Effect of short-ranged spatial correlations on the Anderson localization of phonons in mass-disordered systems
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Bull. Mater. Sci., Vol. , No. , , pp. c (cid:13)
Indian Academy of SciencesDOI E ff ect of short-ranged spatial correlations on the Anderson localization ofphonons in mass-disordered systems WASIM RAJA MONDAL , and N. S. VIDHYADHIRAJA
1, * Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560 064, India.
Abstract. We investigate the e ff ect of spatially correlated disorder on the Anderson transition of phonons in threedimensions using a Greens function based approach, namely, the typical medium dynamical cluster approximation(TMDCA), in mass-disordered systems. We numerically demonstrate that correlated disorder with pairwise correla-tions mitigates the localization of the vibrational modes. A correlation driven localization-delocalization transition canemerge in a three-dimensional disordered system with an increase in the strength of correlations.Keywords. Anderson localization; Phonon localization.
1. Introduction
Anderson introduced an ideal theoretical model containingthe essential ingredients for studying the nature of one-electronstates in disordered systems [1]. The model assumed non-interacting electrons moving through a lattice and allowedto hop only to nearest-neighbor sites. Disorder was intro-duced in the local orbital energies, which were independentquenched random variables distributed according to somespecified probability distribution. Anderson predicted thatthe wave function may become exponentially localized witha characteristic localization length depending on the strengthof disorder. Scaling theory[2] bolstered Anderson’s idea oflocalization[1] by considering non-interacting electron sys-tems with uncorrelated disorder. It found that all one-electronstates are exponentially localized in one and two dimensionseven for infinitesimal amount of disorder, with a true metal-insulator transition occurring only in three dimensions (3D)whence the single-particle states may survive as extendedstates for weak disorder. A series of analytical, numericaland experimental results find strong agreement with one pa-rameter scaling theory of localization. However, the charac-teristics of the disorder potential can have a strong impact onAnderson localization. In particular, spatial correlations inthe disorder can markedly change the conventional physicsof Anderson localization.Such correlated disorder is relevant to transport prop-erties of binary solids, DNA[3,4], graphene [5], quantumHall wires[6], topological insulators[7] and so on. Recently,there has been a growing interest in understanding the ef-fect of spatial correlations on Anderson localization due totremendous experimental progress. Clement et al.[8] devel-oped a experimental technique for creating correlated disor-der through the laser speckle method. In this method, onecan accurately control the spatial correlation length. A spa-tial correlation induced localization-delocalization transitionhas been experimentally observed in GaAs-AlGaAs superlat-tices[9]. Very recently, a transition between algebraic local-ization and delocalization in a 1D disordered potential with a ∗ Author for correspondence ([email protected]) bias has been reported[10]. Such experimental observationscall for an in-depth theoretical analysis of the e ff ect of short-range correlations on Anderson localization.We describe briefly the theoretical investigations that haveincorporated short-range as well as long-range spatial corre-lations in the diagonal as well as o ff -diagonal disorder. Aseries of one-dimensional versions of the Anderson modelhave been used to demonstrate a breakdown of Andersonslocalization driven by spatial correlations on the disorder dis-tribution [11,12,13,14,15]. Also, e ff ort has been made todemonstrate the strong e ff ect of o ff -diagonal correlated dis-order on Anderson localization. For example, a number ofstudies have employed correlated o ff -diagonal interactionsand found delocalized states [16,17,18]. Besides short rangecorrelations, several investigations have been performed con-sidering long range correlations in the disorder distribution.Carpena et al [19] find a long-range correlation-induced metal-insulator transition using a one-dimensional tight-bindingmodel. Francisco et al [20] obtain an Anderson-like metal-insulator transition studying a one-dimensional tight-bindingmodel with long-range correlated disorder. All these studiessuggest that localization properties are greatly renormalizedwhen some kind of spatial correlation is introduced in the dis-order distribution. However, most of the studies are limitedto electronic problems and Anderson localization of phononsin the presence of spatially correlated disorder has receivedscant attention, both theoretically and experimentally.Being a general wave phenomenon, Anderson localiza-tion is ubiquitous. Sajeev John et al [21], using field theoretictechniques, investigated phonon localization in the presenceof long range correlated random potential. However, meth-ods like exact diagonalization (ED), transfer matrix method(TMM), multifractal analysis, diagrammatic techniques, itin-erant coherent-potential approximation (ICPA) have not beenemployed for studying phonon localization in the presenceof correlated disorder. Most of the mentioned methods havebeen confined to simple models of lattice vibrations, wherethe diagonal matrix elements M ( l ) of the Hamiltonian are in-dependent random variables.In our previous study[22], we provided a detailed de-scription of a typical medium dynamical cluster approxima-tion (TMDCA), that yields a proper description of the An-derson localization transition in 3D. It adopts the typical den-sity of states (TDOS) as a single particle order parameter forthe Anderson localization transition (ALT) which makes itcomputationally less expensive compared to other numericalmethods like ED and TMM. It satisfies all the essential re-quirements expected of a successful quantum cluster theory.We have also been able to extend the formalism for study-ing Anderson localization of phonons in the presence of bothdiagonal and o ff -diagonal disorder[22,23].In this work, we investigate the nature of the Andersontransition for phonons in the presence of spatially correlateddisorder in 3D. This paper is organized as follows. In sectionII, we give a brief description of the model and method thatare used in this work. In section III, we present results anddiscussions. We conclude our work in section IV.
