LOCAL FIELDS' LOCALIZATION AND CHAOS AND NONLINEAROPTICAL ENHANCEMENT IN COMPOSITES MARK I. STOCKMAN Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303 E-mail:
[email protected] http://www.phy-astr.gsu.edu/stockman The paper is devoted to linear and nonlinear optical properties of disordered clusters and nanocomposites. Linear and nonlinear optical polarizabilities of large disordered clusters, fractal clusters in particular, and susceptibilities of nanocomposites are found and calculated numerically. A spectral theory with dipole interaction is used to obtain quantitative results. Major properties of systems under consideration are giant fluctuations and chaos of local fields that cause strong enhancement (by many orders of magnitude) of nonlinear optical responses. The enhancement and fluctuations properties of the local fields are intimately interrelated to the inhomogeneous localization of the systems’ eigenmodes (“plasmons”). Due to these fluctuations, mean-field theory completely fails to describe optical polarizabilities.
1. Introduction Clusters and nanocomposites belong to so-called nanostructured materials. Such materials typically are nanoparticles either bound to each other by covalent or van der Waals bonds, or dispersed in a host medium. Description of electromagnetic properties of such system is a long-standing problem going back to such names as Maxwell Garnett1, Lorentz2 and Bruggeman3. Properties of such materials may be dramatically different from those of bulk materials with identical chemical composition. A characteristic property of such systems is confinement of electrons, phonons, electric fields, etc., in small spatial regions. Such a confinement, in particular, modifies spectral properties (shifts quantum levels and changes transition probabilities), and also changes the interaction between the constituent particles. As we will be discussing in this paper, local (near-zone) electromagnetic fields are strongly fluctuating in space. Their magnitude is greatly (by orders of magnitude) enhanced with respect to the external (exciting) fields. A phenomenon closely related to the enhancement and fluctuations of local fields is localization of elementary excitations (eigenmodes) in the composites4-8. The relevant excitations are polar waves that are traditionally called plasmons (this term originates from theory of metallic nanoparticles containing electron plasma, but is now often used in application to other nanocomposites). Plasmon-resonant properties leading to enhancement of local fields are especially pronounced in some metallic (especially silver, gold, or platinum) colloidal clusters, metal
nanocomposites and rough surfaces. A typical example of such responses is surfaceenhanced Raman scattering (see, e.g. a review of Ref. 9 and reference therein). The most pronounced effect of the fluctuating local fields is on nonlinear optical susceptibilities. The reason for that can be understood qualitatively. Imaging two fields with the same average intensity I 1 ∝ E
2
. For the sake of argument, let us
say, the first field has the same constant intensity I 1 in N >> 1 points, and the second is strongly localized at one point where its intensity then should be
I 2 = NI 1 . Consider a n -th order nonlinearity where the nonlinear response is proportional to E
2n
∝ I n . The ratio of the nonlinear response for the first
(constant) field is proportional to
1 ( NI 1n ) = I 1n . In contrast, the response to the N
second (strongly localized) field is
1 ( NI 1 ) n = N n−1 I1n . In such a way, the N
enhancement coefficient (the ratio of the nonlinear response in the second case to n −1
that in the first case) is N . Hence the localization has a potential to bring about strongly enhanced nonlinear responses where the enhancement increases with the order of nonlinearity and the degree of localization (spatial fluctuations). To maximize this effect, our goal is to find systems with the maximum spatial fluctuations of the local densities. We certainly expect that the density fluctuations will cause correspondingly large fluctuations of the local fields. There exists a class of systems that stands out in this respect. These are self-similar (fractal) systems, which (on average) repeat themselves at different scales. In other words, looking at such a system and not seeing its boundaries (neither at the maximum or a minimum scales), one cannot say what fraction of the system is observed, and what is the actual size of the objects seen. For such systems, the number N of constituent particles (monomers) contained within a radius R scales as D
R N ≅ , R0
(1)
where R0 is a typical distance between monomers, and D is the fractal (Hausdorff) dimension of the cluster. The density of monomers as given by Eq. (1) is asymptotically zero for large clusters,
2
R ρ≅R R0 −3 0
D− 3
→0
. (2) However, this does not mean that the interaction between monomers can be neglected. The underlying reason for that is a strong correlation between monomers in a cluster, with the pairpair correlation function scaling similar to Eq. (2). Thus we have a unique system whose macroscopic density is asymptotically zero, but the interaction inside the system is strong. This idea has been proposed by us in an earlier papers4,10 An example of a fractal cluster, obtained by cluster-cluster aggregation11 (CCA) is shown in Fig. 1. This figure illustrates Fig. 1 Cluster-cluster aggregate (CCA) of many properties of fractal clusters and N = 1000 monomers. composites, including the low overall density and strong correlation in the positions of monomers. We will use CCA clusters (composites) throughout the paper as a model of self-similar (fractal) systems. To avoid possible misunderstanding, we point out that other, non-fractal systems also possess enhanced optical responses, especially those that are tailored to have optimally-chosen dielectric properties changing in space12. The physical origin of the enhancement in this case is the same: the local fields change in space, and consequently the higher moments of their intensity are increased. As an example of such systems we will consider a random Maxwell Garnett composite (dielectric or metallic spheres embedded in a host medium at random positions). Such a composite has earlier been considered in a mean-field approximation13. We will consider below a model of such composites where the inclusion spheres are positioned on a cubic lattice and call it a random lattice gas (RLG). We will use RLG as a model of random but not fractal composites.
