We study the transport of a quantum particle through square lattices of
various sizes by employing the tight-binding Hamiltonian from quantum
percolation. Input and output semi-infinite chains are attached to the lattice
either by diagonal point to point contacts or by a busbar connection. We find
resonant transmission and reflection occuring whenever the incident particle's
energy is near an eigenvalue of the lattice alone (i.e., the lattice without
the chains attached). We also find the transmission to be strongly dependent on
the way the chains are attached to the lattice.Comment: 4 pages, 6 figures, submitted to Phys. Rev. 

D. Stauffer

E. Cuansing

H. Nakanishi

S. Datta

W. Press

English

arXiv

We study the transport of a quantum particle through square lattices of various sizes by employing the tight-binding Hamiltonian from quantum percolation. Input and output semi-infinite chains are attached to the lattice either by diagonal point-to-point contacts or by a busbar connection. We find resonant transmission and reflection occurring whenever the incident particle\u27s energy is near an eigenvalue of the lattice alone (i.e., the lattice without the chains attached). We also find the transmission to be strongly dependent on the way the chains are attached to the lattice

Cuansing, E.

Nakanishi, H.

Purdue E-Pubs

PhysicsPhysics Research PublicationsPurdue University Year Quantum interference effects in particletransport through square latticesE. Cuansing H. NakanishiThis paper is posted at Purdue e-Pubs.http://docs.lib.purdue.edu/physics articles/541Quantum interference effects in particle transport through square latticesE. Cuansing* and H. NakanishiDepartment of Physics, Purdue University, West Lafayette, Indiana 47907, USA(Received 26 June 2004; published 28 December 2004)We study the transport of a quantum particle through square lattices of various sizes by employing thetight-binding Hamiltonian from quantum percolation. Input and output semi-infinite chains are attached to thelattice either by diagonal point-to-point contacts or by a busbar connection. We find resonant transmission andreflection occurring whenever the incident particle’s energy is near an eigenvalue of the lattice alone(i.e., thelattice without the chains attached). We also find the transmission to be strongly dependent on the way thechains are attached to the lattice.DOI: 10.1103/PhysRevE.70.066142 PACS number(s): 05.60.Gg, 05.50.1q, 05.10.2a, 73.23.2bQuantum interference effects are important in the trans-port of particles in mesoscopic systems. Consider, for ex-ample, a particle traversing a square array of quantum dots.Assume the distance between dots is close enough so that theparticle can hop between nearest-neighbor dots. Consideringonly the effect of quantum interference, will the particle gothrough the lattice? Classically, the particle has a multitudeof paths to go from one end of the lattice to the other, de-pending on the size of the lattice. Quantum mechanically,however, constructive or destructive interference can occurbecause of the different path lengths. Thus, the transmissionof a particle is not assured even when there are classicallywell-defined paths for it to go through the lattice. In thiswork we investigate the effects of quantum interference inthe transport of a particle in discrete and finite square lat-tices.We consider the particle to be governed by the tight-binding Hamiltonian from quantum percolation[1,2]. ThisHamiltonian has the formH = oki j lvi jsuilk j u + u jlki ud, s1dwhereuil and u jl represent tight-binding basis functions cen-tered on sitesi and j , respectively, andvi j =1 if i and j arenearest neighbors andvi j =0 otherwise. The sum is only overnearest neighbors. In quantum percolation the particle is con-fined to traverse through disordered clusters constructedfrom the methods of percolation theory[3] with some occu-pation probabilityp. For p,1 there is disagreement whetherparticle states are localized or extended. In a review byMookerjeeet al. [4], they concluded that all states are local-ized and transport is dominated by statistically exceptionalnecklacelike resonant states. Daboulet al. [5], by calculatingthe moments of distances between pairs of lattice sites usingseries expansion methods, found evidence of a transitionfrom exponentially localized to extended or power-law de-caying states with an energy-dependent occupation probabil-ity thresholdpqsEd. Recent numerical studies of the scalingof the conductanceg by Hałdaś et al. [6], however, found allstates to be localized and no indication of a localization-delocalization transition. To properly interpret the results ofany calculation on the question of the existence of such atransition as a function