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Valence loss spectra from SiO2 polymorphs of different density
Nuclear Instruments and Methods in Physics Research B 96 (1995) 569-574
NOMB
Boom Intaraotlons with Materials A Atoms
ELSEVIER
Valence loss spectra from SiO, polymorphs of different density D.W. McComb
I, A. Howie *
Catlendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK
Abstract Dielectric response functions for stishovite, coesite, a-quartz and silacalite, obtained from electron energy loss spectra are studied and compared. The data for stishovite conforms well to simple dielectric theory for an insulator. However the behaviour of the other materials, including the relative insensitivity of the plasmon peak to density, is less well explained either on this basis or using more elaborate models based on Maxwell Garnett or Clausius-Mossotti effective medium theory. Possible improvements to effective medium theory are discussed.
1. Introduction
across a defect, such as a semiconductor interface separately resolvable in the elastically scattered signal. For further references on this approach to valence excitation and an illustration of its potential power in the case of the Si-SiO, interface, see Walls and Howie [6]. Via the concept of an effective medium, this dielectric excitation theory also enables us to address more complex, three-dimensional situations where, for example, defects are present and are not resolvable in the electron microscope, either because they are too small or because their images overlap one another too severely. Four such examples which have been studied are the free electron alloys mentioned above [2], the colloidal dispersion of small Al particles formed by electron beam irradiation damage in an AlF, matrix [7] the presence of small voids in NiO/Al,O, mixed oxide catalysts [B] and the case of zeolites which we discuss in more detail here. The usefulness of valence loss spectroscopy for chemical characterisation of zeolites was demonstrated by McComb and Howie [9] who showed that the process of dealumination of mordenite could be followed much more easily by the consequent shift in the plasmon energy than by any core loss spectroscopy technique. On a purely empirical basis, using suitable calibration samples of known and uniform composition, it is thus possible to monitor spatial variations in the dealumination process. In addition, the valence electron density n (or more accurately the density of the electrons participating in the measured response) can in principle be determined from the sum rule
The simple dependence of the collective excitation energy of the free electron gas on the square root of the electron density is perhaps demonstrated most dramatically by the surface plasmon [I] if we somewhat crudely assume that it samples half of the bulk density. At a much milder level, this density dependence was exploited almost thirty years ago by electron microscopists [2] to measure local variations in the composition of Al + Mg and similar alloys using shifts in the bulk plasmon loss. Since that time, the development of scanning transmission electron microscopy with improved facilities for spatially localised electron energy loss spectroscopy has meant that most attention has been devoted to the relatively weaker, but more easily interpretable energy loss features associated with core excitations. Using this approach, it is now possible to map (in two-dimensional projection) chemical composition, with supplementary information about empty state configurations, on a scale of 0.2 nm and in exact correlation with well defined defects such as interfaces or small precipitates clearly visible in the electron microscope image (for a recent summary see Ref. [3]). Spatially localised valence loss spectroscopy has not been totally up-staged by these developments. Fermi’s theory of dielectric excitation, originally based on a classical charged particle projectile travelling in a homogeneous medium, has been adapted largely by Ritchie [4,5] to describe the inelastic scattering of a coherent electron wave, focussed as in the STEM probe and scanning with an impact parameter controllable on the 0.2 nm level
* Corresponding author. Tel. +44 223 337334, fax +44 223 63263. ’ Institute for Materials Research, McMaster University, 1280 Main Street West, Hamilton Ontario L8.5 4M1, Canada.
/0
WC OE?( w) do = nn e2/2E,m,
w, -+ M.
(1)
However the sum rule is often rather slowly convergent so McComb and Howie [9] fitted their measured dielectric data to a standard expression
0168-583X/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00237-5
t.(w)-1=
-EjAiF(w,
.n,, r,),
IV. ELECTRONIC
(2)
RESPONSE
570
D. W. McComb, A. Howie / Nucl. Instr. and Meth. in Phys. Res. B 96 (I 995) 569-574
where F (o, 0, r)=l/{w(w+ir)-fl’)is theclassical oscillator response function. The density of the electrons participating in the measured dielectric response is then directly related tci C,A,, the total oscillator strength. On this basis it was possible to determine the change in electron density as a result of dealumination and thus to make useful inferences about the electronic and atomic structure of the defects in dealuminated zeolite. Used in this context, Eq. (2) is evidently no more than a convenient extrapolation device, implicitly containing the high frequency part of the response needed for full evaluation of the sum rule but not necessarily included directly in the measured data. It would thus perhaps have made little difference to the changes in electron density n which were found had the well known alternative expression, based on Clausius-Mossotti theory, been used instead of Eq. (21, since both forms should be consistent with the sum rule and they become identical at high frequencies. 3
= -CjA;F(
o, a;,
r;).