2. Method
As before [22], we consider the following Hamiltonian forthe ionic degrees of freedom of a disordered lattice within theharmonic approximation in the momentum ( p ) and displacement( u )basis, as H = X α il p i α ( l )2 M i ( l ) + X αβ ll ′ i j Φ αβ i j ( l , l ′ ) u i α ( l ) u j β ( l ′ ) , (1)where the symbols have their usual meaning as described inRef [22]. In this work, we again restrict ourselves to a singlebranch ( α ) and single basis atom ( i =
1) case, hence we dropthe indices, α, β, i , j . The unit cell index ( l ) is retained. Thespatial dependence of the ionic masses M ( l ) is incorporatedthrough a local disorder potential V asˆ V ll ′ = h − M ( l ) / M i δ l , l ′ . (2)In the previous work[22], we had considered a uniform boxdistribution, where the quantity (cid:16) − M ( l ) / M (cid:17) ∈ [ − V , V ] cantake any value in that interval with equal probability and 0 ≤ V ≤ V ′ s from siteto site were taken to be uncorrelated with each other. Asmentioned in the introduction, the objective of this work is toinvestigate the e ff ect of short-range correlations in the massdisorder.We begin with nearest-neighbour correlations. We firstdistribute masses randomly on the odd indexed sites and onthe even indexed sites, exactly as was done previously, ac-cording to a uniform distribution with the same mean andvariance. The disorder potential at the odd indexed sites isdenoted as V and that on the even indexed sites is denotedas V . Therefore, the following initial correlations hold: h V i = h V i = . (3) h V i = σ ; h V i = σ ; h V V i = . (4) Now, since V and V are independent, ρ V V =
0. Fromthese two uncorrelated random sequences, we want to gen-erate correlations between consecutive sites of the odd andeven sequences with a specified correlation coe ffi cient ρ . Theresulting new sequences for the odd and even indexed sites,denoted as V odd and V even , should be correlated pairwise. So,the site 2 n + n should be correlated. ρ V odd V even = (cid:10)(cid:0) V odd − h V odd i (cid:1)(cid:0) V even − h V even i (cid:1)(cid:11) σ , (5)where σ is the variance. Let us construct V odd and V even using linear combinations of V and V as V odd = aV + bV V even = cV + dV , (6)where the unknown coe ffi cients, a , b , c and d will be chosenso that the odd and even sequences get correlated with eachother. So, h V odd V even i = h ( aV + bV )( cV + dV ) i = ac h V i + bd h V i + ( ad + bc ) h V V i . (7)Using Eq.(4) in Eq(7), we write h V odd V even i = ( ac + bd ) σ . (8)Using Eq.(4) in Eq.(5), we write ρ V odd V even = h V odd V even i σ . (9)Using Eq. (8) in Eq. (9), we get ρ V odd V even = ac + bd (10)From Eq(6), we write h V i = h ( aV + bV )( aV + bV ) i = a h V i + b h V i + ( ac + bd ) h V V i . (11)Using Eq.(4), we get h V i = ( a + b ) σ ; h V even i = ( c + d ) σ . (12)We impose the condition h V i = σ ; h V i = σ . (13)From the above, it is easy to see that a + b = c + d = . (14)So, the transformation that yields the desired correlations canbe chosen as a = cos φ = d ; b = sin φ = c . (15)Hence, the expression ac + bd = φ sin φ = sin 2 φ . (16)Thus, random V odd and V even are correlated with ρ V odd V even whichis equal to sin 2 φ , where φ =
12 sin − ( ρ V odd V even ) . (17)We can verify that this method does induce correlationsbetween the even and the odd sequences. For vanishing cor-relation, i.e. for ρ V odd V even →
0, from Eq. 17, φ → a , d → b , c → V odd → V V even → V . (18)Since V and V are anyway uncorrelated, the new sequences, V odd and V even , in this limit are also uncorrelated. While inthe other extreme, namely ρ V odd V even →
1, we get φ → π/ a , b , c , d → / √
2, and hence V odd ≃ V even ≃ ( V + V ) / √
2. Thus, in this limit, V odd and V even becomealmost equal and are hence fully correlated. We illustratethis in Fig. 1, where for four di ff erent correlation coe ffi cients, ρ V odd V even = . , . , . .