of the degree of disorderp, however,we need to know the behavior of the transmittance in thelimit p→1. In this work we thus consider the limiting casep=1 wherein all sites in the lattice are occupied. We aregoing to show that even in this limit, the transport throughthe lattice is very sensitive to the incident particle’s energy,varying from complete transmission to complete reflection,and also to the type of coupling between the lattice and theinput and output chains.To determine the transport properties of a particle travers-ing the square lattice, we attach semi-infinite chains to theleft and right sides of that lattice. Call the left semi-infinitechain the input chain and the right semi-infinite chain theoutput chain. The particle is made incident to the lattice viathe input chain. If the particle goes through the lattice, then itexits via the output chain. Following the Landauer-Büttikerformalism [7], the conductance of the system can then bedetermined from the resulting transmission and reflectionamplitudes. Because of the semi-infinite chains, the corre-sponding matrix equation resulting from Eq.(1) is also infi-nite. Daboulet al. [5] recently described a method to trans-form the infinitely sized Hamiltonian matrix in Eq.(1) into areduced matrixH8 that is finite and involves only the latticeand its connections to the semi-infinite chains using an an-satz. We are implementing this method in this work.There are various ways of attaching the semi-infinitechains to the square lattice. In this work we consider twocomplimentary ways. One is by point-to-point contacts andthe other is by a busbar connection. In point-to-point con-tacts the input chain is singly attached to the top-leftmost sitewhile the output chain is singly attached to the bottom-rightmost site of the square lattice. In a busbar connectionthe input chain is attached to all the sites in the left edge ofthe lattice while the output chain is attached to all the sites inthe right edge of the lattice. This connection roughly corre-sponds to a physical configuration where the contacts aregood conductors and are large enough to span a macroscopicfraction of the sites along the edges of the lattice. The twotypes of couplings we have chosen are complimentary. Thisis because the single connections in point-to-point diagonalcontacts preserve the bipartite symmetry of the square latticewhile the multiple connections in the busbar connection de-stroy that symmetry.*Department of Physics and Astronomy, University of Pittsburgh,Pittsburgh, PA 15260.PHYSICAL REVIEW E 70, 066142(2004)1539-3755/2004/70(6)/066142(4)/$22.50 ©2004 The American Physical Society066142-1Daboulet al. [5] proposed the following ansatz:c−sn+1d = e−inq + reinq,s2dc+sn+1d = teinq,where n=0,1,2, . . . . Thec−sn+1d represent components ofthe wave function along the input chain and thec+sn+1d rep-resent components along the output chain.c−1 and c+1 arefor sites in the input and output chains, respectively, that aredirectly connected to the lattice. The ansatz restricts solu-tions to Eq.(1) in the form of incident and reflected planewaves along the input chain and transmitted plane wavesalong the output chain. Because of this ansatz, the energy ofthe incident particle is also restricted to be withinE=−2 andE=2. The transmission and reflection coefficients can be de-termined from thet and r in Eq. (2) by T= utu2 andR= ur u2.Once the Hamiltonian matrix in Eq.(1) is reduced toH8,the resulting problem can then be cast into the form of alinear equationsH8−Edc=g, whereg is solely a function ofE. This linear equation can then be solved forc onceE ischosen. We determinet andr from c by numerically solvingthe above linear equation exactly—i.e., fromc=sH8−Ed−1g.The matrixsH8−Ed is sparse and is numerically very close tobeing singular, making the use of standard methods such asthe LU decomposition, wherein the matrix is decomposedinto a lower triangular matrixL and an upper triangular ma-trix U, fail in some instances. As such, we implement thetechnique called singular value decomposition[8] to care-fully determine the inverse ofsH8−Ed.Shown in Fig. 1 is a plot of the transmission coefficientagainst the incident particle’s energy for a 10310 latticewith point-to-point contacts to the input and output chains.Also shown are the locations of the doubly degenerate eigen-values of the isolated square lattice. An isolated lattice is onewhere the input and output chains are not attached. The sys-tem is highly transmitting except at some values of energywhere there are sharp dips and the system becomes highlyreflecting. Notice that the dips occur near the eigenvalues ofthe isolated lattice. This phenomenon is analogous to reso-nant tunneling[9] of an incident particle through, for