Magnetic metamaterials with correlated disorder
MMagnetic metamaterials with correlated disorder
Mario I. Molina
Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Dated: August 24, 2020)We examine the transport of magnetic energy in a simplified model of a magnetic metamaterial,consisting of a one-dimensional array of split-ring resonators, in the presence of correlated disorderin the resonant frequencies. The computation of the average participation ratio (PR) reveals thaton average, the modes for the correlated disorder system are less localized than in the uncorrelatedcase. The numerical computation of the mean square displacement of an initially localized magneticexcitation for the correlated case, shows a substantial departure from the uncorrelated (Anderson-like) case. A long-time asymptotic fit (cid:104) n (cid:105) ∼ t α reveals that, for the uncorrelated system α ∼ α >
0, spanning a whole range of behavior ranging from localizationto super-diffusive behavior. The transmission coefficient of a plane wave across a single magneticdimer reveals the existence of well-defined regions in disorder strength-magnetic coupling space,where unit transmission for some wavevector(s) is possible. This implies, according to the randomdimer model (RDM) of Dunlap et al., a degree of mobility. A comparison between the mobilities ofthe correlated SRR system and the RDM shows that the RDM model has better mobility at lowdisorder while our correlated SRR model displays better mobility at medium and large disorder.
Introduction . The subject of metamaterials has con-tinue to attract interest in the scientific community, forits great potential for applications to many different tech-nologies. We can briefly describe them as a class ofman-made materials that are characterized by having en-hanced thermal, optical, and transport properties thatmake them attractive candidates for current and futuretechnologies. Among them, we have magnetic metama-terials (MMs) that consist of artificial structures whosemagnetic response can be tailored to a certain extent.A simple realization of such a system consists of anarray of metallic split-ring resonators (SRRs) coupledinductively[1–3]. This type of system can, for instance,feature negative magnetic response in some frequencywindow, making them attractive for use as a constituentin negative refraction index materials[4–7]. The maindrawback of this system is the existence of large ohm-mic and radiative losses. A possible way out that hasbeen considered is to endow the SRRs with external gain,such as tunnel (Esaki) diodes[8, 9] to compensate for suchlosses. The theoretical treatment of such structures re-lies mainly on the effective-medium approximation wherethe composite is treated as a homogeneous and isotropicmedium, characterized by effective macroscopic parame-ters. The approach is valid, as long as the wavelength ofthe electromagnetic field is much larger than the lineardimensions of the MM constituents.The simplest MM model utilizes an array of split-ringresonators (Fig.1), with each resonator consisting of asmall, conducting ring with a slit. Each SRR unit inthe array can be mapped to a resistor-inductor-capacitor(RLC) circuit featuring self- inductance L , ohmic resis-tance R , and capacitance C built across the slit. We willassume a negligible resistance, and thus each unit willpossess a resonant frequency ω = 1 / √ LC . Under thiscondition and in the absence of driving and dissipation,the dimensionless evolution equations for the charge q n Figure 1. One-dimensional split-ring resonator arrays. Top:All SRRs lying on a common plane ( λ < λ > residing at the nth ring are d dt ( q n + λ ( q n +1 + q n − )) + ω n q n = 0 (1)where q n is the dimensionless charge of the nth ring, λ isthe coupling between neighboring rings which originatesfrom the dipole-dipole interaction, and ω n is the resonantfrequency of the nth ring, normalized to a characteristicfrequency of the system, such as the average frequency: (cid:104) w n (cid:105) = (1 /N ) (cid:80) n w n . The frequency ω n can be changedby varying the capacitance of the ring by altering theslit width or by inserting a dielectric in the slit. For ahomogeous array, ω n = 1.The dimensionless stationary state equation is ob-tained from Eq.