðDÞ ¼ sL rMgO t exp
0 3
63
60
80
100
Energy loss (eV) Fig. 1. Comparison of EELS spectra between nanoporous and bulk MgO.
120
ð2Þ
Here we assume that the electrons contributing to the Mg L23-edge undergo a single ionization event. This approximation is correct only for a thin specimen, in which the thickness is much smaller than the mean-free-path of the ionization event (lL), which should be much larger than the total inelastic scattering MFP, i.e. lL b l. It is noticed that C K-edge EELS has been used to measure specimen thickness in biological samples [18]. It should be noted that we use the density rMgO of bulk MgO instead of the density of nanoporous MgO because t is the mass-thickness instead of the physical thickness of nanoporous MgO. It is then clear that the calculated t/l in nanoporous MgO has a very similar thickness dependence on the normalized intensity of the Mg L23-edge. This therefore
64
N. Jiang et al. / Ultramicroscopy 111 (2010) 62–65
0.12 nanoporous MgO bulk MgO
(IMg/IT) / ln (Itot/I0)
0.1
0.08
0.06
0.04
0.02
0.09 1.5
0
1 0.06
t /λ
0.5
Normalized intensity
Relative thickness
ADF
0.03 0 IMg (Δ)/IT (Δ) 0 50
100
150
Distance (nm) Fig. 2. (a) ADF image showing nanoporous MgO. The white line indicates the path of STEM probe running across the sample while acquiring EELS spectra. (b) Comparison of relative thickness calculated by log-ratio method with the corresponding normalized intensity of the Mg L23-edge.
suggests that nanopores are homogeneously distributed in the MgO matrix. In order to apply the log-ratio method to measure thickness, one needs to obtain the effective MFP for nanoporous materials. As mentioned above, the MFP in porous material is different from that in the parent bulk material. In porous materials, MFP depends not only on the material, but also on porosity, which is usually an unknown parameter, and this is one of the major reasons why mass thickness needs to be measured. From an experimental point of view, however, the MFP of nanoporous material can be estimated by carefully comparing thickness measurements in both nanoporous and bulk materials. According to Eqs. (1) and (2), one can obtain the ratio of the normalized intensity of the Mg L23-edge to the relative thickness obtained from the log-ratio method: R¼
IMg ðDÞ=IT ðDÞ ¼ sL rMgO , lnðItot =I0 Þ
l ¼ constant
ð3Þ
We expect this to be a straight line if one plots R versus relative thickness t/l. In reality, this is not true as shown in Fig. 3. This is because of experimental errors in measuring t/l when the specimen is very thin and the assumption of linear relationship between integrated core-edge intensity and thickness breaks down when the specimen is very thick. The total scattered intensity Itot
0
0.4
0.8
1.2
1.6
2
t/λ Fig. 3. Comparison of the value R between nanoporous and bulk MgO.
consists of thickness-dependent and thickness-independent components. The latter is mainly due to surface excitations, whose contribution to Itot decreases with increase in thickness. In a very thin specimen, however, bulk plasmon excitation can be largely suppressed by surface excitations. According to Eq. (2), the exponential term cannot be ignored if the specimen becomes thick. Therefore, Eq. (3) may be correct only for a range of thicknesses that is neither too small nor too large. As shown in Fig. 3, R decreases with increase in thickness in bulk MgO. Closer examination shows that there is a relatively ‘‘flat’’ range for t/l between approximately 0.5 and 1.0, and its value is between 0.075 and 0.080. For a specimen thinner than 0.5l, the data are scattered and have significantly larger values, indicating the strong effect of surface excitations on the bulk plasmon. On the other hand, for specimens thicker than 1.0l, R decreases rapidly with increase in thickness, indicating a violation of thin specimen requirement in Eq. (2). In contrast to the bulk, nanoporous MgO has a relatively flat R, although the overall trend is similar to that in the bulk. The data are also scattered in the specimen thinner than 0.5l, and slightly decreased in the specimen thicker than 1.0l. In between, R has an approximately constant value of about 0.035. The different values of R indicate the different values of MFP in nanoporous and bulk materials, since both materials have the same sL and rMgO, respectively. In other words, MFP of nanoporous MgO is about half of that in bulk MgO, i.e. lporous(MgO)E l(MgO)/2. Therefore, the measured thickness in nanoporous MgO is only about half of that in bulk MgO at the same relative thickness, as measured by the log-ratio method.
