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The optical absorption of island films at oblique incidence
Short Communications The optical absorption of island films at oblique incidence The first stage in the formation of a thin metallic film by deposition from the vapor is the growth of small, isolated particles. The optical properties of these island films often have been interpreted by assuming that the particles are nonspherica11-3. However, it is also possible to describe these results with the Maxwell Garnett theory4 for spherical particles, and a comparison of the optical transmission and electron micrographs of island films of gold supports this description’. To interpret their results on the optical absorption of island films of silver at oblique incidence, Emeric and Emeric also assumed nonspherical particles6. The purpose of this paper is to show that these results can also be interpreted in terms of spherically symmetrical particles, and to give reasons for preferring this interpretation. First the application of the Maxwell Garnett theory to island films is briefly discussed, based on previous work5. In this theory the complex dielectric constant E, of a thin island film is related to the dielectric constant E of the islands by the equation: (1) where n, is the refractive index of the non-absorbing substrate of the film, and Q is the area fraction of particles in the film. The derivation of eqn. (I) and justifications for the definitions of Q and qSwere given previously’. The real and imaginary parts of the complex dielectric constant E, = elc -isZc are from eqn. (1):
[El(gf) +2n:][cl+(ff)%j +E2qg Elc =
-___
[El+ ~)n~]z+~22
-
(2)
in terms of 8 = sl-ia2. The optical transmission T of a very thin film for normal incidence is approximately’,’ : y-c
-t 1+
4n,.______ f&Y+ E2cY(1+n,)
(4)
where y = 4nd/A; d is the film thickness and 11the wave length. From eqns. (3) and ?%n Solid
Films, 1 (1967/68)
379-382
-- Elsevier,
Amsterdam
- Printed
in the Netherlands
380
SHORT
COMMUNICATIONS
(4) it can be seen that when et = -(2+ Q)n,‘/( I- Q), &2c is a maximum and T shows a minimum (an absorption band). By substituting in the free electron values for E it can be shown that this absorption band results from a plasma oscillation of the electrons in the particles 5s8. When Q becomes small the band is at or near E, = - 2ns2, as deduced also from Mie’s theory’ for the absorption and scattering of small particles. sorption
As Q becomes
larger (closer packing
band shifts to longer wave lengths,
as determined
of the particles)
the ab-
by the factor (2+Q)/
(1 -Q). The validity of these equations was confirmed by measuring Q from electronmicrographs of island films of gold and comparing the band position calculated from this Q to the measured position. Emeric and Emeric measured the optical transmission of their films for light polarized parallel and perpendicular to the plane of incidence (the plane containing the direction of propagation of light and the surface normal) at angles of incidence of O”, 4.5” and 70”. For normal incidence, and for perpendicular polarization (electric vector of the light parallel to the substrate surface) at all angles of incidence, they found the usual absorption band resulting from plasma oscillations of the electrons in the silver, at a wave length above 0.4 p. However, for parallel polarization and non-normal incidence they found in addition to this band another absorption band between 0.3 and 0.4 ,u. They interpreted the presence of this second band and shifts of the other band to longer wavelengths as resulting from the nonspherical shape of the particles. In the next paragraphs it is shown that this second band results from a certain type of anisotropy of the film, so that its presence is not definitive evidence for nonspherical particles. The optical transmissions of a thin film for an angle of incidence 4 with the surface normal and perpendicularly polarized light is:
(cos 4 + n, cos &)* +EZCy (cos 4 + n, cos 4,)
where 4, is the angle of transmission in the substrate with its surface normal, and is related to 4 by sin b/sin 4, = n, 2. For light polarized parallel to the incident plane, the transmission is 4n, cos 4 cos 4,
?I = (cos 4, + n, cos
4)”+ .zZcy (cos4, + n,
n, sin’ C/I cos C#J) cos 4, cos C$ + 21
[
Elc
+EZc
I
(6) The only difference between eqns. (5) and (4) are terms little dependent upon wave length that take account of the reduced intensity at greater angles of incidence. Thus for perpendicularly polarized light no additional absorption bands are expected over those found for normal incidence. Physically this is reasonable, since with this polarization the electric vector of the light has the same orientation with Thin Solid fYlms,
I
(1967168)
379-382
381
SHORT COMMUNICATIONS
respect to the film (is parallel
to it) for all angles of incidence.