3
2. Equations Governing Optical (Dipolar) Responses We concentrate on the dipole-dipole interaction, which is a universal interaction between polarizable particles. We consider a cluster (or, a composite) whose particles (called below monomers) are positioned at points ri . We assume that the system (a cluster or composite) is subjected to the electric field E of the incident optical wave. This field induces the dipole d iα at an i-th
monomer (here α = x , y , z denotes the Cartesian components of the vector, and similar notations will be used for other vectors). The dipole moments satisfy wellknown system of equations
−1 iα 0
α d Here E
( 0) iα
=E
( )( )
rij rij α − ∑ δαβ − 3 rij2 j =1 N
( 0) iα
β
d jβ r 3 ij
.
(3)
is the wave-field amplitude at the i-th monomer, rij = ri − r j is the
relative vector between the i-th and j-th monomers, and α 0 is the dipole polarizability of the monomer. We assume that the size of the system is much less ( 0)
than the wavelength of the exciting wave, and therefore the exciting field E α
is
the same for all the monomers of a cluster. We note that the dipole interaction is not valid in the close vicinity of a monomer. Our choice of interaction is justified if intermediate-to-large scales predominantly contribute to the properties under consideration. 3. Scaling and Spectral Representation The scaling theory of the optical response of fractal clusters has been developed in Refs. 5 and 6. We note that a similar spectral approach has been independently introduced R.Fuchs and collaborators.14,15 There exists also a general spectral approach16 that can be shown to reduce to our approach if the dipole approximation is applied. The material properties of the system enter Eq. (3) only via the combination Zrij3 , −1
where we have introduced the notation Z ≡ α 0 . This along with the (approximate) self-similarity of the system is a prerequisite for scaling in terms of the spectral variable Z . A principal requirement for the scaling of a certain
4
physical quantity F is that the system eigenmodes contributing to F should have their localization radii L intermediate between the maximum scale (size of the cluster R c ) and the minimum scale, R0 . Then this quantity will not depend on any external length, leading to scaling. Because the quantity F is not sensitive to the maximum scale R c , it should have the
(
)
functional dependence F = F ZR0 . On the other hand, the eigenmodes 3
contributing to F are insensitive to a much smaller minimum scale. Therefore the dependence on R0 can be only power (scaling), and consequently
F (Z ) ∝ (ZR03 ) , γ
where
(4)
γ is some scaling index.
In accord with the above arguments, it is convenient to express all results not in terms of frequency, but in terms of Z , separating the imaginary and real parts, Z = − X − iδ . The choice of signs in this expression makes the dissipation parameter δ positive, while the spectral parameter X is positive when the frequency is blue-shifted from the plasmon resonance, and negative otherwise. For the sake of reference, we give here the expressions for X and δ for a metallic nanosphere in the Drude model,
1 (ε + 2 εa ) X = 3 0 Rm 3εaω p
32
where
ε 0 is the
(ε + 2ε ) (ω − ω s ); δ = R13 0 3ε ωa m a p
32
intersubband dielectric constant of the metal,
dielectric constant,
ωp
is the metal plasma frequency,
surface plasmon frequency, and
ωs =
εa
γ 2
, (5) is the ambient
ωp ε 0 + 2εa
is the
Rm is the nanosphere’s radius.
The spectral dependence of X and δ for silver is illustrated in Fig. 2. The most important feature in the figure is that in the yellow-red region of visible light, the real part of the polarizability greatly exceeds its imaginary part. Their ratio has the meaning of the quality factor of the surface-plasmon resonance Q ,
5
Q= X δ
(6) This factor shows how many times on the order of magnitude the amplitude of the local field in a vicinity of a resonant monomer exceeds that of the exciting field. The fact that for many metals Q may be large (as large as ≅ 10 ) plays an important role in the theory since the enhancement of the optical responses is a resonant phenomenon, and strong dissipation would completely suppress it. 2
Fig. 2. Wavelength dependence of the spectral parameter
X
and the dissipation parameter
δ.
To introduce the spectral representation,5,6 we will rewrite Eq.(3) as one equation in the 3N-dimensional space. To do so, we introduce 3N-dimensional vectors
) , whose projections give the physical vectors
d ), E ( 0 ) ,
(iα d ) = d iα , (iα E ( 0 ) ) = E i(α0 ) ,
.
(7)
Equations (3) then acquire the form
(Z + W ) d ) = E (0) )
, where W is the dipole-dipole interaction operator with the matrix elements
( )( )
(8)
rij rij 1 α β (iα W jβ ) = δαβ − 3 rij2 rij3 , i ≠ j i= j. 0, , (9) We introduce the eigenmodes (plasmons) | n ) ( n = 1,....,3N ) as the eigenvectors of the W-operator,
6
W n) = w n n) ,
(10) where w n are the corresponding eigenvalues. Practically, the eigenvalue problem (10) can be solved numerically for any given cluster. Having done so, one can calculate the Green’s function 3N
G iα , jβ = ∑
(iα n )( jβ n ) Z + wn
n =1
, (11) which carries the maximum information on the spectrum and linear response of the system. 4. Linear Optical Responses The polarizability of a cluster or finite volume of a composite eigenmodes ν are expressed in terms of G as
α
and its density of
α αβ = ∑ G iα , jβ , ρ = ∑ G iα ,iα i, j
i, j
, (12) where summation over repeated vector indices is implied. The dielectric constant of a cluster (composite) is given by (13) where ε h is the dielectric constant of a host, V is the volume occupied by the cluster (composite) and
α =
1 3N
∑G
iβ , jβ
is a polarizability of a monomer
i, j
in the cluster (composite). Below we will consider results of numerical computations using Eqs.(13) and (14) and will compare them to some analytical predictions. First, let us consider scaling predictions. For this purpose we have to invoke a large magnitude of the quality factor of the optical resonance (6), Q >> 1 . In this case,
(
a dependence of type (4) becomes F ( X ) ∝ R0 X a dependence for
α αβ ( X )
and
ρ( X )
the same scaling,
7
3
)
γ
. We have introduced5,6 such
and argued that the two quantities have
Im α (X) ≅ Im ρ(X) ≅ R03 R 03 X
where
do
d o −1
, (14) is an index that we called the optical spectral dimension. We have also
argued that the physical range of
d o is 1 ≥ d o ≥ 0 .