4. Conclusion Caution should be exercised when applying the log-ratio method to the measurement of thickness in porous materials. If the nanopores are heterogeneously distributed in the matrix, the strength of excitations at interfaces between pores and the matrix must be examined in order to estimate the error introduced in a thickness measurement. On the other hand, if the nanopores are homogeneously distributed in the matrix, the log-ratio method is in principle still applicable, but MFP in the nanoporous materials will be different from that in the parent bulk material. The approach introduced here in nanoporous MgO can also be extended to other systems. However, the conversion factor between physical
N. Jiang et al. / Ultramicroscopy 111 (2010) 62–65
thickness and apparent thickness is sample dependent. The value given in this work applies only for the particular sample of MgO present.
Acknowledgements This work is supported by NSF Award DMR0603993. The use of facilities within the Center for Solid State Science at ASU is also acknowledged. References [1] P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.J. Whelan, Electron Microscopy of Thin Crystals, 2nd edition, Krieger, New York, 1977. [2] G.W. Lorimer, G. Cliff, J.N. Clark, Developments in Electron Microscopy and Analysis, EMAG 75, in: D.L. Misell (Ed.), Institute of Physics, Bristol1976, p. 153. [3] Z. Pan, C.K.L. Davies, R.N. Stevens, J. Mater. Sci. 29 (1994) 1920.
65
[4] P.M. Kelly, A. Jostsons, R.G. Blake, J.G. Napier, Phys. Status Solidi (a) 31 (1975) 771. [5] J.M. Zuo, Y.F. Shi, Microsc. Microanal. 7 (Suppl. 2) (2001) 224. [6] P.L. Morris, M.D. Ball, P.J. Statham, Electron Microscopy and Analysis, in: T. Mulvey (Ed.), Institute of Physics, London1979, p. 413. [7] Z. Horita, K. Ichitani, T. Sano, M. Nemoto, Phil. Mag. A 59 (1989) 939. [8] R.F. Egerton, Electron Energy-Loss Spectroscopy, 2nd edition, Plenum Press, New York, 1996. [9] D.B. Williams and C.B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science (IV), Springer-Verlag, New York,1996. [10] P.E. Batson, Ultramicroscopy 50 (1993) 1. [11] C. Lee, Y. Ikematsu, D. Shindo, J. Electron Microsc. 51 (2002) 143. [12] R.D. Leapman, C.E. Fiori, C.R. Swyt, J. Microsc. 133 (1984) 239. [13] P.A. van Aken, F. Langenhorst, Eur. J. Mineral. 13 (2001) 329. [14] K.J. Abrams, S.E. Donnelly, J. Phys.: Conf. Ser. 26 (2006) 272. [15] N. Jiang, D. Su, J.C.H. Spence., A. Howie., Appl. Phys. Lett. 94 (2009) 253105. [16] D. Su, N. Jiang., J.C.H. Spence, F. He, W. Pertusky, J. Appl. Phys. 104 (2008) 063514. [17] A. Howie, Ultramicroscopy 11 (1983) 141. [18] L. Zhao, Y.Y. Wang, R. Ho, Z. Shao, A.V. Somolyo, A.P. Somolyo, Ultramiscroscopy 46 (1993) 406.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
On the measurement of thickness in nanoporous materials by EELS Nan Jiang n, Dong Su 1, John C.H. Spence Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 November 2009 Received in revised form 10 August 2010 Accepted 29 September 2010
This work discusses thickness measurements in nanoporous MgO using the log-ratio method in electron energy-loss spectroscopy (EELS). In heterogeneous nanoporous systems, the method can induce large errors if the strength of excitations at interfaces between pores and the matrix is large. In homogeneous nanoporous systems, on the other hand, the log-ratio method is still valid, but the inelastic scattering mean-free-path is no longer equal to that in the same bulk system. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nanoporous Thickness EELS
1. Introduction Specimen thickness is one of the most crucial parameters for quantitative analysis in transmission electron microscopy (TEM). Several methods have been used to measure relative or absolute specimen thickness. The simplest and the most accurate method involves the use of oxide nanocrystals, which grow naturally as perfect single-crystal cubes, such as MgO, whose orientation can be determined from diffraction data. Other methods include straightforward measurements of planar features intersecting both upper and lower surfaces [1] and contamination spots [2] or lines [3]. More accurate methods, however, are based on convergent beam electron diffraction (CBED), including two-beam CBED [4] and zone-axis-CBED [5] methods. Due to the requirement for strong diffraction, however, these are not applicable to very