ex-ample, a double-barrier system. In such a system a peak inparticle transmission occurs whenever the incident particle’senergy is the same as the energy of one of the well’s boundstates. In this work, however, although there is no tunnelinginvolved, we do see resonant reflection whenever the energyof the incident particle falls near an eigenvalue of the iso-lated lattice.Notice as well that there is symmetry between theE.0side and theE,0 side. The square lattice has bipartite sym-metry and point-to-point contact connections preserve thatsymmetry. Maintaining bipartite symmetry can in turn beshown to ensure the symmetry inT aboutE=0.As the size of the isolated lattice is increased the numberof its associated eigenvalues will also increase. For the latticewith point-to-point contacts to the chains, we also see moredips in the transmission coefficient as we increase the size ofthe lattice. These dips are also located near the doubly de-generate eigenvalues of the corresponding isolated squarelattice.Shown in Figs. 2(a) and 2(b) are sample states that arehighly transmitting and highly reflecting, respectively, for aparticle traversing a 20320 lattice. The lattice is at thexyplane. The input chain is attached to the site in the latticelocated at(1, 1). The output chain is attached to the site inthe lattice located at(20, 20). The z axis is the absolutesquare of the components of the wave function at each cor-responding lattice site,ucsx,ydu2. For the highly transmittingstate we see a diagonal line of nonzeroc going from theinput to the output chains. Though this is not always true forall highly transmitting states, those with this feature are al-ways highly transmitting. In the highly reflecting state, onthe other hand, we see large fluctuations and destructive in-terference is manifest at the input and output sites.Shown in Fig. 3 is the transmissionT versus the incidentparticle’s energyE plot for a 10310 lattice with busbar con-nections to the input and output chains. In contrast to theFIG. 1. Plot of the transmission coefficientT against the inci-dent particle’s energyE for a 10310 lattice with point-to-pointcontacts to the input and output chains. The diamonds(L) are thelocations of the doubly degenerate eigenvalues of the isolatedsquare lattice.FIG. 2. Sample states of a particle traversing a 20320 latticewith point-to-point contacts to the input and output chains.(a)Highly transmitting state withE=0.39. (b) Highly reflecting statewith E=0.41.E. CUANSING AND H. NAKANISHI PHYSICAL REVIEW E 70, 066142(2004)066142-2case for point-to-point contacts, the system is mostly reflect-ing but with sharp peaks in transmission at certain values ofthe incident particle’s energy. Also shown in Fig. 3 are thelocations, as diamonds, of the doubly degenerate eigenvaluesof the corresponding isolated 10310 square lattice. Noticethat for theE,0 side the locations of the transmission peaksare near the eigenvalues of the isolated lattice. This is similarto the case with point-to-point contacts but instead of reso-nant reflection we see resonant transmission. For theE.0side, however, some of the peaks do not coincide with thelocations of the eigenvalues of the isolated lattice.From the ansatz shown in Eq.(2), the wave vectorq ofthe particle is related to its energy byE=2 cossqd, whereq=2p /l. For negative energies, the particle’s wavelength isconstrained to be within43 ,l,4. For positive energies, thewavelength should be within eitherl, 43 or l.4. Unlikethe case for point-to-point contacts, the lack of symmetrybetween theE,0 andE.0 sides of the plot in Fig. 3 indi-cates the significance of the incident particle’s wavelengthwhen undergoing through a busbar connection. Mathemati-cally, the multiple connections of the busbar destroys thebipartite symmetry of the square lattice and, consequently,destroying the symmetry inT aboutE=0.Let us call those sites at the sides of the lattice that aredirectly connected to the input and output chains as belong-ing to the input and output connection boundaries, respec-tively. Because of the multiple connections in a busbar, de-structive interference can occur at the connection boundaries,resulting in a vanishingly small transmission through the lat-tice. Some of the minima in transmission in Fig. 3 appear tobe consistent with rules analogous to optical-interferenceminima-maxima conditions on the boundary. For example,the condition that an integer number of wavelengths fitwithin the boundary of a lattice of sizeL3L—i.e., the con-dition L−1=nl—would suggest that certain values ofl re-sult in destructive interference. This would includel=1—i.e., E=2 from E=2 coss2p /ld—for all L and l=2sE=−2d for all odd values ofL. In actuality, completely de-structive interference