The strong localization has been essential for the derivation of Eq.(12). It implies that all eigenmodes (at least all contributing eigenmodes) of a cluster are strongly localized The strong localization, as discussed by Alexander17, means that for any given frequency parameter X there exists only one characteristic length L X of these eigenmodes playing the role of simultaneously their wavelength and their localization length. Using scale invariance arguments, we have shown5,6 that L X should scale as
L X ≅ R R 03 X
d o −1 3− D
. (15)
We have subjected the scaling predictions of Eqs. (14) and (15) to an extensive comparison with the results of large-scale computations.7 The results were quite unexpected. One of those, a polarizability and eigenmode Fig. 3. Numerically obtained polarizability (upper panel) density for cluster-cluster and density of eigenmodes (lower panel) for clusteraggregates (CCA), is shown in cluster aggregates. Fig. 3. The conclusion that one can draw from the figure is that neither the polarizability, nor density of eigenmodes scale. Interestingly enough, they still appear to be quite close to each other, supporting the conclusion of Refs.5 and 6 that all eigenmodes of a fractal cluster contribute (almost) equally to its optical absorption. Similar results have been obtained7 for other types of clusters.
8
Another relation to check is that of Eq.(15). First, one has to formulate how to calculate the localization radius We use the definition of Ref.7,
∑ρ L = ∑ρ n
LX
n
n
n
, where Ln = ∑ r (n iα ) iβ
2 i
where
2
2 − ∑ ri (n iα ) iβ
[
2
,(16)
ρ n = ( X − wn ) + δ 2
]
2 −1
,
Ln is the localization radius of a given eigenmode, and L X is the localization radius at a given frequency. The computed dependence of L X is shown in Fig. 4. As one can clearly see, there is no scaling in these data too. This finding is in contradiction with the conclusion of Ref.18 (precision of our calculations is much higher than that of Ref.18). Evident failure of the scaling implies that at least one of the assumptions as a function of the spectral parameter lying at its foundation is incorrect. Because we used the same model (dipole-dipole) for both the scaling theory and the numerical computations, the nonapplicability of the model to system is out of question. In our consideration we consistently used high values of Q , so that the condition Q >> 1 is also satisfied. The only cause of the failure of scaling appears to be the strong localization assumption. Fig. 4. Localization (coherence) radius
LX X.
Now we will compare the results of our theory to a mean field approximation known as Maxwell Garnett formula (or an equivalent Lorentz-Lorenz formula), see, e.g., Ref. 13. This comparison is shown in. Fig. 5. As we see, the mean field theory gives a poor description of actual dielectric constant, especially in the region of the resonant absorption of the inclusions, where ε p < 0 . Undoubtedly, there should be reasons for the failure of both scaling theory and mean field theory. As we understand now, two interrelated phenomena can be
9
blamed for this failure. These are inhomogeneous localization of eigenmodes and
giant fluctuations of the local fields. In the case of the inhomogeneous localization there are eigenmodes of all localization radii from the minimum distance between monomers (inclusions) R0 to the total size of the system Rc . Both of these two extremes render the scaling theory inapplicable. Obviously, the strong fluctuations contradict to the basic assumption of the mean field theory.
5. Inhomogeneous Localization of Eigenmodes 1
C CA, X> the typical size of the scattering inhomogeneities) sees almost homogeneous medium and propagates almost freely. In contrast, a short wave is strongly scattered from inhomogeneities with sizes on the order of λ (strongly here means that the scattering length itself is on the order of λ ). Thus, there exists the mobility edge, i.e., a frequency above which waves are localized and below which they propagate. This logic leading to the existence of the mobility edge is obviously inapplicable to fractals. They are self-similar systems and, therefore, do not possess any characteristic scattering length. For any eigenmode wavelength λ , there always exist inhomogeneities of the sizes comparable to λ . This may suggest that all the eigenmodes are strongly localized, as assumed in Refs.5 and 6. However, the result presented above and all numerical modeling of Refs.7, 8, 19, and 20 have shown that the strong localization is not the case. In actuality, the inhomogeneous localization8 takes place, where the eigenmodes in a wide range of the localization radii coexist at any given frequency. To examine statistical properties of an eigenmode distribution, we introduce the distribution function P ( L , X ) , which is the probability density that an eigenmode at a given
X has the localization radius of L , P ( L, X ) =
∑ δ (L − L )δ (X − w ) n
n
n
, (17) We show this distribution calculated for CCA clusters in Fig. 8 and for random lattice gas (RLG) composites in Fig. 9.
12
The most conspicuous feature of the distribution of Fig. 8 is its very large width. This width extends from almost the total size of the system Rc to some
100 L 10 1 10
minimum cut-off size l X
1 P+ L,X/0.1
function of frequency ω ( X ) . The cutoff is clearly seen in Fig. 8 where it is also indicated in the lower panel by a thick dotted line. The distribution width is so large that its characterization of the by a single dispersion relation L X [see Eq. (16)] is absolutely insufficient. For most of the spectral region, the cut-off length l X by magnitude is intermediate between the maximum and minimum scales, Rc and R0 . This, along with
102 103 10 4 3 10 102 «X« 0.1
that is a
1 10
CCA N = 1500 L 100
the self-similarity of the clusters, suggests that l X scales with X , i.e.,
10
λ
l X ∝ X . Indeed, Fig. 8 supports the 1 4
10
3
10
2
10
possibility of such a scaling with the corresponding index λ ≈ −0.25 . This illustrates general property of the inhomogeneous localization of eigenmodes for fractal (self-consistent) clusters and composites. A different situation exists for nonfractal composites, as one can see with an example of the RLG shown in Fig. 9.
