- Email: [email protected]
New effective medium models and their applications
Vacuum/volume
Pergamon
46/numbers &IO/pages 935 to 938/1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved
0042-207x/95
0042-207x(95)00075-3
New effective
medium
$9.50+.00
models and their applications
S Dligatch, M W Ng, G B Smith, A J Reuben, A V Radchik and A V Vagov, Depattment University of Technology, Sydney, PO Box 123, Broadway NSW2007, Australia
of Applied Physics,
Thin films and many surfaces often have complex structures which are conveniently analysed for overall optical response by effective medium models. Examples from new deposition techniques and new exact theory are presented that highlight effects that are not predictable by many of the widely used simple effective medium models. High order multipoles and associated multiple resonances play a key role and >alter spectral response and transport properties markedly. The new theories can discriminate the short range forces between neighbouring pairs and longer range forces in chains and arrays. Experimental data on composite films show how changes in component distribution can strongly effect spectral absorptance in ways that are not predictable by standard dipole theories.
Introduction Composites made by co-deposition of two species from different angles of incidence’ and by different techniques, such as sputtering, vacuum arc’ and thermal evaporation3 display different spatial distributions and sizes of the included component. Such systems can be technically important4 and also provide useful means of assessing the ability of various effective medium models to predict optical or other transport properties. Exact results5-9, involving quite simple expressions and any materials, are now available for the polarisabilities of pairs of cylinders and spheres covering all separations of centres from large values where the isolated particle result applies through to the totally merged situation where the single particle result again applies. Such results provide a far better basis for extended effective medium models than the standard theories such as MaxwellGarnett and Bruggeman, which are special limiting cases. The need to improve on simple models when clustering of small particles occurs has been known for some time”. A wide range of spectral response in the quasistatic limit is found as a function of the centre-to-centre gap. The touching and merging cases are especially interesting. Generally there is a significant red shift of absorption ability as multipoles come into play. One of the features that does not appear to be explained by conventional theories that we find in several anisotropic dilute composite structures is the variation of absorption strength as one moves to longer wavelength (1). While it usually falls as /z increases across the visible and near IR it can also rise over a broad band, as if the metal were percolating. While this behaviour is more likely as concentration increases it is surprising for it to occur well below the percolation threshold. A key motivation for this work has been our observation that metal volume fractions of a few percent can give quite strong absorption provided the deposition conditions favour a nonrandom distribution of the included metal component. Both experimental examples in this paper employ the same bi-oblique
deposition geometries that we know ensure an inhomogeneous spread of metal in the cermet (Ag/Al,O,) but use different deposition techniques (sputtering and evaporation). The difference in techniques is shown to give distinct microstructures which in turn cause noticeable differences in absorption spectra. We then introduce and demonstrate some of, the new theories that help explain both the enhancement of absorption that occurs with clustering, the sensitivity of these spectra to details of that structure, and the strong optical anisotropies that occur in these films.
Experimental
examples of thin film cluster composites
Columnar thin film composites deposited by oblique deposition of one or both components develop a complex array of microstructures depending on the details of the deposition geometries (especially the angle of incidence from the various sources to the substrate), the melting points and reactivities of deposited materials, and the energetics and degree of collimation of the incoming species. Absorbing composites are particularly interesting since the absorption spectra relate strongly to the microstructure. Using reactive deposition it is possible to produce cluster composites of metal particles in a matrix of the same metal’s oxide. A very different structure occurs with energetic arc deposition as compared to low energy evaporation. These differences are not only in the more obvious area of density but more importantly in the array structure the columns adopt and in the metal distribution which is non-random. A good example is oblique aluminium in A&O,‘. Sputtering is also of interest and in this report we compare evaporation and sputtering results in a system where it is easier to observe the different components by electron microscopy, namely silver in oxides such as A&O, or SiOZ1. Two high resolution scanning electron micrographs obtained from a field emission system are shown in Figures l(a) and (b). Figure 1 (a) shows a sputtered composite, thickness 1500 A, 935
S Dligafch et a/r New effective medium models and their applications
0.6 , 0.5
I- i:
0.4 8
I
I
I
----__
cls --_. -----____
*= :
0.3
‘\
\
0.2
L *-a_ -_
0.1
I< 0
I
1
-
: I
0.5
I
1 Wavelength
I
1.5 (pm)
I
2
Figure 2. Spectra of absorption coefficients for s and p polarised light at near normal incidence to the sputtered film of Figure l(a). Deposition geometry is in the inset and the plane of incidence for light is the deposition direction.
