Random matrix approach to plasmon resonances in the random impedance network model of disordered nanocomposites
Random impedance networks are widely used as a model to describe plasmon resonances in disordered metal-dielectric and other two-component nanocomposites. In the present work, spectral properties of resonances in random networks are studied within the framework of the random matrix theory. We have shown that the appropriate ensemble of random matrices for the considered problem is the Jacobi ensemble (the MANOVA ensemble). The obtained analytical expressions for the density of states in such resonant networks show a good agreement with the results of numerical simulations in the wide range of metal filling fractions . A correspondence with the effective medium approximation is observed.
pacs:78.67.Sc, 73.20.Mf, 87.10.Hk
Introduction.—Disordered metal-dielectric nanocomposites form a type of optical metamaterials which are relatively simple in fabrication (Fig. 1(a)). Their geometries vary from a dielectric medium with metallic inclusions of a submicron size to a metallic medium with dielectric holes, depending on a metal fraction. Such systems demonstrate a lot of interesting optical phenomena assisted by surface plasmon resonances in metallic regions, namely surface-enhanced Raman scattering (SERS) Le_Ru_2009 (), high-harmonic generation 2001_Breit () and Purcell effect 2015_Carminati (). Various nonlinearities in such composites especially increase near the percolation threshold 2000_Sarychev ().
A number of classical models of the percolation theory are based on random impedance networks (Fig. 1(b)) which have been widely applied to study transport properties 1973_Kirkpatrick (); 1990_Clerc () and resonances 1992_Bergman (); 2000_Sarychev () in disordered nanocomposites. Fluctuations of local electric fields responsible for SERS have been considered in the framework of the random impedance network model 2000_Sarychev (); 1997_Brouers (); 1998_Brouers (), as well as the density of states (DOS) 1998_Jonckheere (); 2000_Albinet (); 2012_Murphy () and the optical absorption 1987_Koss (); 1993_Zhang (); 1995_Zhang (). Such models demonstrate a presence of the Anderson transition 1999_Sarychev (); 2017_Murphy () and possess multifractal properties of electric field distributions 1998_Jonckheere (); 2002_Gu (). However, the main part of the results is obtained numerically.
In the present paper, we propose to apply the random matrix theory for a unified description of the DOS in random impedance networks, which are widely used as a model of plasmon resonances in disordered nanocomposites 2000_Sarychev (). The random matrix theory has found numerous applications in different branches of physics. For example, in nuclear physics 1962_Dyson (), quantum chaos Haake (), description of the conductance of disordered channels Oxford_Handbook (); 2000_Mirlin (), coherent perfect absorbers 2017_Li (), and mechanical properties of disordered solids 2015_Beltukov (). The random matrix theory was also applied to study the statistical properties of financial markets and computer networks Oxford_Handbook (). Each of the mentioned problems has some important symmetries, which lead to different symmetry classes of random matrices (the so-called random matrix ensembles) 2008_Evers ().
Random impedance network model.—We consider a widely used quasistatic approach when the electric field is assumed to be curl-free (), and the electrostatic potential can be introduced such that . Since a characteristic size of inclusions is about tens of nanometers, this condition can be satisfied in THz, infrared and even visible optical region. It is well known that Maxwell’s equations are reduced to the equation for an eddy current within the quasistatic approach 1973_Kirkpatrick (); 1977_Webman (). For a given frequency , the current and the electric field are related by the material equation . The conductivity is related to the permittivity of the same region as LL8 (). The obtained equations are discretized on a mesh. For that reason, the square 2000_Sarychev (); 1996_Clerc (); 1998_Jonckheere (); 1977_Webman () and the simple cubic 2000_Albinet () lattices have been used. After a discretization, equations and transform to the first and the second Kirchhoff’s rules respectively. One can simultaneously represent both Kirchhoff’s rules as a linear system
with being the electric potential at site and being a complex conductance of the bond between sites and in a network with sites 1973_Kirkpatrick (). Diagonal entries are defined as .