tion, however,
there is an additional term in the denominator ). This term arises because the loss function E~~/(E~~~-t- &2C2 the light changes its orientation with respect to the substrate incidence or perpendicular polarization. In this paragraph the behavior dielectric
constant
E, of eqns.
of the loss function
(2) and (3). As Q approaches
For parallel
polariza-
of eqn. (6) involving the electric vector of from that for normal is examined
for the
zero (low particle
density) sic approaches one and a2c 4 1, so that ~~~~~~~~~ +E~~‘) approaches Ed=. Therefore, in this case there is no additional absorption band caused by the loss function. This result is physically reasonable, since the spherical particles should absorb independently at low densities, with no orientation effects. As the particle density becomes higher in the film, the particles interact and polarize one another. This interaction provides the anisotropy needed for development of another plasma absorption band at a shorter wave length. If s2= is reasonably smaller than sic, as is the case in the experiments of Emeric and Emeric, the loss function is a maximum when Erc = 0. This occurs when ~1(1+2Q)/(l-Q)
= -2ns2.
This condition, therefore, gives approximately the position of the new absorption band or minimum in transmission that results for parallel polarization at oblique incidence. It is possible to test this contention for the results of Emeric and Emeric, because the value of Q can be found from the position of the main plasma absorption band for a particular island film. In these calculations the best optical properties from measurements on bulk silver samples are used, as modified to be consistent with the position of the plasma band for small silver particles”, and n, is taken as 1.468. A critical discussion of the optical properties of silver was given previously”. From the main plasma band for Fig. 4 (c) of Emeric and Emeric6, at 0.450 1-1,Q = 0.310; then from eqn. (7) the second band should be at 0.346 p, which is fortuitously close to the measured result of 0.345 /J for this same film. This calculation shows that the second absorption band found by Emeric and Emeric can equally well be interpreted as resulting from interacting spherical particles as from nonspherical particles. In the next paragraph arguments are presented to support the assumption of spherical particles. In their treatment Emeric and Emeric ignore completely the effects of interactions between particles. However, the width of the plasma absorption band in their Fig. 4 shows that the particles in this film were quite small, certainly below 100 A in dimension, as deduced from the dependence of the band width on particle size found by Doremus”. Since the “bulk thickness” of the film was between 10 and 20 A, an appreciable area fraction of the substrate must be covered with particles, and their interactions cannot be ignored. These deductions are consistent with the value of 0.31 found for Q above. Silver films do not adhere strongly to Thin Solid Films, 1 (1967168) 379-382
382
SHORT COMMUNICATIONS
fused silica, so from surface particles.
The optical
tension
properties
considerations
one would
and electronmicrographs
expect
spherical
of island films of silver
deposited from the vapor are consistent with spherically symmetrical particles, so it seems likely that the films of Emeric and Emeric also contained such particles. Schopper l1 has shown that the absorption band between 0.3 and 0.4 p found by Fleischman and his coworkers for thin films of alkali metals at oblique incidence and parallel
polarization
results from the loss function.
He also suggested
shift of this band to longer wave lengths as compared to the position bulk metal resulted from the openness of the film structure. General Electric Company, Research and Development Center, Schenectady, N. Y. (U.S.A.)
I
2 3 4 5 6 7 8 9 IO I1
E. DAVID,
that the
calculated
R.H.