99, the di ff erence of the twosequences, V odd − V even is plotted as a function of the site-index. It is seen that for small correlation coe ffi cients, thedi ff erence is large, and hence the odd and even sequences areuncorrelated. While for large correlation coe ffi cient ( & . ff erence is very small, and hence the two sequences arestrongly correlated.An algorithm that implements the described formalismfor creating correlated disorder potential is stated below:1. The algorithm for generating correlated disorder poten-tial starts with creating local disorder potential V l , which weinitially consider as spatially independent random variablesdistributed according to uniform (box) distribution as P v ( V l ) = Θ ( V − | V l | ) / V , (19)where V l is the disorder potential defined in Eq.(2) and V isthe width of the distribution that corresponds to the disorderstrength.2. Identify the V l at lattice sites l that are labeled by the evennumber or odd number. We define V ( l ) as the disorder po-tential at the odd indexed lattice sites and V ( l ) as the disorderpotential at the even indexed lattice sites.3. We set ρ as correlation strength parameter which can bevaried from 0 to 1. For a given value of ρ , we calculate φ using Eq.(17).4. The unknown coe ffi cients a , b , c , d are calculated usingEq. 15 and the normalization is maintained by imposing thecondition given in Eq.(14).5. The spatial correlations among the V odd and V even are in-troduced depending on the strength ρ according the relationgiven in Eq.(6).The rest of the algorithm is the same as described in ourprevious publication[22]. -0.400.4 V odd ( l ) - V eve n ( l ) ρ=0.2 -0.400.4 ρ=0.5 -0.400.4 ρ=0.8 Site index, l -0.400.4 ρ=0.99
Figure 1 . A plot of the di ff erence of two correlated random se-quences V odd − V even for four di ff erent values of correlation coef-ficient ( ρ ) using cluster size N c =
64. Notice that the two randomvariables are strongly spatially correlated for ρ = .
99, whereasthey are uncorrelated for ρ = . T D O S ρ =0.00ρ=0.50ρ=0.99 T D O S ω T D O S ω V=0.2 V=0.7V=0.5V=0.3 V=0.9V=0.8
Figure 2 . The evolution of the TDOS, calculated from the TMDCA,as a function of the square of the frequency ( ω ) with increasingdisorder strength ( V ) considering a box distribution in three dimen-sions using cluster size N c =
64 for the uncorrelated ( ρ = .
00) andcorrelated ( ρ = . , .
99) spatial disorder. The arrows shows themobility edges.
3. Results and discussion
As we have already discussed, a true delocalization-localizationtransition occurs in 3D depending on the strength of disor-der ( V ). We investigate this Anderson transition of phononsusing the TMDCA in the presence of short range order. Inour previous study [22], we have already established that theTDOS is a valid order parameter for studying phonon lo-calization. So, we first observe the evolution of the TDOSwith increasing disorder strength V for correlated strength V T S W ρ=0ρ=0.1ρ=0.2ρ=0.8ρ=0.90ρ=0.99 Figure 3 . Total spectral weight (TSW) of the TDOS as a func-tion of increasing disorder strength ( V ) for correlated strength ρ = . − .
99. We observe that the rate of decrease of the total spec-tral weight (TSW) of the TDOS decreases with increasing spatialcorrelation. ω V ρ=0ρ=0.1ρ=0.20ρ=0.5ρ=0.7ρ=0.8ρ=0.9ρ=0.99 Figure 4 . Mobility edge trajectory for a box distribution of mass-disordered system in three dimensions. We find that the re-entrancebehavior of the mobility edge shifts towards high-frequency regionwith intermediate correlation strength. Further increase of the cor-relation strength completely destroys the re-entrance behavior andmobility edges keeps on moving towards high-frequency regionindicating a localization-delocalization transition driven by spatialcorrelations. ρ = ρ = .