epared by bioolique aeposmon:
(a) sputtelGu,
produced by depositing silver at 30” to the substrate normal and simultaneous rf sputtering of A&O3 at 70” to the normal. The oxide columns shown are typically 25 nm across and lined up approximately diagonally from the bottom left hand corner. This direction is normal to the deposition plane. The columns are tilted at about 40” to the vertical, from edge micrographs. The silver particles are very fine and not obvious in this micrograph. They are clearly present from their optical signature (see below) which also indicates they are clustered or ordered more than in the evaporated case. Other work has shown in a similar system that very fine silver spheres align along each column’. In the evaporation case, thickness 950 A, of Figure l(b) the brighter silver particles are more obvious and are comparable in size to the oxide columns with diameters from 25 to 50 nm. This film has a higher metal density than the sputtered film. The columnar structure is less ordered in these lower energy techniques. Correlated with this appears to be a lower degree of ordering of the included metal particles. We found even more ordering in arc evaporation’ where the incident ions have very high energy.
light when the plane of incidence parallels the deposition direction. Experimental optical results, namely reflectance, transmittance and absorptance from 250 to 2000 nm, for the two films shown in Figures l(a) and (b) have been processed using thickness t, to extract the effective absorption coefficient CI(taking intensity falling off as exp( --47&/l). These spectral properties of 01are presented in Figures 2 and 3, respectively. The two sets of spectra for each film are for light polarised perpendicular (s) and parallel (p) to the plane of incidence which is the column or deposition direction. Near normal incidence results only are discussed here ; s-polarisation thus defined means the electric field is along the direction from bottom left to top right corners in Figure 1 (a). Strong absorption anisotropy is evident in both films but in the evaporated case (Figure 3) this shows up primarily via magnitudes of CI,and ccPwhile the general shapes of the curves are the same. The silver absorption resonance is quite broad relative to that for isolated particles. The sputtered sample in Figure 2 displays qualitative differences between CI,and tlP as well. Of note is the relative widths of the silver resonance for each polarisation and the double resonance for p-polarised light. The optical differences between sputtered and evaporated films are in accord with the microstructural differences in Figure 1 and the theories that follow. It is significant that both display a strong IR absorption tail with the s-case stronger than the p-case in each. The underlying cause is that columns and their associated
8
0.5 Microstructure and optical absorption The details of how the inclusions and the columns are arranged gives distinct optical signatures which show up when one compares the absorption spectra for different polarisation of incident 936
1 Wavelength
1.5
2
(pm)
Figure 3. Spectra of absorption coefficients for s and p polarised light at near normal incidence to the evaporated film of Figure l(b). Deposition geometry direction.