«X« 0.1
1
Fig. 8. Localization-length
10
distribution
P ( L, X ) of eigenmodes for CCA clusters ( N = 1500 ). The position of the lower X cutoff is qualitatively illustrated by the dashed bold line.
The distribution for X ≥ 0.1 is similar to that of CCA (Fig. 8) characteristic of inhomogeneous localization. The major distinction from Fig. 8 is that the distribution in Fig. 9 shows the complete delocalization of the eigenmodes for
X ≤ 0.01 that appears in a narrow range. Such a delocalization is expected for the low- X part of the spectrum, i.e., at frequencies close to the plasmon resonance
13
of the individual inclusions (monomers). In contrast, there is no such delocalization for fractal (CCA) clusters, as seen in Fig. 8.
100 L 10 1 100 10 P+ L,X/ 1 0.1 102 103 10 4 3 10 102 «X« 0.1
6. Giant Fluctuations of Local Fields and Enhancement of NonRadiative Photoprocesses The picture of the intensities in any individual eigenmode (see Fig. 7) shows very large random changes of the intensity from one monomer to another, i.e., fluctuations in space. When a cluster is subjected to an external exciting radiation, its response is due to the excitations of eigenmodes. Therefore, we may expect that the eigenmode fluctuations will cause strong fluctuations of the local fields at individual monomers. We have investigated this phenomenon in Ref.21.
1 10
RLG N = 1500 L 100
10
The local field at an ith monomer is expressed in terms of the Green’s function (12),
1 104 103 102 0.1
«X« 1
10
E iα = Z ∑ G iα , jβ E β( 0 )
Fig. 9. Same as in Fig. 8 but for a random lattice gas (RLG) composite.
enhancement coefficient G i distribution function
jβ
. (18) Then the local field-intensity for an ith monomer and the corresponding
P (G ) are defined as
14
Gi =
2
Ei E
(0) 2
, P (G ) =
1 N
∑ δ (G − G ) i
i
.
(19)
We introduce also the n th moment of this distribution:
Mn = Gn
∫ P (G )G
=
n
dG
. (20) By its physical meaning, M n is the enhancement coefficient of an n th -order nonradiative (i.e., without emissions of photons) nonlinear photoprocess. If, for instance, a molecule is attached to a monomer of a cluster, then M n shows how many times the rate of its n -photon optical excitation exceeds such a quantity for an isolated molecule. A similar n , R 30 estimate is valid for the enhancement 2 , 0 .0 0 0 1 of a composite consisting of the 3 , 0 .0 0 0 1 nonlinear matrix and resonant 1 , 0 .0 0 1 inclusion clusters. 2 , 0 .0 0 1 The spectral dependencies of M n for different combination of the degree of nonlinearity n and the dissipation parameter δ are shown in Fig. 10. The data in this figure are scaled by
1 , 0 .0 1 2 , 0 .0 1
the factor R0 δ Q , with the resonant quality factor given by Eq. (6) The most remarkable feature of Fig. 10 is an almost perfect collapse of the data into a universal curve in an intermediate region of X for the case of very low dissipation 3
3
R 0X
Fig. 10. Normalized enhancement factors
G n δ Q − 2 ( n −1)
as functions of
X
for
− 2 ( n − 1)
( R0 δ < 0.01 ). Moreover, this curve is actually close to a straight line in the intermediate region indicating a scaling behavior of M n ( X ) . We conjecture CCA clusters for the values of shown.
this scaling as the dependence
δ
and
3
n
M n ≅ Q 2 ( n −1 ) M 1 .
15
For the first moment
M 1 we have previously obtained5,6 the exact relation
M 1 = ( X 2 + δ 2 ) Im α δ . Because the absorption Im α does not scale in X (see Sec.4), the enhancement coefficient M n should not scale either. However, the dependence Im α ( X ) in the intermediate region of X is flat (see Fig. 3). Therefore, for Q >> 1 the apparent scaling in X takes place with a trivial index of 2n , Mn ≅ Q
2 n −1
X 2n X Im α = 2 n −1 Im α ∝ X 2 n δ ,
(21)
in agreement with Fig. 10. The major result of Ref. 21, given by Eq.(21) is that the excitation rate of a nonradiative n th -order nonlinear photoprocess in the vicinity of a disordered cluster is 2 n −1
. This quantity can be understood resonantly enhanced by a factor of M n ≅ Q qualitatively in the following way. For each of the n photons absorbed by a resonant monomer, the excitation probability (rate) is increased by a factor of
≅ Q 2 (proportional to the local field intensity), therefore the total rate is increased 2n by a factor of ≅ Q . However, the fraction of monomers that are resonant is −1 2 n −1 small, ≅ Q . Consequently, the resulting enhancement factor is M n ≅ Q , in agreement with Eq. (21). For instance, for silver in the red spectral region Q ≅ 30 (see, e.g., Ref. 22), so each succeeding order of the nonlinearity gives enhancement by a factor of
Q 2 ≅ 1000 . We emphasize that the origin of this enhancement is the highquality optical resonance in the monomers modified by the cluster. Among interesting effects related to the enhanced non-radiative excitation, we mention one, the selective photomodification of silver clusters.23,24 We have considered above the moments (averaged powers) of the local fields. Now we consider another characteristic of the fluctuations, the distribution function P ( G ) (19) of the local-field intensity. Because the change of the minimum scale
R0 implies the change of the local fields, the distribution function is likely to scale in some intermediate range,
16
P(G ) ≅ G ε , (22) Due to scale invariance, the index ε does not depend on the minimum scale R0 . Consequently, ε does not depend on frequency (the spectral parameter X ) either. A simple model that allows one to calculate the scaling index ε is the binary approximation.5,6 In this approximation, the eigenmode is localized at only a pair of the monomers. In this case we have found21 the enhancement factor G i for a pair separated by a distance field,
G i (r , Θ ) =
r , located at an
angle
δ 2 sin 2 Θ
(X − r )
−3 2
+δ2
+
Θ to the direction of the exciting
δ 2 cos 2 Θ
( X + 2r )
−3 2
+δ2
.