(1) after posing q n ( t ) = q n exp[ i (Ω t + φ )]: − Ω ( q n + λ ( q n +1 + q n − )) + ω n q n = 0 (2)On the other hand, the topic of Anderson localization isan old one, but its importance has no waned throughoutthe years, given its consequences for transport in dis-ordered systems. Roughly speaking, it 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]. Now, Anderson localization is based on the as-sumption that the disorder is “perfect” or uncorrelated. a r X i v : . [ c ond - m a t . d i s - nn ] A ug However it has been noted that in systems with cor-related disorder, a degree of transport is still possible.Such is the case of the random dimer model (RDM) forthe discrete Schr¨odinger equation, that consists on a bi-nary alloy where one of the site energies is assigned atrandom to pairs of lattice sites. This leads to a meansquare displacement of an initially localized excitationthat grows asymptotically as t / at low disorder levels,instead of the saturation behavior predicted by Ander-son theory[13–15]. An experimental demonstration of theRDM prediction has been made in an optical setting[16].A straightforward extension of these ideas to random ar-rays of larger units (n-mers), has also been theoreticallyexplored[17]. Magnetic metamaterials constitute yet an-other setting in which to test all these ideas, whose resultscould have an impact on the design of future materialsof technological importance.In this work, we will explore the stationary modes andthe transport of excitations in the context of a magneticrandom dimer model. As we will see, the presence of un-correlated disorder leads to fully localized modes and tothe absence of transport. On the contrary, the presenceof a short-range correlation in the disorder distributionleads to an improved transport of magnetic energy, inqualitative agreement with the standard RDM. The re-sults of these studies give us some inkling as to the mag-netic energy transport in a correlated disordered SRRarray, as well as to checking the universality of Andersonlocalization.The ‘size’ or extent of the distribution of electric charge { q n ( t ) } stored in the capacitors, can be monitored via theparticipation ratio (PR), defined as P R = ( (cid:88) n | q n ( t ) | ) / (cid:88) n | q n ( t ) | (3)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 propagating inside a SRR array, we resort to themean square displacement (MSD) of the charge distribu-tion, defined as (cid:104) n (cid:105) = (cid:88) n n | q n ( t ) | / (cid:88) n | q n ( t ) | . (4)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 . In the absence of disorder, ω n =1, and the discrete translational invariance leads to anenergy band, obtained from Eq.(2) after assuming a planewave profile q n ∼ exp( ikn ):Ω k = 11 + 2 λ cos( k ) (5)This implies that these magneto-inductive waves can onlyexist for | λ | < /
2. From Lenz law, it can be shown that when all SRRs lie on a common plane, λ <
0, whilewhen all SRRs are centered about a common axis, λ > q n (0) = A δ n and no currents, ( dq n /dt )(0) = 0, we have formally q n ( t ) = ( A/ π ) (cid:90) π − π e i ( kn − Ω k t ) dk +( A/ π ) (cid:90) π − π e i ( kn +Ω k t ) dk (6)where Ω k is given by Eq.(5). After replacing this formfor q n ( t ) into Eq.(4), one obtains after some algebra, aclosed form expression for (cid:104) n (cid:105) : (cid:104) n (cid:105) = (1 / π ) (cid:82) π − π dk ( d Ω k /dk ) (1 − cos(2 Ω k t )) t / π ) (cid:82) π − π dk cos(2 Ω k t ) (7)As we can see from the structure of Eq.(7), as time t increases, the contributions from the cosine terms to theintegrals decrease and, at long times, (cid:104) n (cid:105) approaches aballistic behavior (cid:104) n (cid:105) = (cid:34) π (cid:90) π − π (cid:18) d Ω( k ) dk (cid:19) dk (cid:35) t ( t → ∞ ) , (8)while at short times, (cid:104) n (cid:105) = (cid:34) π (cid:90) ππ (cid:18) Ω k d Ω k dk (cid:19) dk (cid:35) t ( t → . (9)Thus, the transport in our system is ballistic: (cid:104) n (cid:105) = g ( λ ) t , where we can identify (cid:112) g ( λ ) as a kind of char-acteristic ‘speed’ for the ballistic propagation. Insertingthe specific form for Ω k from Eq.(5), we obtain (cid:104) n (cid:105) = ( λt ) − λ ) / t → ∞ , (10) (cid:104) n (cid:105) = λ (1 + λ ) t − λ ) / t → . (11) Disordered case . In this case, we choose the reso-nant frequencies ω n from a random binary distribution { ω a , ω b } . As mentioned before, this can be achieved byaltering the space between the slits of each SRR, or byfilling the space in the slit with different dielectrics. Letus look at the stationary modes obtained from Eq.