thin (or porous), fine grained polycrystalline or amorphous specimens. For thin crystals or amorphous materials, the differential characteristic X-ray absorption method has been used, which makes use of the difference in absorption between K and L (or L and M) characteristic lines emitted simultaneously from the same [6] or different [7] elements. This method, unfortunately, is not applicable for specimens containing only light elements. A method that may, in principle, overcome all the above mentioned limitations is based on electron energy-loss spectroscopy (EELS) [8]. The fundamental assumption of this EELS method is that multiple inelastic scatterings consist of a series of independent events that are thickness dependent, and therefore obey Poisson
n
Corresponding author. Tel.: + 1 480 7277169. E-mail address: [email protected] (N. Jiang). 1 Present address: Center for Functional Nanomaterials, Brookhaven National Laboratory, USA. 0304-3991/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2010.09.011
statistics Pn ¼ ð1=n!Þðt=lÞn expðt=lÞ for n-fold scattering probability [8]. The thickness t can then be determined from the unscattered (n¼0) component (i.e. zero-loss peak): t=l ¼ lnðItot =I0 Þ
ð1Þ
In this log-ratio formula, l is the total inelastic scattering meanfree-path (MFP), and I0 and Itot are integrated intensities of the zeroloss peak and the entire energy-loss spectrum, respectively. The absolute thickness can be obtained if the effective total inelastic MFP is known. In practice, l is sensitively dependent on experimental conditions, such as collection semiangle and beam accelerating voltage [8]. Details of this log-ratio method have been given in many review articles and textbooks [8,9]. Strictly speaking, the total inelastic scattering Itot also includes a contribution from scattering that is independent of specimen thickness, e.g. surface excitations [10]. As a good approximation, intensities due to surface excitations can be ignored if the specimen is not very thin, generally t/l 4 0.1 [9]. One of the advantages of using the log-ratio formula is that the method does not require crystalline perfection. It is therefore widely applied to amorphous materials [11]. In the study of porous materials, the effective thickness of the specimen is crucially important in determining porosity. In specimens that may vary in porosity, thickness is usually measured in terms of mass-thickness (or effective thickness) [12]. Due to high density of nanopores, the effective thickness (or mass thickness) along the path of the electron beam must be shorter than the physical thickness of the specimen. The difference can be used to estimate porosity if the size of the nanopores can be obtained from an image. One may consider the log-ratio EELS technique because the incident electron loses its energy only in the path that contains matter [13,14]. In nanoporous materials, however, there are many
N. Jiang et al. / Ultramicroscopy 111 (2010) 62–65
interfaces between material and nanopores. The interfacial excitations alter the effective dielectric response to the electron beam [15]. Although the physical thickness of a porous specimen may not be small, the large portion of surface area may affect the applicability of the EELS method to porous materials. In this note, we re-evaluate the validity of this method in nanoporous materials.
2. Experimental The samples used in this study were nanoporous and bulk MgO, and were dehydrated from Mg(OH)2 by thermal annealing at 450 and 1000 1C for 2 h, respectively. The samples dehydrated at 450 1C consist of nanopores, but these are sintered to bulk MgO nanoparticles at 1000 1C. The detailed procedure can be found elsewhere [16]. The TEM specimens were prepared by grinding the samples into powders in dry air, and picking them up using a Cu grid covered with a lacy carbon thin film. The specimens were then immediately transferred into and observed in a JEOL 2010F (S)TEM with a fieldemission gun operating at 200 keV and a Gatan Enfina parallel EELS system. The energy resolution is about 1.0 eV measured by the full width at half maximum of the zero-loss peak. The samples were examined using both TEM and STEM modes. The STEM probe was about 2 nm in diameter, and the semiangle of the collection aperture 15 mrad. All spectra were acquired and processed using Digital Micrograph.