The effect which electron density changes have on the complete dielectric response function E(W), rather than simply on the sum rule, is however clearly quite different in Eqs. (2) and (3). For more than a century there has been intermittent discussion [lo-131 about the appropriate circumstances under which each should be used. Empirically, Eq. (2) certainly gives the correct behaviour for free electron metals but Eq. (3) appears to be more appropriate in situations where the polarisable units are more clearly separated. Even in aluminum however, (to which Eq. (2) undoubtedly applies) it may be relevant to note that the root mean square displacement of the electrons in the zero point plasmon modes is only about 20% of the interatomic distance. The observed value of the plasmon energy ho, is a particular quantity which we might hope to estimate with a simple dielectric model whereby the right hand side of Eq. (2) or (3) is reduced to a single term, setting h.0 equal to the band gap and taking A = (ne*/q,m). In the case of small damping, where the plasmon frequency wP can be defined by E( w,,) = 0, we find o.$= cr(,
e*/&,m)
+ a*,
(4)
with cx = 1 for Eq. (2) and LY= 2/3 for Eq. (3). Neither of these possibilities seems very successful in accounting for the fact that in a typical zeolite the plasmon energy is about 23 eV [9] which is not dramatically different from the value of about 24 eV found in quartz 1141 where the density can be as much as a factor of 1.4 greater and the band gap of 9 eV is very similar. It thus seems relevant to measure the dielectric response functions of a number of silica polymorphs of different density and to compare them more carefully in the light of Eqs. (2) and (3) and other theories.
2. Experiment The four SiO, polymorphs used are listed in Table 1. Stishovite is the most dense of the silica polymorphs, and has the rutile structure. In this material, silicon atoms are found in octahedral coordination with oxygen, in contrast to all other known forms of silica where silicon is found in tetrahedral coordination. Coesite is also a high density silica, but has a structure based on four-membered rings of SiO, tetrahedra while a-quartz, the most commonly found silica polymorph has a compact structure based on sixmembered rings of SiO, tetrahedra. The least dense polymorph studied was silicalite, which is a pure silica form of the molecular sieve zeolite ZSMJ widely used in hydrocarbon catalysis. It has a well-characterised structure, based again on SiO, tetrahedra but including a three-dimensional system of intersecting channels. Samples of each material were crushed and ground with a mortar and pestle to a fine powder. The powder was suspended in ethanol, then a droplet of the suspension was placed on a copper grid covered with a holey carbon film, and allowed to dry. Valence loss spectroscopy was carried out using a VG HB501 dedicated scanning transmission electron microscope (STEM) with a CCD-based parallel recording system. The spectra were acquired in image mode, using an effective collection angle of 8.3 mrad with an energy dispersion less than 0.2 eV per channel. This resulted in an energy-resolution, measured by the FWHM of the zero-loss peak, of about 0.5 eV. Dark current spectra were recorded immediately after each acquisition and were subtracted from the experimental spectra. The spectra were then deconvoluted to remove the contributions of multiple inelastic scattering. The zero loss peak shape was obtained from a spectrum, covering the same energy range as the experimental data, acquired while the electron probe was positioned over a hole in the carbon support film. After appropriate scaling, the zero loss spectrum was subtracted from the experimental data to remove the effect of the tail of this peak on the experimental spectrum. Kramers-Kronig analysis of the data was then performed to determine the real and imaginary part of the dielectric function for each sample. For data normalisation the squares of the refractive indices given in Table 1 were used. The complex dielectric functions thus obtained were employed in all subsequent calculations.
Table 1 Coordination number
Specific gravity
Mean refractive
Stishovite Coesite a-quartz
6 4 4
4.30 2.93 2.66
1.81 1.60 1.55
Silicalite
4
1.79
1.39
index
D. W. McComb, A. Howie /Nucl.
571
Instr. and Meth. in Phys. Res. B 96 (1995) 569-574
3. Results
12
The bulk energy loss functions Im[ - I/E(W)], calculated from the experimental spectra are shown in Fig. 1 for each silica polymorph investigated. While the spectra from coesite, o-quartz and silicalite are all remarkably similar, it is clear that the spectrum of stishovite differs considerably. This is also apparent in the plots of the real and imaginary parts of the dielectric function (Fig. 2). It is interesting to note that the dielectric function data of Fig. 2 show differences between coesite, a-quartz and silicalite more clearly than the loss spectra of Fig. 1. The number of electrons per SiO, unit contributing to the response has been investigated by evaluating the sum rule integral of Eq. (1) as a function of its upper cut-off limit E, = r?wc. From the results, shown in Fig. 3, it can be seen that the convergence of the curves towards the expected value of 16 is rather slow. To the extent that the different curves agree, the electron density estimated from the sum rule with any convenient cut-off energy, such as 50 eV, will be a monotonic function of its true value or of the material density. Clearly however the curves for stishovite and for silicalite agree quite well with one another but not with those for the other two materials. Coesite and a-quartz also agree with one another but distinctly less well. Since the real parts of the dielectric functions generally do not pass through zero (see Fig. 2a), the interpretation of the main bulk loss is less simple than was assumed in the introduction above. Nevertheless the prominent peak near 22 eV in all cases is evidently related to a nearby minimum in the real part of E which occurs at a point where the imaginary part is also small. This peak can therefore clearly be identified with a collective or plasmon loss rather than with a single electron loss which would be associated with a maximum in the imaginary part of E. Table 2 gives the experimental values Eexp for these losses in the different cases, compared with the values expected on the basis of Eq. (4) firstly for a free electron material (a = 1; 0 = 0) and then for an insulator with a = 1 and a = 2/3 together with values of fi R corresponding to the
t
(4
Fig. 2. Measured real (a) and imaginary (b) parts of the dielectric functions for stishovite, coesite, o-quartz and silicalite.
