A two-dimensional disordered magnetic metamaterial
AA two-dimensional disordered magnetic metamaterial
Mario I. Molina
Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Dated: December 23, 2020)We examine the effect of resonant frequency disorder on the eigenstates and the transport ofmagnetic energy of a two-dimensional (square) array of split-ring resonators (SRRs). In particular,we focus on two kinds of binary disorder: an uncorrelated one (Anderson) and a correlated one,where tetramers units are assigned at random to lattice sites. This is a direct extension of therandom dimer model (RDM) in one-dimensional systems. We find that the absence of any resonantmechanism for transmission of two-dimensional plane waves (which do exist in the one-dimensionalcase) prevents the SRR array from exhibiting a net transport of magnetic energy. The mapping ofthe system to a tight-binding model shows that for SRR arrays, there exists a complete correlationbetween energy disorder and coupling disorder, with a site energy and coupling that are the ‘inverse’of the usual tight-binding model. The computed shape of the density of states suggests that ourSRR system is akin to an effective off-diagonal Anderson system.
Introduction . The recent ability to tailor materialproperties at will has lead to a whole class of artificial ma-terials, termed metamaterials, characterized by unprece-dented thermal, optical, and transport properties thatmake them attractive candidates for current and futuretechnologies. Among them, we have magnetic metamate-rials (MMs) that consist on an array of metallic split-ringresonators (SRRs) coupled inductively[1–3]. This type ofsystem can, for instance, feature negative magnetic re-sponse in some frequency window, making them attrac-tive for use as a constituent in negative refraction indexmaterials[4–7]. The usual theoretical treatment of suchstructures is an effective medium approximation wherethe composite is treated as a homogeneus and isotropicmedium, characterized by effective macroscopic param-eters. Of course, this approach is valid, as long as thewavelength of the electromagnetic field is much largerthan the linear dimensions of the MM constituents.One of the simplest two-dimensional MM model con-sists of a periodic square array of split-ring resonators(SRRs) lying on a common plane, where each resonatorconsists of a small, conducting ring with a slit (Fig1).Each SRR unit in the array is equivalent to a resistor-inductor-capacitor (RLC) circuit featuring self- induc-tance L , ohmic resistance R , and capacitance C builtacross the slit. If we assume a negligible resistance, eachunit will possess a resonant frequency ω = 1 / √ LC . Un-der this condition and in the absence of driving, the evo-lution equations for the charge Q n residing at the nth ring are given by dQ n dt = I n (1) L dI n dt + Q n C = − M (cid:88) m dI m dt (2)where M is the mutual inductance and the sum is re-stricted to nearest-neighbors of n . These equations canbe cast in dimensionless form as d dt (cid:32) q n + λ (cid:88) m q m (cid:33) + ω n q n = 0 (3) Figure 1. Two-dimensional split-ring resonator array. where q n denotes the dimensionless charge of the nthring, λ is the coupling between neighboring rings thatoriginates from the dipole-dipole interaction, and ω n isthe resonant frequency of the nth ring, normalized to acharacteristic frequency of the system, such as the aver-age frequency: (cid:104) w (cid:105) = (1 /N ) (cid:80) n w n . We assume thatthe magnetic component of any incident electromagneticwave is perpendicular to the SRRs’ plane, and that theelectric field of the incoming wave is transverse to theslits. Under these conditions, only the magnetic compo-nent of the incoming wave creates an electromotive forceon the rings, giving rise to an oscillating current in eachSRR and to an oscillating voltage difference across theslits. Also, it is a good idea to minimize electric dipole-dipole effects coming from the strong electric fields at theslits by a judicious placing of the SRRs in the commonplane so that the slit-to-slit distance is kept as large aspossible (Fig.1). The main drawback of the SRR arrayis the existence of large ohmmic and radiative losses. Apossible way to deal with this problem that has been con-sidered is to endow the SRRs with external gain, such astunnel (Esaki) diodes[8, 9] to compensate for such losses.The dimensionless stationary state equation is ob- a r X i v : . [ n li n . PS ] D ec tained from Eq.