(23)
Using this, we calculate the distribution function as
P (G ) = ∫ δ (G − G i (r , Θ )) C ( r ) d 3 r
.
(24) D −1
Here C ( r ) ∝ r is the density correlation function of the cluster. Taking into account those large values of G are of interest, we obtain from Eq. (24) that
3 R X 0 0
CC A 0.5 1 3
P(G ) ∝ G
(25) Thus, in the binary approximation we obtain a universal scaling index of − 3 2 . Surprisingly, this index does not depend on the cluster’s dimension D . Its value is determined merely by the vector nature of the fields.
R 30 X
RW
0
It is interesting to compare both the scaling prediction and the calculated value of the index ε = − 3 2 with the numerical results. These are shown in Fig. 11 for CCA clusters and for random walk (RW) clusters. The main
0.5 1 3
Fig. 11. Distribution function of the local field P (G ) intensity calculated for
R 03δ = 0.001 ,
− 32
for the values of
X
shown. The data on the upper panel are for CCA clusters and on the lower panel for random walk (RW) clusters.
17
feature is an unusually wide distribution. The local intensities are on the order of the exciting intensity ( G = 1 ), as well as three orders of magnitude smaller or greater. This feature is referred to as giant fluctuations of the local field. The regions of high intensity are responsible for enhanced nonlinear responses. They have also been observed directly with the scanning photon-tunneling microscope.25 The value of the index ε is indeed almost independent from the frequency (parameter X ), as expected from the scaling theory. Interestingly enough, these values ( ε = 1.45 for CCA and ε = 1.44 for RW) are quite close to the prediction of the binary theory ( ε = − 3 2 ). This agreement is unexpected because there are no grounds to believe that the binary theory is applicable in a wide range of frequencies. 7. Chaos of Eigenmodes The eigenmode equation (10) has the same form as the quantum-mechanical Schrödinger equation. In quantum mechanics it is not uncommon that highlyexcited states or states of complex systems possess chaotic behavior (see, e.g., Refs. 26,27). Similar situation one may expect for eigenmodes of large disordered clusters and composites. The extreme sensitivity of the individual eigenmodes to a very small change of their frequency that is discussed in Sec. 5 and illustrated by Fig. 7 is a direct indication of such chaos. Even more than individual eigenmodes, statistical properties of chaotic eigenstates are of great interest. The giant fluctuations of the local fields discussed above in Sec. 6 provide one of the statistical descriptions. In this section we will consider spatial correlations of the chaotic eigenmodes.28,29 A principal property that distinguishes this problem from quantum-mechanical chaos is a long-range nature of the dipole-dipole interaction. A similar tightbinding problem of quantum mechanics (Anderson model) is usually formulated with only next-neighbor hopping. In studied quantum-mechanical problems, chaotic quantum states do not possess long-range spatial correlations.27,30 The long ranged interaction on one hand tends to induce the long-range spatial correlations. On the other hand, it may tend to establish a mean field, suppress fluctuations, and eliminate the chaos. As we demonstrate below in this section, either of those trends may dominate, depending on the system geometry and spectral region. We expect that chaos is the most pronounced in clusters and composites with fractal geometry. The rationale for it is the following. A mean field is established and spatial chaos is eliminated when the correlation range of eigenmodes exceeds a characteristic size of the density variations in the system. However, fractal (self-
18
similar) geometry implies that the system repeat itself on all spatial scales and, consequently, there exists no such characteristic spatial scale. This is a prerequisite for the coexistence of chaos and long-range correlations. To characterize the spatial correlations of eigenmode amplitudes, we introduce the amplitude correlation function (also called dynamic form factor),
19
r 10 100 10 0.01
1 10 1 0.1 «X« 0.01 0.001 0.000
104
«X«
1
102
10
1
100 10 1 0.1 C+r,X/ 0.01
S+r,X/
0.01 10
r
r
100
100
10
10
1
1
«X«
104 103 102 0.1
1
Fig. 12. Dynamic form factor
10 4 10 3 10 2 0.1
10
S (r , X )
Fig.
for
The
S (r , X ) ≤ 10 −4
function
is removed.
plotted
log[10 4 S (r , X ) ] sgn[S (r , X )] .
Intensity
1
correlation
10
«X«
function
for CCA composite
over ensemble of 300 systems. The upper panel is the function plotted in the triple-logarithmic scale and the lower panel is the corresponding contour map.
S (r , X ) . To obtain it, a small region
of the plot for
13.
C (r , X ) calculated ( N = 1500 ) averaged
CCA composite (the number of inclusions N = 1500 , averaged over an ensemble of 300 composites. The upper panel shows a 3dimensional representation of the function, while the lower panel is the corresponding contour map. The vertical scale is pseudo-logarithmic to show simultaneously positive and negative values of
r
is The
horizontal scales are logarithmic.