is in the inset and plane of incidence
for light is the deposition
S Dligarch et a/: New effective medium models and their applications silver particles are found on average to be closer in directions normal to the deposition planeZA. Features that the new models are needed for include the second peak in Figure 2 for 01~.This is due to multipole effects arising from close but separate particles. Touching or merging, however, merges all multipole resonances into a single broad continuous spectrum as found in the other curves. The first peak in clp, Figure 2, is also narrower than all others, which also follows from the lower degree of touching in the deposition direction. However there must be some touching or merging otherwise the absorption tail into the IR would not be so broad. We now turn to some of the theories that help explain and quantify these effects and highlight the inadequacy of dipole only models. New models and example results on cermet array In this section we will demonstrate new models with cylinders and spheresM that enables an understanding of some of the resonant features just displayed. Exact theories for the polarisability of pairs and arrays of cylinders for any centre-to-centre separation (including intersection) and with complex refractive indices have recently been formalised based on a new approach to the use of conformal transformations. These techniques lead more efficiently to solutions because the effects of the array are built into the transformed coordinate frame from the outset5.6. Previous techniques required additional boundary equations based on lattice sums. Separate cylinders give polarisability p which can be expressed for any array in terms of the spectral parameter
can remain near 400 nm when many multipoles (i.e. many nonzero wn) are present. Touching and intersecting particles give continuous, not discrete eigenvalue spectra. A red shift of absorption occurs due to interactions, which maximises in extent at the touching case. An example using equation (1) showing discrete resonances is in Figure 4 for silver particles (optical constant data from Palik”) in an oxide matrix. We compare an isolated pair with a pair in a chain and a lattice of the same pairs. The spacing between pairs is such that the overall metal fill factor is low in this example (0.3). This spectral range is of considerable interest in solar energy applications. We next look at chains using silver spheres. For spheres until recently only separate and intersecting pairs could in principle be solved exactly by transformation methods since the resulting frames allow solution of the Laplace equation by separation of variables. In practice the intersecting and touching sphere problem has onli recently been solved7z8 this way for arbitrary dielectric constants. The result which includes the touching limit is given in equation (2). For intersecting spherical particles the result for longitudinal polarisability is
na.
p1 = 8717.3 * A2Bj*‘)tan h - d;l 2a s0 where Bj’) is defined as
l+s V=l-_E -1 with e the complex dielectric constant ofeach unit radius cylinder. p, and J+ are for fields along and normal to a given pair axis and come from the discrete sum
PlJ =
nj?l &
3
v,
=
-r3sinhs2
1
Also
+ =
e-*’
sinhg
Id 2a
-ae,,
d s2
=
2a
-17%
-n
where for circular cross-sections weights are given by -497 w n =----a2v n
in the (x,~) plane, the spectral
and
n n
with
CIis an easily determined geometric boundary parameter for each array5,7. The complex variable t = x-iv, log F, determines the conformal mapping and C is a contour which can be one particle’s surface. Thus once F is determined the problem is essentially solved. Each n gives a pole at v = vn and hence a resonance and these show up as multiple peaks in absorption spectra with IZ= 1 the dipole. From equation (1) at large n we see that higher order poles give overlapping resonances centred near the spectral position found from Real(v) = 0, that is Real(s) = - 1. This is precisely the dipole resonance position for an isolated cylindrical particle and helps explain why the dominant peaks in our data
Wavelength (p) Figure 4. Predicted absorption spectra for unit radius silver cylinders embedded in an A&O, matrix. Results are shown for a pair, a chain and a lattice of the cylinders. The pair centre separation is 2.3 units, and the gap to the edge of the next pair is 3 units in the horizontal and vertical directions. 937
S Dligatch et a/: New effective medium models and their applications
23 =-
E-l &+2’
second peak in Figure 2 for p waves is near that predicted chain model for spheres.
a=JEE
in terms of centre separation d and 8, = sin-‘a/a. This spectrum is continuous with index L for each pole in the continuum. For separate spherical particles and chains we have developed a new technique’ based on the three-dimensional hypercomplex variable X(U) = xficos (u)u+ isin (u)z, that retains the advantages of the conformal approach and is in fact utilised by making direct analogies with two-dimensional transformations. It is however necessary to solve a matrix equation for potential but convergence is much more rapid than in any previous methods so that we are able to study quite dense chains exactly. These were in practice insoluble by established techniques. An example result is given in Figure 5 for a chain of separate silver spheres. The remarkable feature here is the very strong enhancement of the near-infra-red absorption relative to the single sphere resonance at a much lower wavelength. The IR effects we see in the samples discussed earlier are believed to be related to such enhancements, with peaks washed out by slight variations in particle separations and by touching or merging where from equation (2) the tail is broad but falls off monotonically. The
by our
Conclusion Strong absorption anisotropy occurs in thin films of metal-insulator composites deposited using bi-oblique deposition even when the metal content is low. This anisotropy and the fine details of the associated absorption spectra can be attributed to multipolar resonant effects. Multiple peaks or broad absorption bands extending well into the infra-red result. New expressions incorporating proximity effects between neighbouring particles help understand these observations and demonstrate that absorption can be very sensitive to the fine details of microstructure. Dipole only models are unable to explain such effects. Since high order multipolar resonances accumulate near the same wavelengths as the isolated particle dipole resonance, peak position alone is not enough to determine whether the advanced theories are needed. The whole spectrum, especially the infra-red tail should be examined to evaluate the importance of multipolar interactions. Due to variations in particle spacing or to a predominance of touching or intersecting effects the obvious multipole signature of multiple peaks may not always show up. Acknowledgement The support
2.5
of an ARC grant is gratefully
acknowledged.