Fig. 3. Sum rule convergance plots, showing number of electrons per SiO, unit as a function of cut-off energy fro, for stishovite, coesite, o-quartz and silicalite.
measured band gap energies Eg given in the table. The simple insulator model with a = 1 gives excellent agreement with the measured loss for stishovite but none of
Table 2 Values (in eV) of the band gap energy Es and the experimental plasmon energy EeXp compared with the free electron plasmon energy Ehee and the insulator plasmon energies E, and E,,, for (Y= 1 and 2/3 resnectivelv
Fig. 1. Experimental loss functions Im( - l/cc coesite, o-quartz and silicalite.
0)) for stishovite,
Stishovite Coesite o-quartz Silicalite
5.0 9.3 9.2 8.8
31.2 24.0 23.6 23.2
30.8 25.4 24.2 20.0
31.2 27.0 25.9 21.9
IV. ELECTRONIC
25.7 22.8 21.8 18.6
RESPONSE
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572
these simple models seem to work for the three other materials. We therefore examine these other cases and the relation between them with the aid of different variants of effective medium theory.
4. Effective medium theories The best known form of effective theory is due to Maxwell Garnett [15] and gives the following expression for the effective complex dielectric function of a medium consisting of two components A and B where ideally A is supposed to be present as a low volume fraction f of small particles in the host material B. Here medium A will in all cases eventually be taken to be vacuum. %rr(W) - c*(eJ)
%( 0) - &a( eJ>
&&( 0) + 2Eu( eJ) =L*(
&I) + 2&u( 0) ’
(5)
or 1 _=-Gff
l-f
1
3f 1+ 2f En + 1 + 2f (1+
3 2f)&*
+ 2(1 -f)&,
’ (6)
where the alternative form of Eq. (6) has been pointed out [7] as more transparent in displaying the characteristic energy loss excitation functions. If we interchange the two materials A and B, but still retain f for the volume fraction of A, we obtain a second, nonequivalent form of effective medium theory
%ff(w> - d 0) = (1 -f) %I(~) -%(w) %fr(W)+ 24~) %( WI+ %4(WI’
0
Fig. 4. Comparison between the experimental loss function for coesite and those calculated on the basis of data for stishovite using the three effective medium models (a), (b) and (c).
the three effective medium models just described, putting cA = 1 and choosing appropriate values of f. In the various figures, model (a) is the Maxwell Garnett model based on Eqs. (5) and (6); (b) is the Clausius-Mossotti model based on Eqs. (7) and (8); (c) is based on Eq. (9). The results, illustrated for the single case of coesite in Fig. 4, were all very unsatisfactory. It seems likely that the large discrepancy between the theories and experiment is primarily due to the nature of the stishovite whose experimental loss function is significantly different from those of the other three materials, with a peak close to the value predicted by simple theory (see Table 2). Unlike the other
(7)
or 1 _=-%ff
f 3-2f
1 + -3(1-f) &*
3 - 2f
3 (3 - 2f)EB + 2fE*. (8)
When f is small, it is usually assumed that Eq. (5) is superior to Eq. (7). In the case where Ed = 1 however and the composite material may be regarded as a less dense version of material B formed by mixing in some small empty spaces, Eq. (7) becomes identical with the Clausius-Mossotti theory underlying Eq. (3). A third version of effective medium theory, proposed by Howie and Walsh [7] in order to bring the Maxwell Gamett theory of Eq. (6) into better agreement with their experimental observations, reduces, in the case of very small particles of material A, to the simple form 1 l-f _=-_++ 1+2f %ff
1
3f
En
1+2f
3 EAf2EB’
(9)
Taking stishovite, the most dense of the four materials as the reference material B, we first attempted to reproduce the loss functions of each of the other three materials using
Fig. 5. Loss functions for (a) a-quartz and (b) silicalite calculated on the basis of data for coesite using the effective medium models (a), (b) and (c) and compared with the experimental loss functions.
D. W. McComb, A. Howie/Nucl.
Instr. and Meth. in Phys. Res. B 96 (1995) 569-574
Fig. 6. The real (a) and imaginary (b) parts of the dielectric functions of silicalite computed on the basis of coesite using the three effective medium models and compared with the measured data.
tetrahedrally coordinated materials, each Si atom in stishovite is octahedrally coordinated to six oxygen atoms. With coesite as the starting material, the modelling process was therefore repeated and the results for u-quartz and silicalite are compared with the experimental loss functions in Figs. 5a and 5b respectively. For a-quartz, models (b) and (c) seem to fit the loss peak position fairly well and better than model (a) but in all cases are lower in magnitude than the observed peak. This discrepancy could conceivably arise from some error in the normalisation of the a-quartz data. In the case of silicalite, with a much greater density difference from coesite, all three theories give quite different results and, although none of them agree particularly well with the experimental loss function, the Clausius-Mossotti model (b) is the most successful of the three. The relative superiority of the Clausius-Mossotti model for silicalite is confirmed in more detail in Figs. 6a and 6b where the real and imaginary parts of the dielectric functions are shown.