Next, we briefly consider some of the simplest models. In the optical frequency range, the permittivity of a metal can be described with the aid of the Drude model , with being a plasma frequency of a metal. At the same time, the permittivity of dielectric regions can be taken as a constant . Then, a composite is replaced with a resonant -network, Fig. 1(c) 1989_Zeng (); 1995_Zhang (); 2016_OBP (). At low frequencies another model having a form of an -network can be introduced, Fig. 1(d). Such a model is used to consider transient responses in composites 1990_Clerc (); 2009_Korniss (); 2017_Aouaichia ().
In order to study properties of resonances, we reduce Maxwell’s equations to the eigenvalue problem 1979_Bergman_1 (); 1979_Bergman_2 (). In a generic two-component system, the conductance can be represented as
Matrices and are defined in the following manner. In the general case, we can assume that if sites and are connected by a metallic bond and if sites and are connected by a dielectric bond. The remaining off-diagonal elements of matrices and are zero. Diagonal elements are defined as , . Thus, matrices and represent discrete Laplacians defined on the corresponding metallic and dielectric subsets.
For certain frequencies , the linear system (1) has nontrivial solutions , which represent dielectric resonances in the network 1992_Bergman () corresponding to plasmon resonances of a composite. For a two-component system (2), eigenfrequencies can be found using the generalized eigenvalue problem 1998_Jonckheere ()
where eigenvalues are related to eigenfrequencies as
Eigenvalues and eigenvectors are determined by matrices and , which do not depend on the dielectric functions of constituents . As a result, the dielectric functions affect only eigenfrequencies , which are related to eigenvalues by Eq. (4). It is important to mention that matrices and are positive semidefinite Horn-book (), and thus for networks of an arbitrary geometry Ortega-book ().
Considerable efforts have been put to figure out an analytic description of the DOS in such networks within the framework of the random matrix theory (RMT) 1999_Fyodorov (); 1999_Fyodorov_1 (); 2001_Fyodorov (); 2003_Staring (). In the mentioned papers Gaussian ensembles of random matrices have been applied to describe resonances in long-range networks with a quasi-one-dimensional topology, which show no direct relation to the problem of plasmon resonances in two-dimensional and three-dimensional disordered nanocomposites.
In order to simplify the problem, we will consider a common model of a random network, which assumes that each bond in a lattice with a coordination number is metallic with probability or dielectric with probability 1998_Jonckheere ().
Density of states.—The matrices and are positive semidefinite, so they can be represented in the form and . There are different possibilities to choose the matrices and for the same matrices and . However, there is the most natural form of the matrices and , which is known as the incidence matrix in the graph theory Bollobaas (). In this case, the height of the matrix is the number of sites and the width of the matrix is the number of metallic bonds in the lattice. The nonzero matrix elements are and , where is the index of a bond and and are indices of sites connected by the -th bond. For each pair of and , the choice of and is arbitrary. The definition of the matrix is the same but for dielectric bonds. Therefore, the generalized eigenvalue problem (3) can be written in the form
In the above definition, the matrices and are sparse with a certain structure of nonzero elements. However, it does not play a crucial role for the DOS. Indeed, for any orthogonal matrices , , and , we can introduce matrices and , which leads to the generalized eigenvalue problem
with the same set of eigenvalues as for Eq. (5). As a result, one can assume that the DOS mostly depends on the correlations given by the form of Eq. (6) rather than the internal correlations of the matrices and 2015_Beltukov (). Thus, we assume that the matrices and are Gaussian random matrices. The sizes of the matrices are and respectively, where and are total numbers of metallic and dielectric bonds, and is a number of sites in the lattice. In this case, Eq. (6) defines the so-called Jacobi ensemble of the random matrix theory Forrester (). It is also known as the MANOVA ensemble since Eq. (6) has a special meaning in the multivariate analysis of variance (MANOVA).