DOREMUS
Z. Physik, 114 (1939) 389; ibid., 115 (1940) 514. H. SCHOPPER, Z. Physik, 130 (1951) 565. G. RASIGNI AND P. ROUARD, J. Opt. Sot. Am., 53 (1963) 604. J. C. MAXWELL GARNETT, Phil. Trans. Roy. Sot., 203 (1904) 385; ibid., 205 (1906) 237 R. H. DOREMUS, J. Appl. Phys., 37 (1966) 2775. N. EMERIC AND A. EMERIC, Thin Solid Films, I (1967) 13. H. WOLTER, Z. Physik, 105 (1937) 269. W. T. DOYLE, Phys. Rev., I11 (1958) 1067. G. MIE, Ann. Physik, 25 (1908) 377. R. H. DOREMUS, J. Chem. Phys., 42 (1965) 414. H. SCHOPPER, Z. Physik, 174 (1963) 125.
Received
October
Thin Solid Films,
3,1967
1 (1967/68)
379-382
for
[El(gf) +2n:][cl+(ff)%j +E2qg Elc =
-___
[El+ ~)n~]z+~22
-
(2)
in terms of 8 = sl-ia2. The optical transmission T of a very thin film for normal incidence is approximately’,’ : y-c
-t 1+
4n,.______ f&Y+ E2cY(1+n,)
(4)
where y = 4nd/A; d is the film thickness and 11the wave length. From eqns. (3) and ?%n Solid
Films, 1 (1967/68)
379-382
-- Elsevier,
Amsterdam
- Printed
in the Netherlands
380
SHORT
COMMUNICATIONS
(4) it can be seen that when et = -(2+ Q)n,‘/( I- Q), &2c is a maximum and T shows a minimum (an absorption band). By substituting in the free electron values for E it can be shown that this absorption band results from a plasma oscillation of the electrons in the particles 5s8. When Q becomes small the band is at or near E, = - 2ns2, as deduced also from Mie’s theory’ for the absorption and scattering of small particles. sorption
As Q becomes
larger (closer packing
band shifts to longer wave lengths,
as determined
of the particles)
the ab-
by the factor (2+Q)/
(1 -Q). The validity of these equations was confirmed by measuring Q from electronmicrographs of island films of gold and comparing the band position calculated from this Q to the measured position. Emeric and Emeric measured the optical transmission of their films for light polarized parallel and perpendicular to the plane of incidence (the plane containing the direction of propagation of light and the surface normal) at angles of incidence of O”, 4.5” and 70”. For normal incidence, and for perpendicular polarization (electric vector of the light parallel to the substrate surface) at all angles of incidence, they found the usual absorption band resulting from plasma oscillations of the electrons in the silver, at a wave length above 0.4 p. However, for parallel polarization and non-normal incidence they found in addition to this band another absorption band between 0.3 and 0.4 ,u. They interpreted the presence of this second band and shifts of the other band to longer wavelengths as resulting from the nonspherical shape of the particles. In the next paragraphs it is shown that this second band results from a certain type of anisotropy of the film, so that its presence is not definitive evidence for nonspherical particles. The optical transmissions of a thin film for an angle of incidence 4 with the surface normal and perpendicularly polarized light is:
(cos 4 + n, cos &)* +EZCy (cos 4 + n, cos 4,)
where 4, is the angle of transmission in the substrate with its surface normal, and is related to 4 by sin b/sin 4, = n, 2. For light polarized parallel to the incident plane, the transmission is 4n, cos 4 cos 4,
?I = (cos 4, + n, cos
4)”+ .zZcy (cos4, + n,
n, sin’ C/I cos C#J) cos 4, cos C$ + 21
[
Elc
+EZc
I
(6) The only difference between eqns. (5) and (4) are terms little dependent upon wave length that take account of the reduced intensity at greater angles of incidence. Thus for perpendicularly polarized light no additional absorption bands are expected over those found for normal incidence. Physically this is reasonable, since with this polarization the electric vector of the light has the same orientation with Thin Solid fYlms,
I
(1967168)
379-382
381
SHORT COMMUNICATIONS
respect to the film (is parallel
to it) for all angles of incidence.
tion, however,
there is an additional term in the denominator ). This term arises because the loss function E~~/(E~~~-t- &2C2 the light changes its orientation with respect to the substrate incidence or perpendicular polarization. In this paragraph the behavior dielectric
constant
E, of eqns.