occurs whenE=2 for all L.2 and alsowhenE=−2 for all LÞ2 and 4. There are also several otherminima in T that are consistent with this condition. For ex-ample, forL=5, l=4 sE=0d also satisfies the condition andindeed it is close to a transmission minimum. ForL=6, l=5/n sn=1,2,3,4d, corresponding toE<−1.62 and 0.62,also satisfy the condition and they are also close to a minimaof T. In addition, atl=2 sE=−2d we actually observe com-pletely constructive interference forL=2 and 4, whereL−1=s1/2dl and L−1=s3/2dl, respectively, are satisfied.These observations suggest strong influences of interferenceon or near the connection boundaries on the overall transmis-sion regarding the busbar connection though this boundaryinterference effect is far from providing a satisfactory expla-nation. In fact, since we have a discrete system with unitlattice constant rather than a continuousslit as in an opticalsystem, it is not clear whyl=1 actually leads to destructiveinterference rather than the opposite(except forL=2). Ofcourse, any influence of interference along the connectionboundary must only be a part of the story since interferenceactually occurs throughout the bulk of the system(on mostof which l is not even well defined) and since it must alsocompete with resonant transmission and reflection wheneverthe values of the incident particle’s energies at the inputchain fall near the eigenvalues of the isolated cluster.Two sample states for a particle traversing a 20320 lat-tice with busbar connections to the input and output chainsare shown in Fig. 4. The busbars are connected at they=1andy=20 sides of the lattice. Shown in Fig. 4(a) is a highlytransmitting state while Fig. 4(b) is a highly reflecting state.Notice that the difference in amplitudes between the states isseveral orders of magnitudes. In Fig. 4(b) strong destructiveinterference occurs in such a way that the statec nearlyvanishes within the lattice.In conclusion, we find resonant transmission and reflec-tion in the transport of a particle through finite square latticeswhenever the particle’s energy is near a doubly degenerateeigenvalue of the isolated lattice. The way the input andoutput chains are attached to the lattice influences the trans-port behavior of the incident particle. For point-to-point con-tacts particle transport is mostly transmitting but with trans-FIG. 3. Plot of T againstE for a 10310 lattice with busbarconnections to the input and output chains. The diamonds(L) areagain the locations of the doubly degenerate eigenvalues of theisolated square lattice.FIG. 4. Sample states of a particle traversing a 20320 latticewith busbar connections.(a) Highly transmitting state withE=0.017.(b) Highly reflecting state withE=0.051.QUANTUM INTERFERENCE EFFECTS IN PARTICLE… PHYSICAL REVIEW E 70, 066142(2004)066142-3mission dips whenever resonance occurs. For busbarconnections particle transport is mostly reflecting with trans-mission peaks whenever resonance also occurs. There are,however, peaks in transmission that cannot be accounted forby resonance. These peaks are the results of interferenceoriginating from the multiple connections in a busbar. Thesharp fluctuations of the conductance in thep=1 limit, bothas a function of the energyE of the incident particle and thetype of coupling chosen between the lattice and chains, willdefinitely affect the nature of transport whenever disorder isintroduced, especially in the weak-disorder limit. However,whether these fluctuations have a significant effect on theexistence or nature of the disputed localized-to-delocalizedtransition or whether they only affect transport at the weakdisorder limit still remains to be explored. We are currentlyconducting such a study.We would like to thank Y. Goldschmidt, Y. Lyanda-Geller,G. Baskaran, A. Finkelstein, L. Rokhinson, G. Giuliani, andN. Giordano for fruitful discussions.[1] S. Kirkpatrick and T. P. Eggarter, Phys. Rev. B6, 3598(1972).[2] P. de Gennes, P. Lafore, and J. Millot, J. Phys. Chem. Solids11, 105 (1959).[3] D. Stauffer and A. Aharony,Introduction to PercolationTheory, 2nd revised ed.(Taylor & Francis, Bristol, PA 1994).[4] A. Mookerjee, I. Dasgupta, and T. Saha, Int. J. Mod. Phys. B9, 2989(1995).[5] D. Daboul, I. Chang, and A. Aharony, Eur. Phys. J. B16, 303(2000).[6] G. Hałdaś, A. Kolek, and A. Stadler, Phys. Status Solidi B230, 249 (2002).[7] M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B31, 6207(1985).[8] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery,Nu-merical Recipes in Fortran, 2nd ed.(Cambridge UniversityPress, Cambridge, U.K., 1992).[9] S. Datta,Electronic Transport in Mesoscopic Systems(Cam-bridge University Press, Cambridge, U.K., 1995).E. CUANSING AND H. NAKANISHI PHYSICAL REVIEW E 70, 066142(2004)066142-4