20
S αβ (r, X ) =
(
) ∑ (iα n)( jβ n ) (r − rij ) δ (X − wn )
1 Im ∑ G iα , jβ δ r − rij = π i, j
n,i, j
(
)
This function gives the correlation factor of amplitudes iα n and
. (26)
( jβ n ) of two
eigenmodes with polarizations α and β at two points separated by a spatial interval of r at an eigenvalue (“frequency”) of X , averaged over ensemble of the systems. Similarly, we introduce a correlation function of the intensities of eigenmodes, C (r , X ) =
2 2 ∑ (iα n ) ( jβ n ) δ (r − rij ) δ (X − w n )
n ,i , j
. (27) In a similar way, this function yields a correlation of two eigenmode intensities,
(iα n )
2
(
and jβ n
). 2
We have calculated28,29 the above-defined correlation functions for an ensemble of CCA clusters (composites) with N = 1500 monomers (inclusions). The corresponding result for S (r , X ) =
1 S φβ (r , X ) is shown in Fig. 12. We see a 3
developed pattern of irregular, chaotic correlations in almost the whole spatialspectral region. The landscaped observed in Fig. 12 well deserves the name of a “devil’s hill”, where narrow regions of positive and negative correlations are interwoven, resulting in a turbulence-like pattern. This chaotic behavior is indeed a reflection of the chaos of individual eigenmodes. However, this chaos is in some sense stronger, because it is in a quantity averaged over an ensemble of statistically independent composites (consequently, it is fully reproducible). The deterministically chaotic pattern of Fig. 12 is likely to depend on a specific topology of the composite (CCA clusters). The straight lines shown in Fig. 12 are given by the binary-ternary approximation, see the discussion of Fig. 14 below. To distinguish between fluctuations of phase and amplitude in the formation of the devil's hill in Fig. 12, we will compare it to the second-order correlation function C (r , X ) shown in Fig. 13. We see that that this function differs dramatically from the dynamic form factor. The relief in Fig. 13 is very smooth, in contrast to Fig. 12. This implies that the devil's hill is formed due to spatial fluctuations of the phases of individual eigenmodes, while their amplitudes are smooth functions in spacefrequency domain. Another important feature present in Fig. 13 is the long range of the correlation. The correlation decay in r is indeed weaker than any power.
21
As we discussed above in this section, self-similarity (fractality) of the CCA composite is of principal importance for the coexistence of chaos and long-range correlations of the eigenmodes. To emphasize this point, we show in Fig. 14 the results for the dynamic form factor S ( r , X ) for RLG composites. We see that for extremely large eigenvalues, X ≥ 1 , the form factor S ( r , X ) has a quasi-random structure that is reproducible under the ensemble averaging. This structure is similar to what is observed for CCA composites (cp. Fig. 12). In the middle of the spectral region, 1 ≥ X ≥ .003 , the form factor is dominated by a few branches of excitation. The strongest one is marked on the lower panel by a solid white line
X = −2r −3 that is the corresponding branch of binary approximation5,6. In this approximation, each eigenmode consists of two excited regions (“hot spots”) and the form factor is given by
(
) (
)
(
)
(
)
S (r , X ) = δ ( r ) ρ ( X ) + f ( r )[δ X + 2r −3 − δ X − 2 r −3 + 2δ X − r −3 − 2δ X + r −3 ],
r 10
1
100 10 0.01
(28) where f (r ) is a smooth distribution of the inter-monomer distances,
10 1 0.1 «X« 0.01 0.001 0.0001
f (r ) = N δ (r − rij ) . The weaker
S+r,X/
branch, marked by the dotted line
0.01 10
X = −(1 + 57 )r −3 , is due to the ternary excitations, i.e., eigenvectors consisting of three hot spots. At X ≈ 0.003 , we see a sharp transition to delocalization, where the positive-correlation region becomes uniform, spreading over most of the system. This transition occurs when the correlation length becomes comparable to the size of the system
r 100
10
Rc ,
4
3
10
2
10
0.1
1
10
at
X ≅ Rc−3 ∝ N .
Consequently, this transition is mesoscopic, i.e., it phases out as size of the system becomes very large.
1 10
i.e.
«X«
Fig. 14. The same as in Fig. 12, but for RLG composite.
22
This is in contrast to an Anderson transition where local density of scatters is the determining parameter. This difference from Anderson localization is due to the long-range interaction in our case. As a result, most of the eigenvectors are not propagating waves, but change from binary/ternary excitations to delocalized surface plasmons as X decreases. To briefly conclude this section, chaos of eigenmodes is present for both conventional and fractal geometries of the composites. However, fractal composites demonstrate this chaos in the whole spectral region, while conventional composites do only in the extremes of the spectrum. For conventional geometries (RLG composites in particular), the eigenmodes are dominated by binary/ternary excitations with a mesoscopic transition to uniform surface plasmons near plasmon resonance of the inclusion particles (monomers), i.e. at X → 0 . In the next section we will study the role of the fluctuations and chaos of eigenmodes on the nonlinear radiative photoprocesses.
8. Enhancement of Radiative Photoprocesses Among enhanced radiative photoprocesses, one of the best studied experimentally is surface-enhanced Raman scattering (SERS).9 The intensity of Raman scattering for molecules adsorbed at rough surfaces or colloidal-metal particles is known to be 6
greatly enhanced, by a factor of up to 10 . There are two limiting cases of SERS. In the first case, the Raman shift is very large, much greater than the homogeneous width of monomer’s absorption spectrum. In this case, as we have shown22 the exact expression of the SERS enhancement factor
G
RS
G RS can be obtained as
X2 +δ2 = Im α ≈ Q X Im α δ .