3. Results and discussion Fig. 1 compares two spectra acquired in bulk and nanoporous MgO. Both spectra were normalized to their total integrated intensities [0 120 eV]. The relative thicknesses evaluated by the log-ratio methods are about the same, i.e. t=l ¼ 0:80 and 0.78 for bulk and nanoporous MgO, respectively. The ratios of plasmon to zero-loss peak are also about the same; they are 0.055 and 0.052 for bulk and nanoporous MgO, respectively. Following the criterion given by Williams and Carter [9], the errors in composition measurements for these thicknesses by EELS should be less than 7 22.2
23.0
Mg L23-edge
0.03
6 0.02
Normalized Intensity
5 0.01 4
50
60
70
80
lL
90
2
1
20
40
60
80
nanoporous MgO
100 bulk MgO
0 0
20
40
10%. However, the integral of the background-stripped Mg L23-edge in nanoporous MgO is only about half of that in bulk MgO, and the ratio of their integrated intensities is nearly 1:2. Therefore, in apparently equal mass-thickness samples, the amount of Mg in the nanoporous sample is only half of that in the bulk sample. This cannot be true, because both X-ray and electron diffractions indicate that the lattice parameters of MgO in the two samples are not significantly different from each other [15]. The inconsistent results of the method indicate the failure of the logratio method for measuring thickness in nanoporous MgO. This inconsistency can be reconciled by comparing the plasmon excitations in both samples. As shown in the inset of Fig. 1, the loss intensities before the maximum peak in nanoporous MgO are much stronger than in the bulk MgO. In addition, the peak position of the maximum loss shifts from 23.0 eV in bulk MgO to 22.2 eV in nanoporous MgO. All these differences between nanoporous and bulk MgO have been interpreted as the result of existence of nanopores in the MgO matrix [15]. As a simple model, we can consider a fast electron passing through a composite medium in nanoporous MgO, which consists of nanopores and an MgO matrix. It might seem reasonable to determine the energy loss from the sum of pores (vacuum) and MgO according to their volume fractions. However, such a composite system is full of dielectric interfaces between the nanopores and MgO matrix. According to the effective medium dielectric theory [17], the contribution from interfacial excitations must be included. As for surface excitations, the interfacial term does not depend on specimen thickness, and thus can cause error in thickness measurement using the log-ratio method. In a heterogeneous system, the large contribution from the interfaces between pores and matrix may lead to failure of the method. However, if nanopores are homogeneously distributed in the matrix, or if the density of nanopores is constant throughout the specimen, the total contribution from the interfacial excitations may depend on the thickness of the specimen. In this case, the logratio method may work, but in a different manner. In other words, the effective inelastic scattering MFP in nanoporous material will be different from that of the bulk material. To confirm this idea, we acquired spectra from different thicknesses by running a STEM probe across the sample, as shown in Fig. 2a. The ‘‘thickness’’ profile across the sample evaluated using Eq. (1) is plotted in Fig. 2b. For comparison, Fig. 2b also gives the high angle annular darkfield intensity profile across the sample, and the corresponding normalized intensities of the Mg L23-edge, i.e. IMg(D)/IT(D), in which IMg(D) is the integrated intensity of the Mg L23-edge across an energy window of 53–83 eV, and IT(D) the intensity of zero-loss (direct beam) electrons, combined with the low-loss electrons, over an energy window of 30 eV. In the thin specimen, the normalized core-edge intensity is related to the cross-section of the Mg L23-edge sL, the density of bulk MgO rMgO, and the specimen (mass) thickness t, according to Refs. [8,9]: t sL rMgO t IMg ðDÞ=IT ðDÞ ¼ sL rMgO t exp
0 3
63
60
80
100
Energy loss (eV) Fig. 1. Comparison of EELS spectra between nanoporous and bulk MgO.
120
ð2Þ
Here we assume that the electrons contributing to the Mg L23-edge undergo a single ionization event. This approximation is correct only for a thin specimen, in which the thickness is much smaller than the mean-free-path of the ionization event (lL), which should be much larger than the total inelastic scattering MFP, i.e. lL b l. It is noticed that C K-edge EELS has been used to measure specimen thickness in biological samples [18]. It should be noted that we use the density rMgO of bulk MgO instead of the density of nanoporous MgO because t is the mass-thickness instead of the physical thickness of nanoporous MgO. It is then clear that the calculated t/l in nanoporous MgO has a very similar thickness dependence on the normalized intensity of the Mg L23-edge. This therefore
64
N. Jiang et al. / Ultramicroscopy 111 (2010) 62–65
0.12 nanoporous MgO bulk MgO
(IMg/IT) / ln (Itot/I0)
0.1
0.08
0.06
0.04
0.02
0.09 1.5
0
1 0.06
t /λ
0.5
Normalized intensity
Relative thickness
ADF
0.03 0 IMg (Δ)/IT (Δ) 0 50
100
150
Distance (nm) Fig. 2. (a) ADF image showing nanoporous MgO. The white line indicates the path of STEM probe running across the sample while acquiring EELS spectra. (b) Comparison of relative thickness calculated by log-ratio method with the corresponding normalized intensity of the Mg L23-edge.