of the loss function
(2) and (3). As Q approaches
For parallel
polariza-
of eqn. (6) involving the electric vector of from that for normal is examined
for the
zero (low particle
density) sic approaches one and a2c 4 1, so that ~~~~~~~~~ +E~~‘) approaches Ed=. Therefore, in this case there is no additional absorption band caused by the loss function. This result is physically reasonable, since the spherical particles should absorb independently at low densities, with no orientation effects. As the particle density becomes higher in the film, the particles interact and polarize one another. This interaction provides the anisotropy needed for development of another plasma absorption band at a shorter wave length. If s2= is reasonably smaller than sic, as is the case in the experiments of Emeric and Emeric, the loss function is a maximum when Erc = 0. This occurs when ~1(1+2Q)/(l-Q)
= -2ns2.
This condition, therefore, gives approximately the position of the new absorption band or minimum in transmission that results for parallel polarization at oblique incidence. It is possible to test this contention for the results of Emeric and Emeric, because the value of Q can be found from the position of the main plasma absorption band for a particular island film. In these calculations the best optical properties from measurements on bulk silver samples are used, as modified to be consistent with the position of the plasma band for small silver particles”, and n, is taken as 1.468. A critical discussion of the optical properties of silver was given previously”. From the main plasma band for Fig. 4 (c) of Emeric and Emeric6, at 0.450 1-1,Q = 0.310; then from eqn. (7) the second band should be at 0.346 p, which is fortuitously close to the measured result of 0.345 /J for this same film. This calculation shows that the second absorption band found by Emeric and Emeric can equally well be interpreted as resulting from interacting spherical particles as from nonspherical particles. In the next paragraph arguments are presented to support the assumption of spherical particles. In their treatment Emeric and Emeric ignore completely the effects of interactions between particles. However, the width of the plasma absorption band in their Fig. 4 shows that the particles in this film were quite small, certainly below 100 A in dimension, as deduced from the dependence of the band width on particle size found by Doremus”. Since the “bulk thickness” of the film was between 10 and 20 A, an appreciable area fraction of the substrate must be covered with particles, and their interactions cannot be ignored. These deductions are consistent with the value of 0.31 found for Q above. Silver films do not adhere strongly to Thin Solid Films, 1 (1967168) 379-382
382
SHORT COMMUNICATIONS
fused silica, so from surface particles.
The optical
tension
properties
considerations
one would
and electronmicrographs
expect
spherical
of island films of silver
deposited from the vapor are consistent with spherically symmetrical particles, so it seems likely that the films of Emeric and Emeric also contained such particles. Schopper l1 has shown that the absorption band between 0.3 and 0.4 p found by Fleischman and his coworkers for thin films of alkali metals at oblique incidence and parallel
polarization
results from the loss function.
He also suggested
shift of this band to longer wave lengths as compared to the position bulk metal resulted from the openness of the film structure. General Electric Company, Research and Development Center, Schenectady, N. Y. (U.S.A.)
I
2 3 4 5 6 7 8 9 IO I1
E. DAVID,
that the
calculated
R.H.
DOREMUS
Z. Physik, 114 (1939) 389; ibid., 115 (1940) 514. H. SCHOPPER, Z. Physik, 130 (1951) 565. G. RASIGNI AND P. ROUARD, J. Opt. Sot. Am., 53 (1963) 604. J. C. MAXWELL GARNETT, Phil. Trans. Roy. Sot., 203 (1904) 385; ibid., 205 (1906) 237 R. H. DOREMUS, J. Appl. Phys., 37 (1966) 2775. N. EMERIC AND A. EMERIC, Thin Solid Films, I (1967) 13. H. WOLTER, Z. Physik, 105 (1937) 269. W. T. DOYLE, Phys. Rev., I11 (1958) 1067. G. MIE, Ann. Physik, 25 (1908) 377. R. H. DOREMUS, J. Chem. Phys., 42 (1965) 414. H. SCHOPPER, Z. Physik, 174 (1963) 125.
Received
October
Thin Solid Films,
3,1967
1 (1967/68)
379-382
for