For the Jacobi ensemble, the joint probability distribution of an ascending list of eigenvalues is
where is a nomalization constant 1963_Constantine (). The last product in Eq. (7) wanishes when . This leads to the level repulsion effect which is well-known for the Gausian orthogonal ensemble (GOE) and was also observed for random impendance networks 2017_Murphy (). However, eigenvalue probability density functions (i.e., DOS) for the Jacobi ensemble and for the GOE are different. For the Jacobi emsemble, it has the form Forrester (); 2013_Erdos ()
where the spectral edges are given by
In addition to eigenvalues defined by , there is a number of degenerate eigenvalues and . The relative number of eigenvalues is , and the relative number of eigenvalues is .
One of the fundamental properties of the generalized eigenvalue problem (3) is the homogeneity symmetry 1998_Jonckheere (): the DOS obeys the relation due to the equivalence of statistical properties of matrices and . It is obvious that the Jacobi ensemble satisfies this symmetry.
Comparison with numerical results.—First, we consider networks with a topology of the two-dimensional square lattice. This case is the most studied and widely addressed in the literature (see review 2000_Sarychev () and references therein). The corresponding density of states for the square lattice with different fractions of metallic bonds is shown in Figure 2(a)-(d).
At low filling fractions , the numerically obtained DOS demonstrates a presence of a rich structure with well-resolved resonant peaks. These peaks correspond to resonances of typical clusters which are formed by several metallic bonds embedded into a dielectric lattice – the so-called lattice animals 1998_Jonckheere (). For example, the most salient peak at corresponds to the dipole resonance of a single metallic bond surrounded by a dielectric environment. Potential distribution, in this case, is , which corresponds to a dipole in the two-dimensional electrodynamics 1996_Clerc (). In the dilute case with it is the only remaining peak. Nearby peaks are aligned symmetrically and correspond to resonances of two-bond clusters, and so on. Detailed maps of resonances of animals at the square lattice can be found in 1996_Clerc (); 1998_Jonckheere (); 2003_Raymond ().
Peaks near the edges of the resonance spectrum are associated with complicated clusters formed by many bonds. The probability of such cluster configurations to occur is low, thus, these peaks are much less pronounced than the central ones. Finally, at the very edges of the spectrum, the amount of resonances is exponentially small, because these resonances are associated with long linear chains of connected metallic bonds which arise with an exponentially small probability 1998_Jonckheere (). Such a behavior is referred to as Lifshitz tails after I.M. Lifshitz, who was the first one to describe analogous phenomena in vibrational spectra of binary harmonic alloys 1964_Lifshitz (). As seen from Figure 2, mentioned peaks are absent in the DOS given by the RMT approach. This originates from our neglect by the correlation between the matrices and . Indeed, the concept of a cluster loses its meaning in this case. As a result, Lifshitz tails are absent as well. They are replaced by resonance gaps at and .
At higher fillings , the resonant peaks are less pronounced because of a complication of the geometric structure, which causes typical clusters to be less expressed. Indeed, if is large enough than typical clusters are usually interact with nearby clusters and are unlikely to be positioned in a large gap filled with the dielectric bonds. As a result, the numerical DOS becomes smoother and more similar to the one given by the RMT approach, Fig. 2(b)-(c).
As the filling fraction increases, the bond percolation threshold is reached at some point (Fig. 2(d)). The percolation is a geometric phase transition, which means that at fillings an infinite metallic cluster is formed that connects the opposite sides of the system. Thus, an initially insulating system becomes a conducting one in the stationary () regime at . The percolation threshold can be obtained within the RMT approach as follows. Since resonances correspond to the poles of the conductivity of a system, a non-vanishing dc conductivity corresponds to a non-vanishing DOS at . Hence, 1992_Bergman (), which gives the RMT estimate of the percolation threshold
This result is well known and has a transparent physical interpretation. Indeed, the necessary condition for existence of a path connecting opposite sides of the network is that at least two of bonds which connect a site with its neighbors are metallic. Also, this estimate gives the exact value for the square lattice.