(3) after posing q n ( t ) = q n exp[ i (Ω t + φ )]: − Ω (cid:32) q n + λ (cid:88) m q m (cid:33) + ω n q n = 0 . (4)The frequency ω n can be changed by varying the capac-itance of the ring, which is accomplished by altering theslit width or by inserting a dielectric in the slit. For ahomogeneous array, ω n = 1.On the other hand, the topic of the effect of disorderon the stationary and transport properties of a discrete,periodic system, is an old one, but its importance has nowaned throughout the years given its fundamental im-portance in several fields. The most important result inthis area is Anderson localization which asserts that thepresence of disorder tends to inhibit the propagation ofexcitations. In fact, for 1D systems, all the eigenstatesare localized and transport is completely inhibited[10–12]. This was also proven to be true in two-dimensionalsystems, while in three-dimensions a mobility edge isformed. Now, Anderson localization is based on the as-sumption that the disorder is “perfect” or uncorrelated.However, it has been noted that in one-dimensional lat-tices with correlated disorder, a degree of transport isstill possible. This is the case of the random dimer model(RDM) for the usual tight-binding model. It consists ofa binary alloy for the site energies where each site energyis assigned at random to pairs of lattice sites. This leadsto a mean square displacement of an initially localizedexcitation that grows asymptotically as t / at low dis-order levels, instead of the saturation behavior predictedby Anderson theory[13–15].An experimental demonstration of the RDM predictionhas been made in an optical setting[16]. A straightfor-ward extension of these ideas to random arrays of largerunits (n-mers), has also been theoretically explored[17].In the context of a disordered, one-dimensional SRR ar-ray, it was found that an uncorrelated disorder alwaysleads to localization of magnetic energy at any disorderstrength, with a transmissivity that decreases exponen-tially with the size of the system. For correlated disorderand small and medium disorder level, however, it be-comes possible for a fraction of states to have resonanttransmission, leading to a power-law decrease of the over-all transmissivity, with system size[18].In this work, we examine the localization of the mag-netic modes and the transport of magnetic excitations ina two-dimensional disordered square array of split-ringresonators, where the resonant frequencies ω n in Eq.(4)are taken as random quantities. We will use two dif-ferent types of randomness: A completely uncorrelatedone (i.e., Anderson-like) and a correlated one consistingon a straightforward 2D generalization of the well-knownrandom dimer model used in one-dimensional systems.As expected, in the uncorrelated case we obtain fullylocalized modes and absence of transport. For the cor-related case, we find that unlike the 1D random dimermodel, the absence of transmission resonances also leads to an absence of magnetic transport. These studies giveus some inkling as to the magnetic energy transport inhigher dimensions, as well as to checking the universalityof Anderson localization.A usual indicator of localization is given by the par-ticipation ratio (PR), that measures the extent of theelectric charge distribution stored in the capacitors (ormagnetic energy density stored in the inductors): P R = (cid:32) (cid:88) n | q n ( t ) | (cid:33) / (cid:88) n | q n ( t ) | (5)For a completely localized excitation, P R = 1, while fora complete delocalized state,
P R = N .To monitor the degree of mobility of a magnetic exci-tation we resort to the mean square displacement (MSD)of the charge distribution, defined as (cid:104) n (cid:105) = (cid:88) n n | q n ( t ) | / (cid:88) n | q n ( t ) | . (6)Typically (cid:104) n (cid:105) ∼ t α at large t , where α is known asthe transport exponent. The types of motion are clas-sified according to the value of α : ‘localized’ ( α = 0),‘sub-diffusive’ (0 < α < α = 1), ‘super-diffusive’ (1 < α <
2) and ‘ballistic’ ( α = 2). Homogeneous case . Before embarking into the effectsof disorder on the system, let us begin by examining thephenomenology in the absence of disorder, ω n = 1, inorder to have a proper comparison context.After posing q n ∼ exp( i k · n ) and solving for Ω , weobtain the dispersion relation in d -dimensions asΩ k = 11 + λ (cid:80) m exp( i k · n ) (7)where the sum is restricted to nearest neighbors.The time evolution of a completely localized initialcharge q n (0) = A δ n , , and no currents, ( dq n /dt )(0) = 0,is given by q n ( t ) = ( A/v ) (cid:90) F BZ e i ( k · n − Ω k t ) d k +( A/v ) (cid:90) F BZ e i ( k · n +Ω k t ) d k (8)where v is the volume of the first Brillouin zone (FBZ),and Ω k is given by Eq.(7). After replacing this form for q n ( t ) into Eq.(6), one obtains after some algebra, a closedform expression for (cid:104) n (cid:105) : (cid:104) n (cid:105) = (1 /v ) (cid:82) F BZ ( ∇ k Ω k ) (1 − cos(2 Ω k t ))1 + (1 /v ) (cid:82) F BZ cos(2 Ω k t ) t . (9)At long times (cid:104) n (cid:105) approaches a ballistic behavior (cid:104) n (cid:105) = (cid:20) v (cid:90) F BZ ( ∇ k Ω k ) d k (cid:21) t ( t → ∞ ) (10)while at short times, (cid:104) n (cid:105) = (cid:20) v (cid:90) F BZ Ω k ( ∇ k Ω k ) d k (cid:21) t ( t → . (11) ( a ) ( b )( c ) ( d ) ⌦ k ⌦ k ⌦ k Figure 2. Dispersion relation for homogeneous case: (a) λ =0 .