References
0
0.3
0.4
0.5 Wavelength
Figure 5. Predicted
0.6
0.7
(pm)
absorption spectra for silver spheres embedded in an A1,09 matrix. Results are shown for a single sphere, a pair and an infinite chain of spheres. Centres are 2 units apart in the pair and chain and each radius is 0.497 units.
938
’ M Suzuki and Y Taga, J Appl Phys, 71,284s (1992) *G B Smith, M W Ng, R J Ditchburn, P J Martin and R P Netterfield, Proc SPIE, 1536, 126 (1991). 3 G Niklasson, J Appl Phys, 62,258 (1987). 4 G B Smith, M W Ng, R J Ditchburn, P J Martin and R P Netterfield, J Solar Energy Mater, 25, 149 (1992). 5A V Radchik, G B Smith and A J Reuben, Phys Rat B, 46,61115 (1992). 6 R Ruppin, J Phys Sot Japan, 58, 1446 (1989). ‘A V Paley, A V Radchik, G B Smith and A V Vagov, Appl Phys Lett, 64,3187 (1994) ; J Appl Phys (in press). *A V Paley, A V Radchik and G B Smith, J Appl Phys, 73,3446 (1992). ‘A V Vagov, A V Radchik and G B Smith, Phys Rw Lett, 73, 1035 (1994). “‘P Clippe, R Evrard and A A Lucas, Phys Rev B, 14, 1715 (1976). “E D Palik (Editor), Handbook of Optical Constants of Solids, p 81. Academic Press, Orland, FL (1985).
Pergamon
46/numbers &IO/pages 935 to 938/1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved
0042-207x/95
0042-207x(95)00075-3
New effective
medium
$9.50+.00
models and their applications
S Dligatch, M W Ng, G B Smith, A J Reuben, A V Radchik and A V Vagov, Depattment University of Technology, Sydney, PO Box 123, Broadway NSW2007, Australia
of Applied Physics,
Thin films and many surfaces often have complex structures which are conveniently analysed for overall optical response by effective medium models. Examples from new deposition techniques and new exact theory are presented that highlight effects that are not predictable by many of the widely used simple effective medium models. High order multipoles and associated multiple resonances play a key role and >alter spectral response and transport properties markedly. The new theories can discriminate the short range forces between neighbouring pairs and longer range forces in chains and arrays. Experimental data on composite films show how changes in component distribution can strongly effect spectral absorptance in ways that are not predictable by standard dipole theories.
Introduction Composites made by co-deposition of two species from different angles of incidence’ and by different techniques, such as sputtering, vacuum arc’ and thermal evaporation3 display different spatial distributions and sizes of the included component. Such systems can be technically important4 and also provide useful means of assessing the ability of various effective medium models to predict optical or other transport properties. Exact results5-9, involving quite simple expressions and any materials, are now available for the polarisabilities of pairs of cylinders and spheres covering all separations of centres from large values where the isolated particle result applies through to the totally merged situation where the single particle result again applies. Such results provide a far better basis for extended effective medium models than the standard theories such as MaxwellGarnett and Bruggeman, which are special limiting cases. The need to improve on simple models when clustering of small particles occurs has been known for some time”. A wide range of spectral response in the quasistatic limit is found as a function of the centre-to-centre gap. The touching and merging cases are especially interesting. Generally there is a significant red shift of absorption ability as multipoles come into play. One of the features that does not appear to be explained by conventional theories that we find in several anisotropic dilute composite structures is the variation of absorption strength as one moves to longer wavelength (1). While it usually falls as /z increases across the visible and near IR it can also rise over a broad band, as if the metal were percolating. While this behaviour is more likely as concentration increases it is surprising for it to occur well below the percolation threshold. A key motivation for this work has been our observation that metal volume fractions of a few percent can give quite strong absorption provided the deposition conditions favour a nonrandom distribution of the included metal component. Both experimental examples in this paper employ the same bi-oblique
deposition geometries that we know ensure an inhomogeneous spread of metal in the cermet (Ag/Al,O,) but use different deposition techniques (sputtering and evaporation). The difference in techniques is shown to give distinct microstructures which in turn cause noticeable differences in absorption spectra. We then introduce and demonstrate some of, the new theories that help explain both the enhancement of absorption that occurs with clustering, the sensitivity of these spectra to details of that structure, and the strong optical anisotropies that occur in these films.