Quantum interference effects in particle transport through square lattices

PhysicsPhysics Research PublicationsPurdue University Year Quantum interference effects in particletransport through square latticesE. Cuansing H. NakanishiThis paper is posted at Purdue e-Pubs.http://docs.lib.purdue.edu/physics articles/144Quantum interference effects in particle transport through square latticesE. Cuansing* and H. NakanishiDepartment of Physics, Purdue University, West Lafayette, Indiana 47907, USA(Received 26 June 2004; published 28 December 2004)We study the transport of a quantum particle through square lattices of various sizes by employing thetight-binding Hamiltonian from quantum percolation. Input and output semi-infinite chains are attached to thelattice either by diagonal point-to-point contacts or by a busbar connection. We find resonant transmission andreflection occurring whenever the incident particle’s energy is near an eigenvalue of the lattice alone (i.e., thelattice without the chains attached). We also find the transmission to be strongly dependent on the way thechains are attached to the lattice.DOI: 10.1103/PhysRevE.70.066142 PACS number(s): 05.60.Gg, 05.50.1q, 05.10.2a, 73.23.2bQuantum interference effects are important in the trans-port of particles in mesoscopic systems. Consider, for ex-ample, a particle traversing a square array of quantum dots.Assume the distance between dots is close enough so that theparticle can hop between nearest-neighbor dots. Consideringonly the effect of quantum interference, will the particle gothrough the lattice? Classically, the particle has a multitudeof paths to go from one end of the lattice to the other, de-pending on the size of the lattice. Quantum mechanically,however, constructive or destructive interference can occurbecause of the different path lengths. Thus, the transmissionof a particle is not assured even when there are classicallywell-defined paths for it to go through the lattice. In thiswork we investigate the effects of quantum interference inthe transport of a particle in discrete and finite square lat-tices.We consider the particle to be governed by the tight-binding Hamiltonian from quantum percolation [1,2]. ThisHamiltonian has the formH = okijlvijsuilkju + ujlkiud , s1dwhere uil and ujl represent tight-binding basis functions cen-tered on sites i and j, respectively, and vij =1 if i and j arenearest neighbors and vij =0 otherwise. The sum is only overnearest neighbors. In quantum percolation the particle is con-fined to traverse through disordered clusters constructedfrom the methods of percolation theory [3] with some occu-pation probability p. For p,1 there is disagreement whetherparticle states are localized or extended. In a review byMookerjee et al. [4], they concluded that all states are local-ized and transport is dominated by statistically exceptionalnecklacelike resonant states. Daboul et al. [5], by calculatingthe moments of distances between pairs of lattice sites usingseries expansion methods, found evidence of a transitionfrom exponentially localized to extended or power-law de-caying states with an energy-dependent occupation probabil-ity threshold pqsEd. Recent numerical studies of the scalingof the conductance g by Hałda et al. [6], however, found allstates to be localized and no indication of a localization-delocalization transition. To properly interpret the results ofany calculation on the question of the existence of such atransition as a function of the degree of disorder p, however,we need to know the behavior of the transmittance in thelimit p→1. In this work we thus consider the limiting casep=1 wherein all sites in the lattice are occupied. We aregoing to show that even in this limit, the transport throughthe lattice is very sensitive to the incident particle’s energy,varying from complete transmission to complete reflection,and also to the type of coupling between the lattice and theinput and output chains.To determine the transport properties of a particle travers-ing the square lattice, we attach semi-infinite chains to theleft and right sides of that lattice. Call the left semi-infinitechain the input chain and the right semi-infinite chain theoutput chain. The particle is made incident to the lattice viathe input chain. If the particle goes through the lattice, then itexits via the output chain. Following the Landauer-Büttikerformalism [7], the conductance of the system can then bedetermined from the resulting transmission and