(2).After a simple rearrangement, one can write down anequivalent eigenvalue problem: − (cid:18) (cid:19) q n + (cid:18) ω n (cid:19) q n + λ (cid:18) ω n (cid:19) ( q n +1 + q n − ) = 0 (12)This looks similar to the tight-binding Anderson prob-lem, except that now, we have a completely correlatedsite and coupling randomness. Not only that, but thesite energies and couplings appear all ‘inverted’. Thismeans that the usual, large disorder limit of the tight-binding system corresponds here to the small disorderlimit. � � � � ������������� ω � � << � � >> Figure 2. Mode -and realization average participation ratiofor the correlated SRR (solid line) and uncorrelated SRR case(dashed line), as a function of disorder strength. The dotmarks the value of the participation ratio in the absence ofdisorder ( N = 150 , λ = 0 . , ω b = 1 , realizations = 100.) The rigid correlation between the site ‘energies’ andthe coupling terms will be of lesser importance thanthe other correlation we really have in mind: thatof a random binary alloy for the resonance frequen-cies, as in the RDM. Following the RDM, we as-sign the site frequencies ω n at random to pairs of lattice sites (that is, two sites in succession): ...ω a , ω a , ω b , ω b , ω b , ω b , ω a , ω a , ω b , ω b , ω a , ω a , ... . The fre-quency of each pair is generated according to ω n = ω a + ( ω b − ω a ) × rand, where rand=0 or 1 with fiftypercent probability. It should be emphasized that oursystem cannot be mapped to the original RDM becauseof the the additional correlation between site and cou-pling values.In what follows, we will examine the stationary andtransport properties of a SRR array with binary disorder(“correlated”) and that of a simple Anderson-like (“un-correlated”) disordered SRR array.Figure 2 shows the mode-and realization average of theparticipation ratio, as a function of disorder strength.As we can see, in both cases, the presence of disorder makes the PR smaller than in the periodic case ω a = ω b ,where P R = (2 / N . This last case corresponds in ourplot to ω a = 1. In the correlated case, the PR is alwaygreater than in the uncorrelated case, which means thatthe modes are more extended in space. This, in turn,means that there is more overlap between modes whichleads to an increased mode coupling. An increased modecoupling means greater ease for an excitation to jump tonearby sites, thus increasing the general mobility.A qualitative explanation for this enhanced transportcan be given within the context of the RDM. The idea isthat the presence of the correlated binary disorder createsa finite fraction of modes with localization length equalor greater to the length of the array. Adapting the RDMargument, we consider an array of SRRs with identicalresonance frequencies ω n = 1, that contains an embeddeddimer impurity. Without loss of generality, we place thedimer at n = 0 and n = 1 and set ω = ω = ω . Let usconsider an incoming plane wave and compute its trans-mission coefficient across the dimer. The amplitudes tothe left and right of the dimer impurity are q n = (cid:26) R exp( ikn ) + R exp( − ikn ) n ≤ T exp( ikn ) n ≥ R is the incoming amplitude, R the reflected am-plitude and T the transmission amplitude. The station-ary equations at the dimer sites are (cid:20) − (cid:18) (cid:19) + (cid:18) ω (cid:19)(cid:21) ( R + R )+ (14) (cid:18) λω (cid:19) ( T e ik + R e − ik + Re ik ) = 0 (15) (cid:20) − (cid:18) (cid:19) + (cid:18) ω (cid:19)(cid:21) T e ik + (16)+ (cid:18) λω (cid:19) ( T e ik + R + R ) = 0 (17)together with Ω = 1 / (1 + 2 λ cos( k )).From these equations, it is possible to find the trans-mission coefficient t ( k ) = | T ( k ) | / | R | in closed formas t ( k ) = (cid:12)(cid:12)(cid:12)(cid:12) ( − e ik ) λ e ik λ ( ω −
1) + e ik λ ( ω −
1) + 2 e ik λ ( ω − ω + λ ω + e ik (( ω − + λ ( − ω − ω )) (cid:12)(cid:12)(cid:12)(cid:12) . (18)In the absence of the dimer, ω = 1 and t ( k ) = 1. Figure3 shows some transmission plots for a fixed value of λ and different ω . The most interesting thing to notice isthe existence of resonant cases. That is, the existence ofwavevectors k at which t ( k ) = 1, for some values of λ, ω .To determine the resonant region in parameter space wemake a sweep of t ( k ) in λ, k and ω . Results are shown in Fig.4 that shows regions of possible resonance-no res-onance in parameter space. This region corresponds tothe area enclosed by the curves 1 / (1+2 λ ) and 1 / (1 − λ ).Inside the region, a resonance(s) across a single dimer ispossible, and we will obtain an enhanced transport be-havior when we form a random dimer alloy: Once a planewave goes through a dimer without reflection, it will pass - (cid:1) (cid:1) t r an s m i ss i on - (cid:1) (cid:1) t r an s m i ss i on - (cid:1) (cid:1) t r an s m i ss i on - (cid:1) (cid:1) t r an s m i ss i on (a) (b)(c) (d) Figure 3. Transmission coefficient of magneto-inductive planewaves across a SRR dimer, for different values of the resonantfrequency mismatch. (a) ω = 2, (b) ω = 2 /