5. Discussion It is clear that stishovite is the only one of the four materials studied whose dielectric behaviour, and in particular the position of the main loss peak, seems to be well
573
described by the simple insulator model. In coesite, aquartz and silicalite the position of the main loss peak still varies monotonically with electron density but much less strongly than any of the simple theories would predict. Effective medium theories, including even the ClausiusMossotti theory which is often rather useful in describing the dependence of dielectric response on density, are only moderately successful for these materials. Of the three theories examined, the Clausius-Mossotti theory is the most successful in reproducing the remarkable insensitivity of the position of the loss peak as starting from coesite, the electron density is progressively reduced to the value for a-quartz and then still further to the value for silicalite. It is perhaps surprising that the Clausius-Mossotti theory should perform better than the more orthodox form of Maxwell Garnett theory of Eqs. (5) and (6). Aspnes [16] has pointed out however that for particular microstructures, the minority dielectric component may still effectively act as the host material. A further reason for the comparative success of the Clausius-Mossotti theory in our case is readily apparent from a comparison of Eqs. (6) (8) and (9) which exhibit implicitly the various excitation functions involved. To produce a more or less constant loss peak position, the denominators of these functions should remain as close as possible to .cn (the response function of the host coesite), mixing in as little as possible of E~ = 1. The second loss function in Eq. (8) clearly achieves this at small values of f, the first loss function with zero imaginary part, contributes nothing. This idea can be related to a more general theory of the effective medium based on the poles and zeros of a complex function and due to Bergman [17] and Milton 1181. In terms of this theory the expression for &efl can be written in the form
t=:
cXjc*+(YL.)c
I
B’
where CC,=l.
The constants Cj and CY~,where 0 I cy~I 1, are determined by the geometry of the medium. Clearly the bulk response from either medium A or B, if present, arises from terms with crj = 1 and cyi = 0 respectively. The form of effective medium theory proposed by Howie and Walsh [7] is a special case of Eq. (10). A number of points can be made, if Eq. (10) is to apply successfully to a zeolite. In the first place, since the two regions of the zeolite, corresponding to vacuum A and silica B are continuous, .cen must become infinite (signifying a conducting medium) if either E,_, or cB becomes infinite. This means that terms containing LYE = 0 or crj = 1 do not occur in Eq. (lo), at least at w = 0. On the other hand, from the results obtained here, it can be concluded
IV. ELECTRONIC
RESPONSE
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D. W. McComb, A. Howie/ Nucl. Instr. and Meth. in Phys. Res. 3 96 (1995) 569-574
that terms when oj is close to zero must make a large contribution to Eq. (10). Since the zeolite is perhaps not too far from being a medium with a symmetrical geometry where the two media A and B can be interchanged without affecting ~~n, we may deduce that terms with ‘Y~near 1 should also contribute with equal amplitude Cj. After rationalisation however, these latter terms will make a very small contribution to the loss function Im( - l/&,rrl in the case when eA = 1. Our conclusions must remain largely conjectures at present. More detailed calculations for the zeolite structure would be required to justify the use of Eq. (101 and to determine the values of the constants Cj and oj involved. Perhaps the numerical approach recently opened up by Pendry [19] may offer a route forward.
Acknowledgements D. McComb thanks the Master and Fellows of Corpus Christi College, Cambridge for a Research Fellowship and the Royal Commission for the 1851 Exhibition for financial support.
References [l] R.H. Ritchie, Phys. Rev. 106 (1956) 874. [2] A.J.F. Metherell, Adv. Opt. Electron Microsc., eds V.E. Cosslett and R. Barer, vol. 4 (1971) 171. [3] L.M. Brown, Nature 366 (1993) 721. [4] R.H. Ritchie and A. Howie, Philos. Msg. A 58 (1988) 753. [5] R. Garcia-Molina, A. Gras-Marti, A. Howie and R.H. Ritchie, J. Phys. C 18 (1985) 5335. [6] M.G. Walls and A. Howie, Ultramicrosc. 28 (1989) 40. [7] A. Howie and C.A. Walsh, Microsc. Microanal. Microstruct. 2 (1991) 171. [8] F.J. Cadete Santos Aires, A. Howie and C.A. Walsh, J. Solid State Chem. 106 (1993) 48. [9] D.W. McComb and A. Howie, Ultramicrosc. 34 (1990) 84. [lo] J.W.S. Rayleigh, Philos. Mag. 34 (1892) 481. [ll] C.G. Darwin, Proc. R. Sot. 146 (1934) 17. [12] U. Fano, Phys. Rev. 118 (1960) 451. [13] U. Fano, Rev. Mod. Phys. 64 (1992) 313. [14] A. Howie and R.H. Milne, Ultramicrosc. 18 (1985) 427. [ 151 J.C. Maxwell .Garnett, Philos. Trans. 203 (1904) 385. [16] D.E. Aspnes, Phys. Rev. B 33 (1986) 677. [17] D.J. Bergman, Phys. Rep. 43 (1978) 377. [18] G.W. Milton, J. Appl. Phys. 52 (1981) 5286. [19] J.B. Pendry and L. Martin Moreno, Phys. Rev. B 50 (1994) 5062.