reflectionamplitudes. Because of the semi-infinite chains, the corre-sponding matrix equation resulting from Eq. (1) is also infi-nite. Daboul et al. [5] recently described a method to trans-form the infinitely sized Hamiltonian matrix in Eq. (1) into areduced matrix H8 that is finite and involves only the latticeand its connections to the semi-infinite chains using an an-satz. We are implementing this method in this work.There are various ways of attaching the semi-infinitechains to the square lattice. In this work we consider twocomplimentary ways. One is by point-to-point contacts andthe other is by a busbar connection. In point-to-point con-tacts the input chain is singly attached to the top-leftmost sitewhile the output chain is singly attached to the bottom-rightmost site of the square lattice. In a busbar connectionthe input chain is attached to all the sites in the left edge ofthe lattice while the output chain is attached to all the sites inthe right edge of the lattice. This connection roughly corre-sponds to a physical configuration where the contacts aregood conductors and are large enough to span a macroscopicfraction of the sites along the edges of the lattice. The twotypes of couplings we have chosen are complimentary. Thisis because the single connections in point-to-point diagonalcontacts preserve the bipartite symmetry of the square latticewhile the multiple connections in the busbar connection de-stroy that symmetry.*Department of Physics and Astronomy, University of Pittsburgh,Pittsburgh, PA 15260.PHYSICAL REVIEW E 70, 066142 (2004)1539-3755/2004/70(6)/066142(4)/$22.50 ©2004 The American Physical Society066142-1Daboul et al. [5] proposed the following ansatz:c−sn+1d = e−inq + reinq,s2dc+sn+1d = teinq,where n=0,1 ,2 , . . . . The c−sn+1d represent components ofthe wave function along the input chain and the c+sn+1d rep-resent components along the output chain. c−1 and c+1 arefor sites in the input and output chains, respectively, that aredirectly connected to the lattice. The ansatz restricts solu-tions to Eq. (1) in the form of incident and reflected planewaves along the input chain and transmitted plane wavesalong the output chain. Because of this ansatz, the energy ofthe incident particle is also restricted to be within E=−2 andE=2. The transmission and reflection coefficients can be de-termined from the t and r in Eq. (2) by T= utu2 and R= uru2.Once the Hamiltonian matrix in Eq. (1) is reduced to H8,the resulting problem can then be cast into the form of alinear equation sH8−Edc=g, where g is solely a function ofE. This linear equation can then be solved for c once E ischosen. We determine t and r from c by numerically solvingthe above linear equation exactly—i.e., from c= sH8−Ed−1g.The matrix sH8−Ed is sparse and is numerically very close tobeing singular, making the use of standard methods such asthe LU decomposition, wherein the matrix is decomposedinto a lower triangular matrix L and an upper triangular ma-trix U, fail in some instances. As such, we implement thetechnique called singular value decomposition [8] to care-fully determine the inverse of sH8−Ed.Shown in Fig. 1 is a plot of the transmission coefficientagainst the incident particle’s energy for a 10310 latticewith point-to-point contacts to the input and output chains.Also shown are the locations of the doubly degenerate eigen-values of the isolated square lattice. An isolated lattice is onewhere the input and output chains are not attached. The sys-tem is highly transmitting except at some values of energywhere there are sharp dips and the system becomes highlyreflecting. Notice that the dips occur near the eigenvalues ofthe isolated lattice. This phenomenon is analogous to reso-nant tunneling [9] of an incident particle through, for ex-ample, a double-barrier system. In such a system a peak inparticle transmission occurs whenever the incident particle’senergy is the same as the energy of one of the well’s boundstates. In this work, however, although there is no tunnelinginvolved, we do see resonant reflection whenever the energyof the incident particle falls near an eigenvalue of the iso-lated lattice.Notice as well that there is symmetry between the E.0side and the E,0 side. The square lattice has bipartite sym-metry and point-to-point contact connections preserve thatsymmetry. Maintaining bipartite symmetry can in turn beshown to ensure the symmetry in T about E=0.As the size of the isolated lattice is increased the numberof its associated eigenvalues will