3, (c) ω = 0 . ω = 0 . λ = 0 . - - (cid:1) (cid:1) a low disorderhigh disorder Figure 4. Region in parameter space where a resonant trans-mission(s) across the SRR dimer is possible (shaded region).( ω b = 1) through all of the other SRRs dimers and will reach theends of the system unscattered. Of course this only hap-pens for a single wavevector, so it might be regarded as amarginal effect. However, around the perfectly transmit-ting case, there will be a fraction of states whose trans-mission across the system is finite[14] and, in our case,will give rise to some degree of magnetic transport.Let us also compute the average transmission of a planewave across an extended portion of the disordered ar-ray. Results are shown in Fig.5. For weak disorder( ω a (cid:29) , ω b = 1) we see that for the uncorrelated casethe transmission decreases exponentially with system size (cid:104) T (cid:105) ∼ exp( − αL ), while for the correlated case the de-crease obeys a power-law (cid:104) T (cid:105) ∼ L − β . These resultsare in qualitative agreement with previous studies ontight-binding systems[17]. For large disorder strengths( ω a (cid:28) , ω b = 1), we obtain exponential decrease forboth cases (not shown). T
100 200 300 400 50010 - - - - < T r an s m i ss i on >
50 100 200 5000.020.050.100.20 Length < T r an s m i ss i on > Figure 5. wavevector -and realization average transmissionof plane waves across a weakly disordered segment of finitelength. (a) Uncorrelated case (b) Correlated case. Note thelogarithmic scale in (b). Upper curves: ω a = 1 . , ω b = 1.Lower curves: ω a = 2 , ω b = 1 ( λ = 0 . , realizations=500). M S D M S D M S D M S D M S D M S D Figure 6. Mean square displacement as a function of time,for a SRR array with a localized initial condition and differ-ent disorder widths, for both, uncorrelated (left column) andcorrelated (right column) cases. (a) and (b): ω a = 0 . λ . (c)and (d): ω a = 2 λ . (e) and (f): ω a = 3 λ ( λ = 0 . , ω b = 1 , N =1500 , t max = 900). Transport . Let us compute the mean square displace-ment (MSD) of an initially localized magnetic excitation,and compare the results for the correlated and uncorre-lated cases. Results are shown in figure 6. In this figurewe compare the MSD for the uncorrelated case (left) col-umn with the correlated one (right column), for three ω a M S D e x ponen t Figure 7. Mean square displacement exponent α as a functionof disorder strength, for an initially localized magnetic exci-tation. Squares (circles) denote the SRR correlated (RDMcorrelated) case. ( λ = 0 . , ω b = 1 , N = 1500 , t max = 900). common values of the disorder strength ω a . The plotsare computed for a single realization of a long SRR ar-ray and long evolution times. For all of them we usethe same skeleton random number sequence, and varyonly ω a while keeping ω b = 1. In this way, the valueof ω a controls the disorder width. As we can see, whilefor the uncorrelated case the MSD tends to saturate atlong times, in the correlated case there is a finite frac-tion of propagation at long times. Not only that, but the ω a values separating mobile from localized regions arein agreement with the mobility phase diagram of Fig.4. Assuming a long-time asymptotic dependence of theform (cid:104) n (cid:105) ∼ t α , we computed numerically the α valuesas a function of the disorder strength ω a . Results areshown in Fig.7, and are compared to the tight-bindingRDM model of Dunlap et al.[14]. Interestingly, at lowerdisorder strength the RDM shows higher exponents (i.e.,better propagation). In this range, the exponent val-ues are close to the superdiffusive value: α = 3 /
2. Forthe correlated SRR case, the propagation is closer to thediffusive regime: α = 1 /
2. At intermediate and high dis-order strengths however, this tendency reverts and theMSD for the SRR model shows substantially higher ex-ponents, signaling an enhanced mobility. In the limit oflarge disorder, both sets of exponents converge to zero.Of course, these observations are only valid for a finitearray; the real asymptotic exponents are defined for aninfinite system only.