05 (b) λ = 0 .
1, (c) λ = 0 . For a square lattice ( d = 2),Ω k = 11 + 2 λ (cos( k x ) + cos( k y )) (12)where, k = ( k x , k y ). We see that the system is capableof supporting magnetoinductive waves, if | λ | < /
4. Thebandwidth, defined as | Ω − Ω ± π | depends directly on λ . Figure 2 shows the band Ω k , as well as the band-width. We see that at the edges of the Brillouin zone,the bandwidth diverges at | λ | = 1 /
4, but the increasein bandwidth is mostly concentrated in the immediatevicinity of ( k x , k y ) = ( ± π, ± π ). Disorder . We introduce now disorder into our systemby considering the case when the resonant frequencies ω n are taken as random. This can be done in practiceby altering the spacing between the slits, or by insertingdifferent dielectrics in the slits. Let us get back to thestationary equation (4) which can be rewritten as − (cid:18) (cid:19) q n + (cid:18) ω n (cid:19) q n + λ (cid:18) ω n (cid:19) (cid:88) m q m = 0 (13)We see right away that the equation looks similar toan Anderson tight-binding model, with a site energyterm equal to 1 /ω n , and a site-dependent coupling λ/ω n .Thus, the “diagonal” term and the “coupling” term arecompletely correlated, and their values appear “inverted”when compared to a usual tight-binding model.We will explore two kinds of disorder: an uncorrelatedone, where the site frequencies ω n are assigned at randomfrom a binary distribution { , ω } for a given impurityfraction. Clearly, for ω = 1, we recover the homogeneous ○ ○ ● ○ ○ ○ ○ ○ ● ○ ○ ● ● ○ ● ○ ● ○ ○ ○ ○ ○ ○○ ○ ● ○ ○ ● ○ ○ ○ ● ○ ● ○ ○ ● ○ ○ ● ○ ● ○ ○ ○○ ○ ○ ○ ● ○ ○ ○ ○ ● ○ ○ ● ○ ○ ○ ○ ○ ● ○ ○ ○ ○○ ○ ● ○ ○ ● ● ○ ● ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ● ○○ ○ ○ ○ ○ ○ ● ○ ○ ○ ● ○ ● ○ ○ ● ● ○ ● ○ ○ ○ ○○ ● ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ●● ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ● ● ○ ○ ● ○● ○ ○ ○ ● ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ● ○ ● ○ ○○ ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ●○ ○ ● ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ●○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ● ○ ○ ● ○ ○● ○ ○ ○ ○ ○ ● ○ ○ ● ○ ○ ○ ● ● ○ ○ ● ○ ● ○ ○ ○● ● ○ ○ ○ ● ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○● ○ ○ ○ ○ ○ ● ○ ● ○ ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ● ○ ○ ○ ○ ○ ○ ○ ○● ○ ● ● ○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ● ○ ○ ○ ○ ○ ● ○ ○ ○ ○ ● ○ ○ ○ ○ ● ○ ● ○ ○ ●○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ●○ ○ ○ ○ ○ ○ ● ○ ● ○ ● ○ ○ ○ ○ ● ○ ○ ○ ● ○ ● ○○ ○ ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ● ●○ ○ ○ ○ ● ○ ○ ● ● ○ ○ ○ ○ ○ ● ● ○ ● ○ ○ ● ○ ○○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ○ ● ○ ○ ○ ○ n ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ● ● ○ ○ ○ ● ● ● ● ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ● ● ● ● ○ ○ ○ ● ● ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ● ● ○ ● ●○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ● ○ ● ●○ ○ ○ ○ ○ ○ ● ● ○ ● ● ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○○ ○ ○ ○ ○ ○ ● ● ○ ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ● ● ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ● ● ● ● ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ● ● ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ● ● ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ● ● ○○ ○ ● ● ○ ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ● ● ○ ● ● ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ● ● ○ ○ ○ ● ● ○ ○ ○ ○ ○○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○○ ○ ○ ● ● ○ ○ ○ ● ● ○ ○ ○ ○ ○ ○ ● ● ○ ○ ○ ○ ○ n Figure 3. Random realization of an uncorrelated (left) and acorrelated (right) distribution of random resonant frequencies.(Impurity fraction: 20%). D en s i t y o f s t a t e s D en s i t y o f s t a t e s D en s i t y o f s t a t e s D en s i t y o f s t a t e s D en s i t y o f s t a t e s D en s i t y o f s t a t e s ! = 0 . ! = 0 . ! = 1 ! = 2 ! = 3 ! = 4 Figure 4. Average density of states for the uncorrelated dis-order, for several impurity mismatch values ω ( N = 15 × .