also increase. For the latticewith point-to-point contacts to the chains, we also see moredips in the transmission coefficient as we increase the size ofthe lattice. These dips are also located near the doubly de-generate eigenvalues of the corresponding isolated squarelattice.Shown in Figs. 2(a) and 2(b) are sample states that arehighly transmitting and highly reflecting, respectively, for aparticle traversing a 20320 lattice. The lattice is at the xyplane. The input chain is attached to the site in the latticelocated at (1, 1). The output chain is attached to the site inthe lattice located at (20, 20). The z axis is the absolutesquare of the components of the wave function at each cor-responding lattice site, ucsx ,ydu2. For the highly transmittingstate we see a diagonal line of nonzero c going from theinput to the output chains. Though this is not always true forall highly transmitting states, those with this feature are al-ways highly transmitting. In the highly reflecting state, onthe other hand, we see large fluctuations and destructive in-terference is manifest at the input and output sites.Shown in Fig. 3 is the transmission T versus the incidentparticle’s energy E plot for a 10310 lattice with busbar con-nections to the input and output chains. In contrast to theFIG. 1. Plot of the transmission coefficient T against the inci-dent particle’s energy E for a 10310 lattice with point-to-pointcontacts to the input and output chains. The diamonds (L) are thelocations of the doubly degenerate eigenvalues of the isolatedsquare lattice.FIG. 2. Sample states of a particle traversing a 20320 latticewith point-to-point contacts to the input and output chains. (a)Highly transmitting state with E=0.39. (b) Highly reflecting statewith E=0.41.E. CUANSING AND H. NAKANISHI PHYSICAL REVIEW E 70, 066142 (2004)066142-2case for point-to-point contacts, the system is mostly reflect-ing but with sharp peaks in transmission at certain values ofthe incident particle’s energy. Also shown in Fig. 3 are thelocations, as diamonds, of the doubly degenerate eigenvaluesof the corresponding isolated 10310 square lattice. Noticethat for the E,0 side the locations of the transmission peaksare near the eigenvalues of the isolated lattice. This is similarto the case with point-to-point contacts but instead of reso-nant reflection we see resonant transmission. For the E.0side, however, some of the peaks do not coincide with thelocations of the eigenvalues of the isolated lattice.From the ansatz shown in Eq. (2), the wave vector q ofthe particle is related to its energy by E=2 cossqd, where q=2p /l. For negative energies, the particle’s wavelength isconstrained to be within 43 ,l,4. For positive energies, thewavelength should be within either l, 43 or l.4. Unlikethe case for point-to-point contacts, the lack of symmetrybetween the E,0 and E.0 sides of the plot in Fig. 3 indi-cates the significance of the incident particle’s wavelengthwhen undergoing through a busbar connection. Mathemati-cally, the multiple connections of the busbar destroys thebipartite symmetry of the square lattice and, consequently,destroying the symmetry in T about E=0.Let us call those sites at the sides of the lattice that aredirectly connected to the input and output chains as belong-ing to the input and output connection boundaries, respec-tively. Because of the multiple connections in a busbar, de-structive interference can occur at the connection boundaries,resulting in a vanishingly small transmission through the lat-tice. Some of the minima in transmission in Fig. 3 appear tobe consistent with rules analogous to optical-interferenceminima-maxima conditions on the boundary. For example,the condition that an integer number of wavelengths fitwithin the boundary of a lattice of size L3L—i.e., the con-dition L−1=nl—would suggest that certain values of l re-sult in destructive interference. This would includel=1—i.e., E=2 from E=2 coss2p /ld—for all L and l=2sE=−2d for all odd values of L. In actuality, completely de-structive interference occurs when E=2 for all L.2 and alsowhen E=−2 for all LÞ2 and 4. There are also several otherminima in T that are consistent with this condition. For ex-ample, for L=5, l=4 sE=0d also satisfies the condition andindeed it is close to a transmission minimum. For L=6, l=5/n sn=1,2 ,3 ,4d, corresponding to E<−1.62 and 0.62,also satisfy the condition and they are also close to a minimaof T. In addition, at l=2 sE=−2d we actually observe com-pletely constructive