(29) This result predicts that the on the order of magnitude the enhancement is the same (by the factor of ≅ Q ) as for non-radiative photoprocesses of the second order. Qualitatively, we may interpret this result in the following way. For a large frequency shift, the outgoing photon is out of resonance and does not considerably interact with the system. Correspondingly, the entire enhancement is due to the local field of the absorbed photons, the same as for non-radiative processes [cp. the discussion after Eq.(21)]. Much more dramatic enhancement is found22 for the second case, namely that of Raman frequency shifts smaller than monomer’s linewidth. This situation is
23
characteristic of the most typical and interesting cases of SERS. We have shown22 that in this case there exists an approximate expression for the enhancement coefficient
G RS of SERS, 3
1 1 X G ≈ Q 3 R03 X Im α = R03 X Im α 2 2 δ . (30) RS 3 ≅ Q , where the physical interpretation is the In this case we have G 2 following. Enhancement by ≅ Q RS 3 -3 RS
G
(R 0 )
is due to the first power of the localfield intensity. Another power
0 .5 (R 30 |X |) 4 Im
≅ Q 2 appears because the outgoing photon is not emitted freely, but in the resonant environment. Finally, one power of Q vanishes because only a small fraction ≅ 1 Q of the monomers is resonant to the exciting radiation.
|X | 3 .3
Fig. 15. Scaled enhancement coefficient of SERS from silver colloid clusters in comparison with the prediction of Eq.(30). The theoretical dependence is calculated for CCA clusters.
the dependence on
We compare the functional dependence predicted by Eq. (30) with the numerical simulation in Fig. 15. As we see, this equation describes the numerically found dependencies quite well, especially
X. Equation (30) shows that the Raman scattering is enhanced for an adsorbed molecule by a factor of
T h e o ry E xp e rim e n t
G RS ≅ Q 3 . For noble
Fig. 16. Theoretical and experimental spectral (in terms of wavelength) dependencies of the enhancement coefficient
G RS
for silver colloid clusters.
24
metals in the red region of visible light, we have Q ≅ 30 − 100 , so the predicted enhancement is very large,
G RS ≅ 10 4 − 10 6 . This explains the range of enhancements found
experimentally.9 The comparison22 of theoretical spectral dependence with the experiment is shown in Fig. 16. As we can see, the theory explains the experimental observations quite well. Now, we will very briefly discuss coherent, or parametric, nonlinear photoprocesses. These processes are due to nonlinear wave mixing. One of the most interesting is frequency-degenerate four-wave mixing (a third-order process), responsible for such an interesting effect as phase conjugation. We have found the corresponding enhancement coefficient in Ref.10 in the binary approximation and Ref.31 numerically. Physically, the enhancement of an n th -order parametric photoprocess can be estimated in the following simple way. The enhancement of the amplitude of each of the n + 1 participating photons (including the emitted photon) is ≅ Q . This n +1
yields the enhancement of the amplitude of the process as ≅ Q . For any coherent process, the amplitude, not probability, is the quantity to average. The averaging is done by multiplying by the fraction of resonant monomers, which is
≅ Q −1 , leading to mean amplitude as ≅ Q n . Finally, the amplitude should be (n) ≈ Q 2 n . More precise theory squared, yielding the enhancement coefficient G gives
G ( n ) ≈ Q 2 n ( X Im α ) . 2
For
(31)
n = 3 , this reduces to
2 G ( 3) ≈ ( X δ ) ( X Im α ) 6
(32) in agreement with the corresponding result of Ref.31. The comparison of Eq. (32) to the numerical calculations shown in Fig. 17 supports this result. A remarkable feature of Eq. (32) is that the predicted Fig. 17. Scaled enhancement coefficient for third-order degenerate parametric process, calculated for CCA.
25
enhancement is quite large. For the realistic dielectric parameters of silver in the
≅ 10 − 10 . The visible range, Q ≅ 10 − 30 , and correspondingly G 32 experimental investigation has indeed found a very strong enhancement for the ( 3)
phase conjugation, with
4
7
G ( 3 ) ≅ 10 6 , confirming the theoretical predictions.
The enhancement discussed above in this section is calculated for nonlinear-optical molecules adsorbed at the surface of monomers of a fractal cluster. Below we will consider another important system, namely Maxwell Garnett composite modeled as a RLG. It will be assumed for simplicity that only inclusion particles possess optical nonlinearity, while the host is optically linear. We will compute the nonlinear polarizability from the spectral theory and also compare it with the results of meanfield theory of Ref. 13. For the linear polarization of exciting radiation, the third-order nonlinear (hyper)polarizability of an isotropic medium is completely characterized by one constant,
χ (3) , that enters the expression for induction
D = εE + χ ( 3 ) E E 2
.