Experimental
examples of thin film cluster composites
Columnar thin film composites deposited by oblique deposition of one or both components develop a complex array of microstructures depending on the details of the deposition geometries (especially the angle of incidence from the various sources to the substrate), the melting points and reactivities of deposited materials, and the energetics and degree of collimation of the incoming species. Absorbing composites are particularly interesting since the absorption spectra relate strongly to the microstructure. Using reactive deposition it is possible to produce cluster composites of metal particles in a matrix of the same metal’s oxide. A very different structure occurs with energetic arc deposition as compared to low energy evaporation. These differences are not only in the more obvious area of density but more importantly in the array structure the columns adopt and in the metal distribution which is non-random. A good example is oblique aluminium in A&O,‘. Sputtering is also of interest and in this report we compare evaporation and sputtering results in a system where it is easier to observe the different components by electron microscopy, namely silver in oxides such as A&O, or SiOZ1. Two high resolution scanning electron micrographs obtained from a field emission system are shown in Figures l(a) and (b). Figure 1 (a) shows a sputtered composite, thickness 1500 A, 935
S Dligafch et a/r New effective medium models and their applications
0.6 , 0.5
I- i:
0.4 8
I
I
I
----__
cls --_. -----____
*= :
0.3
‘\
\
0.2
L *-a_ -_
0.1
I< 0
I
1
-
: I
0.5
I
1 Wavelength
I
1.5 (pm)
I
2
Figure 2. Spectra of absorption coefficients for s and p polarised light at near normal incidence to the sputtered film of Figure l(a). Deposition geometry is in the inset and the plane of incidence for light is the deposition direction.
epared by bioolique aeposmon:
(a) sputtelGu,
produced by depositing silver at 30” to the substrate normal and simultaneous rf sputtering of A&O3 at 70” to the normal. The oxide columns shown are typically 25 nm across and lined up approximately diagonally from the bottom left hand corner. This direction is normal to the deposition plane. The columns are tilted at about 40” to the vertical, from edge micrographs. The silver particles are very fine and not obvious in this micrograph. They are clearly present from their optical signature (see below) which also indicates they are clustered or ordered more than in the evaporated case. Other work has shown in a similar system that very fine silver spheres align along each column’. In the evaporation case, thickness 950 A, of Figure l(b) the brighter silver particles are more obvious and are comparable in size to the oxide columns with diameters from 25 to 50 nm. This film has a higher metal density than the sputtered film. The columnar structure is less ordered in these lower energy techniques. Correlated with this appears to be a lower degree of ordering of the included metal particles. We found even more ordering in arc evaporation’ where the incident ions have very high energy.