NOMB
Boom Intaraotlons with Materials A Atoms
ELSEVIER
Valence loss spectra from SiO, polymorphs of different density D.W. McComb
I, A. Howie *
Catlendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK
Abstract Dielectric response functions for stishovite, coesite, a-quartz and silacalite, obtained from electron energy loss spectra are studied and compared. The data for stishovite conforms well to simple dielectric theory for an insulator. However the behaviour of the other materials, including the relative insensitivity of the plasmon peak to density, is less well explained either on this basis or using more elaborate models based on Maxwell Garnett or Clausius-Mossotti effective medium theory. Possible improvements to effective medium theory are discussed.
1. Introduction
across a defect, such as a semiconductor interface separately resolvable in the elastically scattered signal. For further references on this approach to valence excitation and an illustration of its potential power in the case of the Si-SiO, interface, see Walls and Howie [6]. Via the concept of an effective medium, this dielectric excitation theory also enables us to address more complex, three-dimensional situations where, for example, defects are present and are not resolvable in the electron microscope, either because they are too small or because their images overlap one another too severely. Four such examples which have been studied are the free electron alloys mentioned above [2], the colloidal dispersion of small Al particles formed by electron beam irradiation damage in an AlF, matrix [7] the presence of small voids in NiO/Al,O, mixed oxide catalysts [B] and the case of zeolites which we discuss in more detail here. The usefulness of valence loss spectroscopy for chemical characterisation of zeolites was demonstrated by McComb and Howie [9] who showed that the process of dealumination of mordenite could be followed much more easily by the consequent shift in the plasmon energy than by any core loss spectroscopy technique. On a purely empirical basis, using suitable calibration samples of known and uniform composition, it is thus possible to monitor spatial variations in the dealumination process. In addition, the valence electron density n (or more accurately the density of the electrons participating in the measured response) can in principle be determined from the sum rule
The simple dependence of the collective excitation energy of the free electron gas on the square root of the electron density is perhaps demonstrated most dramatically by the surface plasmon [I] if we somewhat crudely assume that it samples half of the bulk density. At a much milder level, this density dependence was exploited almost thirty years ago by electron microscopists [2] to measure local variations in the composition of Al + Mg and similar alloys using shifts in the bulk plasmon loss. Since that time, the development of scanning transmission electron microscopy with improved facilities for spatially localised electron energy loss spectroscopy has meant that most attention has been devoted to the relatively weaker, but more easily interpretable energy loss features associated with core excitations. Using this approach, it is now possible to map (in two-dimensional projection) chemical composition, with supplementary information about empty state configurations, on a scale of 0.2 nm and in exact correlation with well defined defects such as interfaces or small precipitates clearly visible in the electron microscope image (for a recent summary see Ref. [3]). Spatially localised valence loss spectroscopy has not been totally up-staged by these developments. Fermi’s theory of dielectric excitation, originally based on a classical charged particle projectile travelling in a homogeneous medium, has been adapted largely by Ritchie [4,5] to describe the inelastic scattering of a coherent electron wave, focussed as in the STEM probe and scanning with an impact parameter controllable on the 0.2 nm level
* Corresponding author. Tel. +44 223 337334, fax +44 223 63263. ’ Institute for Materials Research, McMaster University, 1280 Main Street West, Hamilton Ontario L8.5 4M1, Canada.
/0
WC OE?( w) do = nn e2/2E,m,
w, -+ M.
(1)
However the sum rule is often rather slowly convergent so McComb and Howie [9] fitted their measured dielectric data to a standard expression
0168-583X/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00237-5
t.(w)-1=
-EjAiF(w,
.n,, r,),
IV. ELECTRONIC
(2)
RESPONSE
570
D. W. McComb, A. Howie / Nucl. Instr. and Meth. in Phys. Res. B 96 (I 995) 569-574
where F (o, 0, r)=l/{w(w+ir)-fl’)is theclassical oscillator response function. The density of the electrons participating in the measured dielectric response is then directly related tci C,A,, the total oscillator strength. On this basis it was possible to determine the change in electron density as a result of dealumination and thus to make useful inferences about the electronic and atomic structure of the defects in dealuminated zeolite. Used in this context, Eq. (2) is evidently no more than a convenient extrapolation device, implicitly containing the high frequency part of the response needed for full evaluation of the sum rule but not necessarily included directly in the measured data. It would thus perhaps have made little difference to the changes in electron density n which were found had the well known alternative expression, based on Clausius-Mossotti theory, been used instead of Eq. (21, since both forms should be consistent with the sum rule and they become identical at high frequencies. 3
= -CjA;F(
o, a;,
r;).