This result is the only analytically established percolation threshold for a lattice. Its derivation is based upon a special symmetry of the square lattice – the self-duality. This peculiar property also causes a mirror symmetry of the DOS under the transform , clearly seen in Fig. 2(a)-(d) 1971_Dykhne (); 1998_Jonckheere (). The DOS for the diamond lattice is shown for a comparison in Fig. 2(e)-(h). This lattice also has the coordination number , hence it is described by the same RMT curve. However, it is not self-dual, and, as a result, the corresponding symmetry of is absent. The percolation threshold in the diamond lattice also slightly differs from that of the square lattice and equals .
Next, we compare the results for lattices with the coordination number . The DOS for the simple cubic and the triangular lattices at different fillings is shown in Fig. 3. The whole situation is similar to the previous case with . However, there is a difference in frequencies of dipole resonances of individual metallic bonds, so that . It also corresponds to the frequency of the dipole localized plasmon resonance in a metallic sphere . This is in agreement with the fact that the dipole potential in the simple cubic lattice decreases with the distance as 1996_Clerc (). Resonances of other lattice animals differ as well, also due to different geometries of typical clusters and different probabilities of their occurrence. A map of the resonances of typical clusters in the simple cubic lattice can be found in 2000_Albinet (). The percolation thresholds in the simple cubic and triangular lattices are and , correspondingly, which is still close to the RMT prediction .
As was pointed out in 1998_Jonckheere (), some of the extensively degenerate eigenvalues of Eq. (3) with and do not correspond to resonances. A number of these non-physical eigenvalues is defined by a number of connected clusters formed by dielectric and metallic bonds, correspondingly 2009_Korniss (), and can be easily obtained for any particular implementation of a network.
Discussion and conclusions.—Let us also point out an interesting interplay between the results given by our RMT approach and by the effective medium approximation (EMA). The latter was introduced by D.A.G. Bruggeman as a self-consistent homogenization scheme for the evaluation of the conductivity of mixtures 1935_Bruggeman (); 1992_Bergman () and is widely applied to systems at finite frequencies 1998_Jonckheere (); 1973_Kirkpatrick (); 1987_Koss (); 1993_Zhang (); 1995_Zhang (). The main equation of the EMA on a hypercubic lattice reads as 1992_Bergman ()
where is an effective conductance of the lattice with randomly arranged bonds of conductances and . The above equation has an explicit solution which is non-vanishing over the interval , with given by exactly the same expressions as in the RMT approach: . Indeed, resonances of the system are poles of its conductance, and thus in a non-dissipating system and should be nonvanishing in the same spectral region. Some correspondence between the random matrix theory and the effective medium description in the case of Gaussian ensembles has been addressed in 1999_Biroli (); 2002_Semerjian (); 2003_Dorogovtsev (); 2017_Circuta ().
Predictions of the considered model are in a qualitative agreement with the results of recent experiments with lithographic networks 2015_Gaio () and disordered nanocomposite films 2014_Hedayati (). In particular, experimentally measured Purcell enhancement and absorption spectra demonstrate the presence of a broad maximum whose width depends on the metal filling , as well as the presence of the optimal filling which maximizes the absorption band.
To conclude, we have considered a description of resonances in random impedance networks based on the Jacobi ensemble of the random matrix theory. The obtained expressions satisfy all natural symmetries of the considered problem and demonstrate a good agreement with the results of numerical simulations as well as a correspondence with the effective medium approximation. A further development of the obtained description, e.g. a comprehensive study of level spacing statistics 2002_Gu_1 (); 2007_Lansey () and properties of eigenvectors can be of major interest in the area of Anderson localization 1999_Sarychev (); 2017_Murphy ().
Acknowledgements. We are grateful to V.I. Kozub, D.A. Parshin and D.F. Kornovan for fruitful discussions. The work is supported by the Russian Foundation for Basic Research (project no. 16-32-00359) and the “Dynasty” Foundation.
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