suggests that nanopores are homogeneously distributed in the MgO matrix. In order to apply the log-ratio method to measure thickness, one needs to obtain the effective MFP for nanoporous materials. As mentioned above, the MFP in porous material is different from that in the parent bulk material. In porous materials, MFP depends not only on the material, but also on porosity, which is usually an unknown parameter, and this is one of the major reasons why mass thickness needs to be measured. From an experimental point of view, however, the MFP of nanoporous material can be estimated by carefully comparing thickness measurements in both nanoporous and bulk materials. According to Eqs. (1) and (2), one can obtain the ratio of the normalized intensity of the Mg L23-edge to the relative thickness obtained from the log-ratio method: R¼
IMg ðDÞ=IT ðDÞ ¼ sL rMgO , lnðItot =I0 Þ
l ¼ constant
ð3Þ
We expect this to be a straight line if one plots R versus relative thickness t/l. In reality, this is not true as shown in Fig. 3. This is because of experimental errors in measuring t/l when the specimen is very thin and the assumption of linear relationship between integrated core-edge intensity and thickness breaks down when the specimen is very thick. The total scattered intensity Itot
0
0.4
0.8
1.2
1.6
2
t/λ Fig. 3. Comparison of the value R between nanoporous and bulk MgO.
consists of thickness-dependent and thickness-independent components. The latter is mainly due to surface excitations, whose contribution to Itot decreases with increase in thickness. In a very thin specimen, however, bulk plasmon excitation can be largely suppressed by surface excitations. According to Eq. (2), the exponential term cannot be ignored if the specimen becomes thick. Therefore, Eq. (3) may be correct only for a range of thicknesses that is neither too small nor too large. As shown in Fig. 3, R decreases with increase in thickness in bulk MgO. Closer examination shows that there is a relatively ‘‘flat’’ range for t/l between approximately 0.5 and 1.0, and its value is between 0.075 and 0.080. For a specimen thinner than 0.5l, the data are scattered and have significantly larger values, indicating the strong effect of surface excitations on the bulk plasmon. On the other hand, for specimens thicker than 1.0l, R decreases rapidly with increase in thickness, indicating a violation of thin specimen requirement in Eq. (2). In contrast to the bulk, nanoporous MgO has a relatively flat R, although the overall trend is similar to that in the bulk. The data are also scattered in the specimen thinner than 0.5l, and slightly decreased in the specimen thicker than 1.0l. In between, R has an approximately constant value of about 0.035. The different values of R indicate the different values of MFP in nanoporous and bulk materials, since both materials have the same sL and rMgO, respectively. In other words, MFP of nanoporous MgO is about half of that in bulk MgO, i.e. lporous(MgO)E l(MgO)/2. Therefore, the measured thickness in nanoporous MgO is only about half of that in bulk MgO at the same relative thickness, as measured by the log-ratio method.
4. Conclusion Caution should be exercised when applying the log-ratio method to the measurement of thickness in porous materials. If the nanopores are heterogeneously distributed in the matrix, the strength of excitations at interfaces between pores and the matrix must be examined in order to estimate the error introduced in a thickness measurement. On the other hand, if the nanopores are homogeneously distributed in the matrix, the log-ratio method is in principle still applicable, but MFP in the nanoporous materials will be different from that in the parent bulk material. The approach introduced here in nanoporous MgO can also be extended to other systems. However, the conversion factor between physical
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thickness and apparent thickness is sample dependent. The value given in this work applies only for the particular sample of MgO present.
Acknowledgements This work is supported by NSF Award DMR0603993. The use of facilities within the Center for Solid State Science at ASU is also acknowledged. References [1] P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.J. Whelan, Electron Microscopy of Thin Crystals, 2nd edition, Krieger, New York, 1977. [2] G.W. Lorimer, G. Cliff, J.N. Clark, Developments in Electron Microscopy and Analysis, EMAG 75, in: D.L. Misell (Ed.), Institute of Physics, Bristol1976, p. 153. [3] Z. Pan, C.K.L. Davies, R.N. Stevens, J. Mater. Sci. 29 (1994) 1920.
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