interference for L=2 and 4, where L−1= s1/2dl and L−1= s3/2dl, respectively, are satisfied.These observations suggest strong influences of interferenceon or near the connection boundaries on the overall transmis-sion regarding the busbar connection though this boundaryinterference effect is far from providing a satisfactory expla-nation. In fact, since we have a discrete system with unitlattice constant rather than a continuous slit as in an opticalsystem, it is not clear why l=1 actually leads to destructiveinterference rather than the opposite (except for L=2). Ofcourse, any influence of interference along the connectionboundary must only be a part of the story since interferenceactually occurs throughout the bulk of the system (on mostof which l is not even well defined) and since it must alsocompete with resonant transmission and reflection wheneverthe values of the incident particle’s energies at the inputchain fall near the eigenvalues of the isolated cluster.Two sample states for a particle traversing a 20320 lat-tice with busbar connections to the input and output chainsare shown in Fig. 4. The busbars are connected at the y=1and y=20 sides of the lattice. Shown in Fig. 4(a) is a highlytransmitting state while Fig. 4(b) is a highly reflecting state.Notice that the difference in amplitudes between the states isseveral orders of magnitudes. In Fig. 4(b) strong destructiveinterference occurs in such a way that the state c nearlyvanishes within the lattice.In conclusion, we find resonant transmission and reflec-tion in the transport of a particle through finite square latticeswhenever the particle’s energy is near a doubly degenerateeigenvalue of the isolated lattice. The way the input andoutput chains are attached to the lattice influences the trans-port behavior of the incident particle. For point-to-point con-tacts particle transport is mostly transmitting but with trans-FIG. 3. Plot of T against E for a 10310 lattice with busbarconnections to the input and output chains. The diamonds (L) areagain the locations of the doubly degenerate eigenvalues of theisolated square lattice.FIG. 4. Sample states of a particle traversing a 20320 latticewith busbar connections. (a) Highly transmitting state with E=0.017. (b) Highly reflecting state with E=0.051.QUANTUM INTERFERENCE EFFECTS IN PARTICLE  PHYSICAL REVIEW E 70, 066142 (2004)066142-3mission dips whenever resonance occurs. For busbarconnections particle transport is mostly reflecting with trans-mission peaks whenever resonance also occurs. There are,however, peaks in transmission that cannot be accounted forby resonance. These peaks are the results of interferenceoriginating from the multiple connections in a busbar. Thesharp fluctuations of the conductance in the p=1 limit, bothas a function of the energy E of the incident particle and thetype of coupling chosen between the lattice and chains, willdefinitely affect the nature of transport whenever disorder isintroduced, especially in the weak-disorder limit. However,whether these fluctuations have a significant effect on theexistence or nature of the disputed localized-to-delocalizedtransition or whether they only affect transport at the weakdisorder limit still remains to be explored. We are currentlyconducting such a study.We would like to thank Y. Goldschmidt, Y. Lyanda-Geller,G. Baskaran, A. Finkelstein, L. Rokhinson, G. Giuliani, andN. Giordano for fruitful discussions.[1] S. Kirkpatrick and T. P. Eggarter, Phys. Rev. B 6, 3598 (1972).[2] P. de Gennes, P. Lafore, and J. Millot, J. Phys. Chem. Solids11, 105 (1959).[3] D. Stauffer and A. Aharony, Introduction to PercolationTheory, 2nd revised ed. (Taylor & Francis, Bristol, PA 1994).[4] A. Mookerjee, I. Dasgupta, and T. Saha, Int. J. Mod. Phys. B9, 2989 (1995).[5] D. Daboul, I. Chang, and A. Aharony, Eur. Phys. J. B 16, 303(2000).[6] G. Hałda, A. Kolek, and A. Stadler, Phys. Status Solidi B230, 249 (2002).[7] M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B31, 6207 (1985).[8] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Nu-merical Recipes in Fortran, 2nd ed. (Cambridge UniversityPress, Cambridge, U.K., 1992).[9] S. Datta, Electronic Transport in Mesoscopic Systems (Cam-bridge University Press, Cambridge, U.K., 1995).E. CUANSING AND H. NAKANISHI PHYSICAL REVIEW E 70, 066142 (2004)066142-4

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Quantum interference effects in particle transport through square lattices

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