The effect which electron density changes have on the complete dielectric response function E(W), rather than simply on the sum rule, is however clearly quite different in Eqs. (2) and (3). For more than a century there has been intermittent discussion [lo-131 about the appropriate circumstances under which each should be used. Empirically, Eq. (2) certainly gives the correct behaviour for free electron metals but Eq. (3) appears to be more appropriate in situations where the polarisable units are more clearly separated. Even in aluminum however, (to which Eq. (2) undoubtedly applies) it may be relevant to note that the root mean square displacement of the electrons in the zero point plasmon modes is only about 20% of the interatomic distance. The observed value of the plasmon energy ho, is a particular quantity which we might hope to estimate with a simple dielectric model whereby the right hand side of Eq. (2) or (3) is reduced to a single term, setting h.0 equal to the band gap and taking A = (ne*/q,m). In the case of small damping, where the plasmon frequency wP can be defined by E( w,,) = 0, we find o.$= cr(,
e*/&,m)
+ a*,
(4)
with cx = 1 for Eq. (2) and LY= 2/3 for Eq. (3). Neither of these possibilities seems very successful in accounting for the fact that in a typical zeolite the plasmon energy is about 23 eV [9] which is not dramatically different from the value of about 24 eV found in quartz 1141 where the density can be as much as a factor of 1.4 greater and the band gap of 9 eV is very similar. It thus seems relevant to measure the dielectric response functions of a number of silica polymorphs of different density and to compare them more carefully in the light of Eqs. (2) and (3) and other theories.
2. Experiment The four SiO, polymorphs used are listed in Table 1. Stishovite is the most dense of the silica polymorphs, and has the rutile structure. In this material, silicon atoms are found in octahedral coordination with oxygen, in contrast to all other known forms of silica where silicon is found in tetrahedral coordination. Coesite is also a high density silica, but has a structure based on four-membered rings of SiO, tetrahedra while a-quartz, the most commonly found silica polymorph has a compact structure based on sixmembered rings of SiO, tetrahedra. The least dense polymorph studied was silicalite, which is a pure silica form of the molecular sieve zeolite ZSMJ widely used in hydrocarbon catalysis. It has a well-characterised structure, based again on SiO, tetrahedra but including a three-dimensional system of intersecting channels. Samples of each material were crushed and ground with a mortar and pestle to a fine powder. The powder was suspended in ethanol, then a droplet of the suspension was placed on a copper grid covered with a holey carbon film, and allowed to dry. Valence loss spectroscopy was carried out using a VG HB501 dedicated scanning transmission electron microscope (STEM) with a CCD-based parallel recording system. The spectra were acquired in image mode, using an effective collection angle of 8.3 mrad with an energy dispersion less than 0.2 eV per channel. This resulted in an energy-resolution, measured by the FWHM of the zero-loss peak, of about 0.5 eV. Dark current spectra were recorded immediately after each acquisition and were subtracted from the experimental spectra. The spectra were then deconvoluted to remove the contributions of multiple inelastic scattering. The zero loss peak shape was obtained from a spectrum, covering the same energy range as the experimental data, acquired while the electron probe was positioned over a hole in the carbon support film. After appropriate scaling, the zero loss spectrum was subtracted from the experimental data to remove the effect of the tail of this peak on the experimental spectrum. Kramers-Kronig analysis of the data was then performed to determine the real and imaginary part of the dielectric function for each sample. For data normalisation the squares of the refractive indices given in Table 1 were used. The complex dielectric functions thus obtained were employed in all subsequent calculations.
Table 1 Coordination number
Specific gravity
Mean refractive
Stishovite Coesite a-quartz
6 4 4
4.30 2.93 2.66
1.81 1.60 1.55
Silicalite
4
1.79
1.39
index
D. W. McComb, A. Howie /Nucl.
571
Instr. and Meth. in Phys. Res. B 96 (1995) 569-574
3. Results
12
The bulk energy loss functions Im[ - I/E(W)], calculated from the experimental spectra are shown in Fig. 1 for each silica polymorph investigated. While the spectra from coesite, o-quartz and silicalite are all remarkably similar, it is clear that the spectrum of stishovite differs considerably. This is also apparent in the plots of the real and imaginary parts of the dielectric function (Fig. 2). It is interesting to note that the dielectric function data of Fig. 2 show differences between coesite, a-quartz and silicalite more clearly than the loss spectra of Fig. 1. The number of electrons per SiO, unit contributing to the response has been investigated by evaluating the sum rule integral of Eq. (1) as a function of its upper cut-off limit E, = r?wc. From the results, shown in Fig. 3, it can be seen that the convergence of the curves towards the expected value of 16 is rather slow. To the extent that the different curves agree, the electron density estimated from the sum rule with any convenient cut-off energy, such as 50 eV, will be a monotonic function of its true value or of the material density. Clearly however the curves for stishovite and for silicalite agree quite well with one another but not with those for the other two materials. Coesite and a-quartz also agree with one another but distinctly less well. Since the real parts of the dielectric functions generally do not pass through zero (see Fig. 2a), the interpretation of the main bulk loss is less simple than was assumed in the introduction above. Nevertheless the prominent peak near 22 eV in all cases is evidently related to a nearby minimum in the real part of E which occurs at a point where the imaginary part is also small. This peak can therefore clearly be identified with a collective or plasmon loss rather than with a single electron loss which would be associated with a maximum in the imaginary part of E. Table 2 gives the experimental values Eexp for these losses in the different cases, compared with the values expected on the basis of Eq. (4) firstly for a free electron material (a = 1; 0 = 0) and then for an insulator with a = 1 and a = 2/3 together with values of fi R corresponding to the
t
(4
Fig. 2. Measured real (a) and imaginary (b) parts of the dielectric functions for stishovite, coesite, o-quartz and silicalite.