light when the plane of incidence parallels the deposition direction. Experimental optical results, namely reflectance, transmittance and absorptance from 250 to 2000 nm, for the two films shown in Figures l(a) and (b) have been processed using thickness t, to extract the effective absorption coefficient CI(taking intensity falling off as exp( --47&/l). These spectral properties of 01are presented in Figures 2 and 3, respectively. The two sets of spectra for each film are for light polarised perpendicular (s) and parallel (p) to the plane of incidence which is the column or deposition direction. Near normal incidence results only are discussed here ; s-polarisation thus defined means the electric field is along the direction from bottom left to top right corners in Figure 1 (a). Strong absorption anisotropy is evident in both films but in the evaporated case (Figure 3) this shows up primarily via magnitudes of CI,and ccPwhile the general shapes of the curves are the same. The silver absorption resonance is quite broad relative to that for isolated particles. The sputtered sample in Figure 2 displays qualitative differences between CI,and tlP as well. Of note is the relative widths of the silver resonance for each polarisation and the double resonance for p-polarised light. The optical differences between sputtered and evaporated films are in accord with the microstructural differences in Figure 1 and the theories that follow. It is significant that both display a strong IR absorption tail with the s-case stronger than the p-case in each. The underlying cause is that columns and their associated
8
0.5 Microstructure and optical absorption The details of how the inclusions and the columns are arranged gives distinct optical signatures which show up when one compares the absorption spectra for different polarisation of incident 936
1 Wavelength
1.5
2
(pm)
Figure 3. Spectra of absorption coefficients for s and p polarised light at near normal incidence to the evaporated film of Figure l(b). Deposition geometry direction.
is in the inset and plane of incidence
for light is the deposition
S Dligarch et a/: New effective medium models and their applications silver particles are found on average to be closer in directions normal to the deposition planeZA. Features that the new models are needed for include the second peak in Figure 2 for 01~.This is due to multipole effects arising from close but separate particles. Touching or merging, however, merges all multipole resonances into a single broad continuous spectrum as found in the other curves. The first peak in clp, Figure 2, is also narrower than all others, which also follows from the lower degree of touching in the deposition direction. However there must be some touching or merging otherwise the absorption tail into the IR would not be so broad. We now turn to some of the theories that help explain and quantify these effects and highlight the inadequacy of dipole only models. New models and example results on cermet array In this section we will demonstrate new models with cylinders and spheresM that enables an understanding of some of the resonant features just displayed. Exact theories for the polarisability of pairs and arrays of cylinders for any centre-to-centre separation (including intersection) and with complex refractive indices have recently been formalised based on a new approach to the use of conformal transformations. These techniques lead more efficiently to solutions because the effects of the array are built into the transformed coordinate frame from the outset5.6. Previous techniques required additional boundary equations based on lattice sums. Separate cylinders give polarisability p which can be expressed for any array in terms of the spectral parameter
can remain near 400 nm when many multipoles (i.e. many nonzero wn) are present. Touching and intersecting particles give continuous, not discrete eigenvalue spectra. A red shift of absorption occurs due to interactions, which maximises in extent at the touching case. An example using equation (1) showing discrete resonances is in Figure 4 for silver particles (optical constant data from Palik”) in an oxide matrix. We compare an isolated pair with a pair in a chain and a lattice of the same pairs. The spacing between pairs is such that the overall metal fill factor is low in this example (0.3). This spectral range is of considerable interest in solar energy applications. We next look at chains using silver spheres. For spheres until recently only separate and intersecting pairs could in principle be solved exactly by transformation methods since the resulting frames allow solution of the Laplace equation by separation of variables. In practice the intersecting and touching sphere problem has onli recently been solved7z8 this way for arbitrary dielectric constants. The result which includes the touching limit is given in equation (2). For intersecting spherical particles the result for longitudinal polarisability is
na.
p1 = 8717.3 * A2Bj*‘)tan h - d;l 2a s0 where Bj’) is defined as
l+s V=l-_E -1 with e the complex dielectric constant ofeach unit radius cylinder. p, and J+ are for fields along and normal to a given pair axis and come from the discrete sum
PlJ =
nj?l &
3
v,
=
-r3sinhs2
1
Also
+ =
e-*’
sinhg
Id 2a
-ae,,
d s2
=
2a
-17%
-n
where for circular cross-sections weights are given by -497 w n =----a2v n
in the (x,~) plane, the spectral
and
n n
with
CIis an easily determined geometric boundary parameter for each array5,7. The complex variable t = x-iv, log F, determines the conformal mapping and C is a contour which can be one particle’s surface. Thus once F is determined the problem is essentially solved. Each n gives a pole at v = vn and hence a resonance and these show up as multiple peaks in absorption spectra with IZ= 1 the dipole. From equation (1) at large n we see that higher order poles give overlapping resonances centred near the spectral position found from Real(v) = 0, that is Real(s) = - 1. This is precisely the dipole resonance position for an isolated cylindrical particle and helps explain why the dominant peaks in our data
Wavelength (p) Figure 4. Predicted absorption spectra for unit radius silver cylinders embedded in an A&O, matrix. Results are shown for a pair, a chain and a lattice of the cylinders. The pair centre separation is 2.3 units, and the gap to the edge of the next pair is 3 units in the horizontal and vertical directions. 937
S Dligatch et a/: New effective medium models and their applications
23 =-
E-l &+2’
second peak in Figure 2 for p waves is near that predicted chain model for spheres.