5, number of realizations=33) case. The second kind of disorder we will explore is a cor-related one, inspired by the RDM. While in the RDM oneassigns random site frequencies at pairs (dimers) of lat-tice sites, in our case we assign (random) site energiesto 4 nearby sites or ‘tetramers’ of lattice sites. Figure 3shows an example of a disorder realization for the uncor-related and correlated cases. We notice the presence of4-sites clusters that constitute the new ‘point impurities’.Is like having fewer, but larger impurities.Figure 4 shows the average density of states (DOS) D (Ω ) = (cid:42) (1 /N ) (cid:88) m δ (Ω − Ω m ) (cid:43) (14) pa r t i c i pa t i on r a t i o pa r t i c i pa t i on r a t i o pa r t i c i pa t i on r a t i o pa r t i c i pa t i on r a t i o pa r t i c i pa t i on r a t i o pa r t i c i pa t i on r a t i o ! = 0 . ! = 0 . ! = 0 . ! = 0 . ! = 2 ! = 2 CorrelatedUncorrelated
Figure 5. Comparison of the average participation ratio forthe uncorrelated (left column) and the correlated case (rightcolumn), for the same single random realization and for sev-eral impurity mismatch values ω ( N = 33 ×
33, impurityfraction=0 . where the sum is over all modes. The figure shows theuncorrelated only, since its correlated counterpart is sim-ilar (not shown). The no-disorder case ( ω = 1) for ourfinite system seems to hint at the existence of van Hove-like singularities. Given the small size of our lattice isnot possible to say more. Also, it seems that some quasi-singularities remain in place at low frequencies and arenot removed by the strength of the frequency mismatch.This behavior is reminiscent of the 2D tight-binding An-derson model with off-diagonal disorder, where the singu-larity is not washed away by the disorder[19]. In our casewe have both types of disorder and this would ordinar-ily mean that the singularity would disappear, however,we have identical values for the site energy and couplingstrength for both types of disorder (Eq.(13)). This mightexplain why our singularities persist even at high mis-match strength. Also, the DOS shows the formation ofa quasi-gap at high mismatch strengths that separatesa low-frequency sector from a high-frequency lobe thatdecreases in amplitud as the mismatch strength is in-creased.In the absence of disorder ( ω = 1) and for a square lattice of N × N sites, the participation ratio is[20]PR = (cid:32) (cid:88) m ,m | q m ,m | (cid:33) / (cid:88) m ,m | q m ,m | = (cid:32)(cid:88) m | q m | (cid:33) / (cid:88) m | q m | , (15)i.e., the square of the one-dimensional participation ra-tio for the 1D lattice. Using q n ∼ sin( kn ), we obtain P R ≈ (4 / N × N in the limit of a large number ofsites[20]. Figure 5 shows the participation ratio PR of theeigenstates for a single random realization, for the uncor-related and correlated cases. In both cases we have em-ployed the same basal binary realization { , } and justchange the amplitudes. In other words, ω n = ω × { , } .The PR in both cases look similar, especially in the low-and medium mismatch disorder strengths. The smallvalue of the PR at low frequencies indicates that local-ization is stronger there. Disordered transport . Let us now compute the spread-ing of an initially localized magnetic excitation and ob-serve its time evolution. A useful quantity to monitor thespreading of an excitation is the mean square displace-ment (MSD), defined in Eq.(6). We will compute andcompare the MSD for the uncorrelated and correlatedcases. Results are shown in Fig. 6, where the MSD iscomputed for several different mismatch values ω . Thespecial case of no disorder ω = 1 is represented by thered curve, and is given in exact form in Eq.(9). As wecan see, the behavior for both kinds of disorder is quitesimilar. In both cases, as ω is increased, the curves be-come higher and higher and seem to converge towards aeventual plateau. Since, because of computational con-straints, we could not set up a large 2D lattice, it is < R M S > ( ) uncorrelated correlated h M S D i ( ) Figure 6. Comparison of the average RMS for the uncorre-lated (left column) and the correlated case (right column), forseveral impurity mismatch values ω ( N = 33 ×
33, impurityfraction=0 .
5, number realizations=33.) not possible to discuss the asymptotic MSD exponent (cid:104)