Fig. 3. Sum rule convergance plots, showing number of electrons per SiO, unit as a function of cut-off energy fro, for stishovite, coesite, o-quartz and silicalite.
measured band gap energies Eg given in the table. The simple insulator model with a = 1 gives excellent agreement with the measured loss for stishovite but none of
Table 2 Values (in eV) of the band gap energy Es and the experimental plasmon energy EeXp compared with the free electron plasmon energy Ehee and the insulator plasmon energies E, and E,,, for (Y= 1 and 2/3 resnectivelv
Fig. 1. Experimental loss functions Im( - l/cc coesite, o-quartz and silicalite.
0)) for stishovite,
Stishovite Coesite o-quartz Silicalite
5.0 9.3 9.2 8.8
31.2 24.0 23.6 23.2
30.8 25.4 24.2 20.0
31.2 27.0 25.9 21.9
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25.7 22.8 21.8 18.6
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572
these simple models seem to work for the three other materials. We therefore examine these other cases and the relation between them with the aid of different variants of effective medium theory.
4. Effective medium theories The best known form of effective theory is due to Maxwell Garnett [15] and gives the following expression for the effective complex dielectric function of a medium consisting of two components A and B where ideally A is supposed to be present as a low volume fraction f of small particles in the host material B. Here medium A will in all cases eventually be taken to be vacuum. %rr(W) - c*(eJ)
%( 0) - &a( eJ>
&&( 0) + 2Eu( eJ) =L*(
&I) + 2&u( 0) ’
(5)
or 1 _=-Gff
l-f
1
3f 1+ 2f En + 1 + 2f (1+
3 2f)&*
+ 2(1 -f)&,
’ (6)
where the alternative form of Eq. (6) has been pointed out [7] as more transparent in displaying the characteristic energy loss excitation functions. If we interchange the two materials A and B, but still retain f for the volume fraction of A, we obtain a second, nonequivalent form of effective medium theory
%ff(w> - d 0) = (1 -f) %I(~) -%(w) %fr(W)+ 24~) %( WI+ %4(WI’
0
Fig. 4. Comparison between the experimental loss function for coesite and those calculated on the basis of data for stishovite using the three effective medium models (a), (b) and (c).
the three effective medium models just described, putting cA = 1 and choosing appropriate values of f. In the various figures, model (a) is the Maxwell Garnett model based on Eqs. (5) and (6); (b) is the Clausius-Mossotti model based on Eqs. (7) and (8); (c) is based on Eq. (9). The results, illustrated for the single case of coesite in Fig. 4, were all very unsatisfactory. It seems likely that the large discrepancy between the theories and experiment is primarily due to the nature of the stishovite whose experimental loss function is significantly different from those of the other three materials, with a peak close to the value predicted by simple theory (see Table 2). Unlike the other
(7)
or 1 _=-%ff
f 3-2f
1 + -3(1-f) &*
3 - 2f
3 (3 - 2f)EB + 2fE*. (8)
When f is small, it is usually assumed that Eq. (5) is superior to Eq. (7). In the case where Ed = 1 however and the composite material may be regarded as a less dense version of material B formed by mixing in some small empty spaces, Eq. (7) becomes identical with the Clausius-Mossotti theory underlying Eq. (3). A third version of effective medium theory, proposed by Howie and Walsh [7] in order to bring the Maxwell Gamett theory of Eq. (6) into better agreement with their experimental observations, reduces, in the case of very small particles of material A, to the simple form 1 l-f _=-_++ 1+2f %ff
1
3f
En
1+2f
3 EAf2EB’
(9)
Taking stishovite, the most dense of the four materials as the reference material B, we first attempted to reproduce the loss functions of each of the other three materials using
Fig. 5. Loss functions for (a) a-quartz and (b) silicalite calculated on the basis of data for coesite using the effective medium models (a), (b) and (c) and compared with the experimental loss functions.
D. W. McComb, A. Howie/Nucl.
Instr. and Meth. in Phys. Res. B 96 (1995) 569-574
Fig. 6. The real (a) and imaginary (b) parts of the dielectric functions of silicalite computed on the basis of coesite using the three effective medium models and compared with the measured data.
tetrahedrally coordinated materials, each Si atom in stishovite is octahedrally coordinated to six oxygen atoms. With coesite as the starting material, the modelling process was therefore repeated and the results for u-quartz and silicalite are compared with the experimental loss functions in Figs. 5a and 5b respectively. For a-quartz, models (b) and (c) seem to fit the loss peak position fairly well and better than model (a) but in all cases are lower in magnitude than the observed peak. This discrepancy could conceivably arise from some error in the normalisation of the a-quartz data. In the case of silicalite, with a much greater density difference from coesite, all three theories give quite different results and, although none of them agree particularly well with the experimental loss function, the Clausius-Mossotti model (b) is the most successful of the three. The relative superiority of the Clausius-Mossotti model for silicalite is confirmed in more detail in Figs. 6a and 6b where the real and imaginary parts of the dielectric functions are shown.