Conclusions . In this work we have examined the sta-tionary modes and the propagation characteristics of aSRR disordered array. We have employed two types ofdisorder: An Anderson-like with randomly distributedresonant frequencies, and a correlated one, where the frequencies distributions follows the RDM pattern, thatis, a disordered binary distribution. In the absence ofdisorder, the mean square displacement is calculated inclosed form, and at long times shows a ballistic behav-ior. In the presence of disorder, the stationary modes fora finite array are mostly localized, with a small fractionpossessing a localization length of the size of the system’slength. However, an examination of the participation ra-tio shows that in the correlated case, the PR is alwaysgreater than in the uncorrelated case. Thus, the cor-related modes are less localized. To better understandthe origin of this affect, we computed the transmissionof a plane wave across a single magnetoinductive dimer,finding the transmission in closed form, as a function ofthe inductive coupling, and the frequency mismatch. Itis found that this transmission can be unity for somewavevectors inside a region in coupling-mismatch spacethat is computed numerically. Thus, for plane waveswhose wavevectors are close to resonance, there is a frac-tion of these modes whose localization length is of the sizeof the system. On the dynamical front, computation ofthe mean square displacement shows that, while for theuncorrelated case there is saturation, in the correlatedcase there is finite propagation for small and mediumdisorder levels, and converging to smaller and smallerpropagation at large disorder level. The correlation inthe disorder also affected the transmission of plane wavesacross a finite disordered segment, where now the trans-mission decrease across the sample following a power-law decrease with sample’s length instead of the typicalexponential decrease. Finally, we compared the MSDfor our system with Dunlap’s tight-binding RDM model,which also displays finite transport. While the MSD forthe RDM model shows larger transport exponents at lowdisorder levels, at medium and high disorder strengths,our correlated model features larger exponent and thusgreater mobility.We conclude that an array of SRRs with uncorrelateddisorder, always displays Anderson localization of mag-netic energy at any disorder strength. For correlateddisorder at small and medium strengths, the SRR arrayis capable of exhibiting a finite degree of mobility. Atlarge disorder level, this mobility ceases and the systembecomes Anderson-like. These results could be usefulfor the design of efficient magnetic energy confinementdevices, and the harvesting and transport of magneticenergy. We are current pursuing an extension of thesestudies to two-dimensional SRR systems.
ACKNOWLEDGMENTS
This work was supported by Fondecyt Grant 1200120. [1] T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R.Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science , 1494 (2004). [2] N. Katsarakis, G. Constantinidis, A. Kostopoulos, R. S.Penciu, T, F, Gundogdu, M. Kafesaki, E. N. Economou,Th. Koschny and C. M. Soukoulis, Opt. Lett. , 1348(2005).[3] M. I. Molina, N. Lazarides, and G. P. Tsironis, Phys.Rev. E , 046605 (2009).[4] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J.Stewart, IEEE Trans. Microwave Theory Tech. , 2075(1999).[5] V. G. Veselago, Sov. Phys. Usp. , 509 (1968).[6] R. A. Shelby, D. R. Smith, S. Schultz, Science , 77(2001).[7] D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S.Schultz, Phys. Rev. Lett. , 4184 (2000).[8] L. Esaki, Phys. Rev. , 603 (1958). [9] T. Jiang, K. Chang, L.-M. Si, L. Ran, and H. Xin, Phys.Rev. Lett. , 205503 (2011).[10] P. W. Anderson, Phys. Rev, 109, 1492 (1958).[11] B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1496(1993).[12] Elihu Abrahams, 50 Years of Anderson Localization(World Scientific Publishing, 2010).[13] J. C. Flores, J. Phys. Cond. Matt. 1, 8471 (1989).[14] D.H Dunlap. H. L. Wu. and P. W. Phillips, Phys. Rev.Lett. 65, 88 (1990).[15] D. Giri, P.K. Datta, and K. Kundu, Phys. Rev. B 48,14113 (1993).[16] U. Naether, S. Stuetzer, R. Vicencio, M. I. Molina, A.T¨unnermann, S. Nolte, T. Kottos, D. N Christodoulidesand A. Szameit, New J. Phys. 15, 013045 (2013).[17] D. Lopez and M. I. Molina, Phys. Rev. E.93