99. It is displayed in Fig.2.As may be expected, the TDOS for ρ = ρ = .
99 for low disorder ( V ≤ . V > .
3, the TDOS for ρ = .
99 starts to deviatestrongly from the TDOS for the uncorrelated disorder. Wenote that the TDOS for ρ = .
99 di ff ers significantly fromthe TDOS for the uncorrelated disorder at V = .
9. We havealready understood that the vanishing of the TDOS impliesthe localization of vibrational modes[22]. Here we repro-duce such behavior for ρ =
0. The overall TDOS for ρ = V which indicates that the vibra-tional modes get localized as disorder increases. This kindof disorder-induced delocalization-localization transition isprevented by the introduction of spatial correlations in the system. Through a direct comparison of the TDOS for ρ = ρ = .
99, such behavior can be easily ob-served. The mobility edges marked by the arrows representthe energy scale demarcating the extended states from thelocalized states. Again, from Fig.2, it is clear that the mobil-ity edge shifts to higher energies with increasing correlationstrength, thus implying that the latter induces delocalizationof the hitherto localized states.An alternative measure of the proximity to the Andersonlocalization transition is total spectral weight of the TDOS.The variation of total spectral weight of the TDOS with in-creasing correlations is shown in Fig3. It clearly shows thatthe total spectral weight of the TDOS for ρ = .
99 decreasesat a much slower rate compared to the uncorrelated disor-der ( ρ = ff ects of spatial corre-lations on the mobility edges for Anderson localization ofelectrons have been studied extensively. However, to best ofour knowledge, it has not been yet reported for the Andersonlocalization of phonons in the correlated disorder case. Wedefine the mobility edge by the boundary of the TDOS anddenote by arrows as indicated in Fig.2.In Fig4, we show calculated mobility edges using theTMDCA with N c =
64 for mass disorder. The phase di-agram implicates that the spatially correlated diagonal dis-order delocalizes the uncorrelated diagonal disorder inducedlocalized vibrational modes. In the phase diagram, we firstobserve the usual behavior of the mobility edges with in-creasing V for ρ =
0. For small disorder V < .
5, the tra-jectory of the mobility edges moves outward with increasing V . But, it starts moving inward for strong disorder V ≥ . V = .
5. We explored this behavior of the mobility edgesin Ref[22]. The spatially correlated disorder destroys thisre-entrant behavior of the mobility edges. As seen in Fig4,the trajectory of the mobility edges for ρ = .
99 is almostthe same as that for ρ = V ≤ .
5. However, in contrast to the uncorrelated case,the trajectory of the mobility edges keeps on moving outwardwith increasing disorder strength V > .
5. It suggests that thespatial correlations drive the system towards delocalization.
4. Conclusions
We have applied the TMDCA formalism for investigating thee ff ects of short-range spatial correlations on phonon localiza-tion in 3D. We have only considered pairwise correlations be-tween the adjacent odd-indexed and even-indexed sites. Thecorrelation strength is varied from 0 to 1. In the weak correla-tion limit, all the sites have completely random masses, whilein the strong correlation limit, the masses of the (2 l − th site and the (2 l ) th site are the same, but as a function of l ,the odd / even sequence of masses is still random. Our mainconclusion is that correlated disorder with just pairwise cor-relations can markedly change the localization transition ofphonons. Such a conclusion is validated by observing thevariation of the TDOS and mobility edges with increasingcorrelation strength. We show that short-range correlated dis-order impedes the localization of the vibrational modes, andeventually, a correlation induced localization-delocalizationtransition of phonons sets in a 3D disordered sample. Itwould certainly be valuable to understand the observed delo-calization transition in the presence of long-range correlateddisorder. For doing so, an extension of the current frameworkincorporating long-range correlations is in progress. Acknowledgement
References [1] Anderson P.W. 1958
Phys. Rev.
Phys. Rev. Lett. Nature
Phys.Rev. E Phys. Rev. Lett.
Phys. Rev. B Phys.Rev. B Nature
Phys.Rev. Lett. arXiv Phys.Rev. Lett. Phys. Rev. Lett. Physics Letters A
Phys. Rev. B Phys.Rev. B Journal of Physics A: Mathematical and Gen-eral [17] Flores J C and Hilke M 1993 Journal of Physics A: Mathe-matical and General L1255[18] Flores J C 1989
Journal of Physics: Condensed Matter Nature [20] de Moura Francisco A. B. F. and Lyra Marcelo L. 1998
Phys.Rev. Lett. Phys. Rev. B Phys. Rev. B Phys. Rev. B11