(33)
9. Concluding Remarks We have considered a variety of the photoprocesses mediated by disordered clusters and composites. We employed models with both fractal (selfsimilar) and conventional geometries to simulate different existing systems. The remarkable feature of the disordered clusters and composites is giant fluctuations of the local fields that bring about their enhanced nonlinearoptical responses. Namely, the enhancement is due not only to the high averaged value of the local fields, but principally due
26
to their fluctuations in space, from one monomer (inclusion particle) to another. Nonlinearity, causing a higher power of the local fields to be an acting parameter, enhances the effects of these fluctuations. Therefore, the enhancements are found to increase dramatically with the order of nonlinearity. The chaos of the local fields bears many similarities to quantum chaos, but differs due to the long range of the dipole interaction. Not only individual eigenmodes are chaotic, but also their spatial correlation factors. These chaos and fluctuations are responsible for the dramatic failure of the mean-field theory to describe the nonlinear polarizability of composites. Finally, many of the effects predicted have been verified experimentally, and we have mentioned some of them. However, many very interesting effects are not discussed due to the space limitations. Among them, we recognize the enhanced laser production of plasma by colloidal-metal clusters.33 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
J.C.Maxwell Garnett, Philos. Trans. R. Soc. London 203, 385 (1906). H.A.Lorentz, Theory of Electrons (Dover, New York), 1952. D.A.G.Bruggeman, Ann. Phys. (Leipzig), 24, 636 (1935). V.M.Shalaev and M.I.Stockman, ZhETF 92, 509 (1987) [Translation: Sov. Phys. JETP 65, 287 (1987)]. V.A.Markel, L.S.Muratov and M.I.Stockman, ZhETF 98, 819 (1990) [translation: Sov. Phys. JETP 71, 455 (1990)]. V.A.Markel, L.S.Muratov, M.I.Stockman, and T.F.George, Phys. Rev. B 43, 8183 (1991). M.I.Stockman, L.N.Pandey, L.S.Muratov and T.F.George, Phys. Rev. B 51, 185 (1995). M.I.Stockman, L.N.Pandey and T.F.George, Phys. Rev. B 53, 2183 (1996). M.Moskovits, Rev. Mod. Phys. 57, 783 (1985). A.V.Butenko, V.M.Shalaev and M.I.Stockman, ZhETF 94, 107 (1988) [translation: Sov. Phys. JETP, 67, 60 (1988)]. T.A.Witten and L.M.Sander, Phys. Rev. Lett. 47, 1400 (1981). G.L.Fischer, R.W.Boyd, R.J.Gehr, S.A.Jenekhe, J.A.Osaheni, J.E.Sipe and L.A.Wellerbrophy, Phys. Rev. Lett. 74, 1871 (1995). J.E.Sipe and R.W.Boyd, Phys. Rev. B 46, 1614 (1992). K.Ghosh and R.Fuchs, Phys. Rev. B 38, 5222 (1988).
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15. R.Fuchs and F.Claro, Phys. Rev. B 39, 3875 (1989). 16. D.J.Bergman and D.Stroud, In: Solid State Physics, v.46, p.148 (Academic Press, Boston, 1992). 17. S.Alexander, Phys. Rev. B 40, 7953 (1989). 18. V.M. Shalaev, R.Botet and A.V.Butenko, B 48, 6662 (1993). 19. M.I.Stockman, L.N.Pandey, L.S.Muratov and T.F.George, Phys. Rev. Lett. 75, 2450 (1995). 20. M.I.Stockman, L.N.Pandey, and T.F.George, Phys. Rev. B 53(5), 2183 (1996). 21. M.I.Stockman, L.N.Pandey, L.S.Muratov and T.F.George, Phys. Rev. Lett. 72, 2486 (1994). 22. M.I.Stockman, V.M.Shalaev, M.Moskovits, R.Botet, and T.F.George, Phys. Rev. B 46, 2821 (1992). 23. A.V.Karpov, A.K.Popov, S.G.Rautian, V.P.Safonov, V.V.Slabko, V.M.Shalaev and M.I.Stockman. , Pis'ma ZhETF 48, 528 (1988) [Translation: JETP Lett. 48, 571 (1988)]. 24. Yu.E.Danilova, A.I.Plekhanov and V.P.Safonov, Physica A 185, 61 (1992). 25. D.P.Tsai, J.Kovacs, Z.Wang, M.Moskovits, V.M.Shalaev, J.S.Suh and R. Botet, Phys. Rev. Lett. 72, 4149, 1994. 26. F.J.Dyson, J. Math. Phys. (N.Y.) 3, 1189 (1962). 27. M.V.Berry, J. Phys. A 10, 2083 (1977). 28. M.I.Stockman, Phys. Rev. Lett. 79, 4562 (1997). 29. M.I.Stockman, Phys. Rev. E 56, 6494 (1997). 30. V.I.Fal’co and K.B.Efetov, Phys. Rev. Lett. 77, 912 (1996). 31. V.M.Shalaev and M.I.Stockman, Physica A 185, 181 (1992). 32. A.V.Butenko, P.A.Chubakov, Yu.E.Danilova, S.V.Karpov, A.K.Popov, S.G.Rautian, V.P.Safonov, V.V.Slabko, V.M.Shalaev and M.I.Stockman, Z.Phys. D 17, 283 (1990). 33. M.M. Murnane, H.C.Kapteyn, S.P.Gordon, J.Bokor, E.N.Glytsis and R.Falcone, Appl. Phys. Lett. 62, 1068 (1993).
28
Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10 Fig. 11 Fig. 12 Fig. 13 Fig. 14 Fig. 15 Fig. 16 Fig. 17 Fig. 18
29
absorption, 16 amplitude correlation function, 19 analytical predictions, 8 binary approximation, 17 CCA clusters, 17 chaos, 1 cluster-cluster aggregates, 9 colloidal-metal clusters, 29 dielectric constant, 7 dipole-dipole interaction, 4 disordered cluster, 16 distribution, 13 distribution function, 14 dynamic form factor, 19 eigenmodes, 1 eigenvalue, 7 enhancement coefficient, 15, 16 fluctuations, 1 giant fluctuations, 1, 10, 18 Green’s function, 7 hyperpolarizability, 26 inclusion particles, 26 inhomogeneous localization, 1 local fields, 1 localization, 5 localization radius, 11 Maxwell Garnett, 3 mean-field theory, 1
mesoscopic, 26 metal nanocomposites, 2 monomers, 3 nanocomposites, 1 nanoparticles, 1 nanostructured materials, 1 nonlinear optical responses, 1 nonlinear photoprocess, 16 numerical computations, 7 optical polarizabilities, 1 optical properties, 1 parametric photoprocess, 25 plasmons, 1 Raman scattering, 23 random lattice gas, 4 random walk clusters, 18 resonant quality factor, 15 scaling, 4 scaling index, 17 self-similarity, 5 spatial correlations, 18 spatial fluctuations, 2 spectral representation, 6 spectral theory, 1 strong localization, 8, 9 surface-plasmon resonance, 6 susceptibilities, 1