5. Discussion It is clear that stishovite is the only one of the four materials studied whose dielectric behaviour, and in particular the position of the main loss peak, seems to be well
573
described by the simple insulator model. In coesite, aquartz and silicalite the position of the main loss peak still varies monotonically with electron density but much less strongly than any of the simple theories would predict. Effective medium theories, including even the ClausiusMossotti theory which is often rather useful in describing the dependence of dielectric response on density, are only moderately successful for these materials. Of the three theories examined, the Clausius-Mossotti theory is the most successful in reproducing the remarkable insensitivity of the position of the loss peak as starting from coesite, the electron density is progressively reduced to the value for a-quartz and then still further to the value for silicalite. It is perhaps surprising that the Clausius-Mossotti theory should perform better than the more orthodox form of Maxwell Garnett theory of Eqs. (5) and (6). Aspnes [16] has pointed out however that for particular microstructures, the minority dielectric component may still effectively act as the host material. A further reason for the comparative success of the Clausius-Mossotti theory in our case is readily apparent from a comparison of Eqs. (6) (8) and (9) which exhibit implicitly the various excitation functions involved. To produce a more or less constant loss peak position, the denominators of these functions should remain as close as possible to .cn (the response function of the host coesite), mixing in as little as possible of E~ = 1. The second loss function in Eq. (8) clearly achieves this at small values of f, the first loss function with zero imaginary part, contributes nothing. This idea can be related to a more general theory of the effective medium based on the poles and zeros of a complex function and due to Bergman [17] and Milton 1181. In terms of this theory the expression for &efl can be written in the form
t=:
cXjc*+(YL.)c
I
B’
where CC,=l.
The constants Cj and CY~,where 0 I cy~I 1, are determined by the geometry of the medium. Clearly the bulk response from either medium A or B, if present, arises from terms with crj = 1 and cyi = 0 respectively. The form of effective medium theory proposed by Howie and Walsh [7] is a special case of Eq. (10). A number of points can be made, if Eq. (10) is to apply successfully to a zeolite. In the first place, since the two regions of the zeolite, corresponding to vacuum A and silica B are continuous, .cen must become infinite (signifying a conducting medium) if either E,_, or cB becomes infinite. This means that terms containing LYE = 0 or crj = 1 do not occur in Eq. (lo), at least at w = 0. On the other hand, from the results obtained here, it can be concluded
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that terms when oj is close to zero must make a large contribution to Eq. (10). Since the zeolite is perhaps not too far from being a medium with a symmetrical geometry where the two media A and B can be interchanged without affecting ~~n, we may deduce that terms with ‘Y~near 1 should also contribute with equal amplitude Cj. After rationalisation however, these latter terms will make a very small contribution to the loss function Im( - l/&,rrl in the case when eA = 1. Our conclusions must remain largely conjectures at present. More detailed calculations for the zeolite structure would be required to justify the use of Eq. (101 and to determine the values of the constants Cj and oj involved. Perhaps the numerical approach recently opened up by Pendry [19] may offer a route forward.
Acknowledgements D. McComb thanks the Master and Fellows of Corpus Christi College, Cambridge for a Research Fellowship and the Royal Commission for the 1851 Exhibition for financial support.
References [l] R.H. Ritchie, Phys. Rev. 106 (1956) 874. [2] A.J.F. Metherell, Adv. Opt. Electron Microsc., eds V.E. Cosslett and R. Barer, vol. 4 (1971) 171. [3] L.M. Brown, Nature 366 (1993) 721. [4] R.H. Ritchie and A. Howie, Philos. Msg. A 58 (1988) 753. [5] R. Garcia-Molina, A. Gras-Marti, A. Howie and R.H. Ritchie, J. Phys. C 18 (1985) 5335. [6] M.G. Walls and A. Howie, Ultramicrosc. 28 (1989) 40. [7] A. Howie and C.A. Walsh, Microsc. Microanal. Microstruct. 2 (1991) 171. [8] F.J. Cadete Santos Aires, A. Howie and C.A. Walsh, J. Solid State Chem. 106 (1993) 48. [9] D.W. McComb and A. Howie, Ultramicrosc. 34 (1990) 84. [lo] J.W.S. Rayleigh, Philos. Mag. 34 (1892) 481. [ll] C.G. Darwin, Proc. R. Sot. 146 (1934) 17. [12] U. Fano, Phys. Rev. 118 (1960) 451. [13] U. Fano, Rev. Mod. Phys. 64 (1992) 313. [14] A. Howie and R.H. Milne, Ultramicrosc. 18 (1985) 427. [ 151 J.C. Maxwell .Garnett, Philos. Trans. 203 (1904) 385. [16] D.E. Aspnes, Phys. Rev. B 33 (1986) 677. [17] D.J. Bergman, Phys. Rep. 43 (1978) 377. [18] G.W. Milton, J. Appl. Phys. 52 (1981) 5286. [19] J.B. Pendry and L. Martin Moreno, Phys. Rev. B 50 (1994) 5062.