a=JEE
in terms of centre separation d and 8, = sin-‘a/a. This spectrum is continuous with index L for each pole in the continuum. For separate spherical particles and chains we have developed a new technique’ based on the three-dimensional hypercomplex variable X(U) = xficos (u)u+ isin (u)z, that retains the advantages of the conformal approach and is in fact utilised by making direct analogies with two-dimensional transformations. It is however necessary to solve a matrix equation for potential but convergence is much more rapid than in any previous methods so that we are able to study quite dense chains exactly. These were in practice insoluble by established techniques. An example result is given in Figure 5 for a chain of separate silver spheres. The remarkable feature here is the very strong enhancement of the near-infra-red absorption relative to the single sphere resonance at a much lower wavelength. The IR effects we see in the samples discussed earlier are believed to be related to such enhancements, with peaks washed out by slight variations in particle separations and by touching or merging where from equation (2) the tail is broad but falls off monotonically. The
by our
Conclusion Strong absorption anisotropy occurs in thin films of metal-insulator composites deposited using bi-oblique deposition even when the metal content is low. This anisotropy and the fine details of the associated absorption spectra can be attributed to multipolar resonant effects. Multiple peaks or broad absorption bands extending well into the infra-red result. New expressions incorporating proximity effects between neighbouring particles help understand these observations and demonstrate that absorption can be very sensitive to the fine details of microstructure. Dipole only models are unable to explain such effects. Since high order multipolar resonances accumulate near the same wavelengths as the isolated particle dipole resonance, peak position alone is not enough to determine whether the advanced theories are needed. The whole spectrum, especially the infra-red tail should be examined to evaluate the importance of multipolar interactions. Due to variations in particle spacing or to a predominance of touching or intersecting effects the obvious multipole signature of multiple peaks may not always show up. Acknowledgement The support
2.5
of an ARC grant is gratefully
acknowledged.
References
0
0.3
0.4
0.5 Wavelength
Figure 5. Predicted
0.6
0.7
(pm)
absorption spectra for silver spheres embedded in an A1,09 matrix. Results are shown for a single sphere, a pair and an infinite chain of spheres. Centres are 2 units apart in the pair and chain and each radius is 0.497 units.
938
’ M Suzuki and Y Taga, J Appl Phys, 71,284s (1992) *G B Smith, M W Ng, R J Ditchburn, P J Martin and R P Netterfield, Proc SPIE, 1536, 126 (1991). 3 G Niklasson, J Appl Phys, 62,258 (1987). 4 G B Smith, M W Ng, R J Ditchburn, P J Martin and R P Netterfield, J Solar Energy Mater, 25, 149 (1992). 5A V Radchik, G B Smith and A J Reuben, Phys Rat B, 46,61115 (1992). 6 R Ruppin, J Phys Sot Japan, 58, 1446 (1989). ‘A V Paley, A V Radchik, G B Smith and A V Vagov, Appl Phys Lett, 64,3187 (1994) ; J Appl Phys (in press). *A V Paley, A V Radchik and G B Smith, J Appl Phys, 73,3446 (1992). ‘A V Vagov, A V Radchik and G B Smith, Phys Rw Lett, 73, 1035 (1994). “‘P Clippe, R Evrard and A A Lucas, Phys Rev B, 14, 1715 (1976). “E D Palik (Editor), Handbook of Optical Constants of Solids, p 81. Academic Press, Orland, FL (1985).