Magnetic field induced metal-insulator transition in a Kagome Nanoribbon
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Magnetic field induced metal-insulator transition in a Kagome Nanoribbon
Moumita Dey, Santanu K. Maiti,
1, 2, ∗ and S. N. Karmakar Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics,Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India Department of Physics, Narasinha Dutt College, 129 Belilious Road, Howrah-711 101, India
In the present work we investigate two-terminal electron transport through a finite width kagomelattice nanoribbon in presence of a perpendicular magnetic field. We employ a simple tight-binding(T-B) Hamiltonian to describe the system and obtain the transmission properties by using Green’sfunction technique within the framework of Landauer-B¨uttiker formalism. After presenting an ana-lytical description of energy dispersion relation of a kagome nanoribbon in presence of the magneticfield, we investigate numerically the transmittance spectra together with the density of states andcurrent-voltage characteristics. It is shown that for a specific value of the Fermi energy the kagomenetwork can exhibit a magnetic field induced metal-insulator transition which is the central investi-gation of this communication. Our analysis may be inspiring in designing low-dimensional switchingdevices.
PACS numbers: 73.63.-b, 73.43.Qt, 73.21.-b
I. INTRODUCTION
Remarkable advances in nanotechnology have enabledus to fabricate different semiconductor superlattices andoptical lattice systems which are promising candidatesto simulate and investigate a lot of rich and exotic quan-tum phenomena in condensed matter physics e.g., Quan-tum Hall Effect (QHE) [1], Spin Hall Effect (SHE) [2],manifestation of topological insulators [3, 4], etc. Un-like the bulk materials, the quantum dot superlatticesystems with different geometry are easier to design bynanolithography technique and the controllability of elec-tron filling in such lattice systems is possible by applyinga gate voltage [5]. Earlier Fukui et al. have fabricatedsquare, triangular and kagome lattices using InAs wireson a GaAs substrate [6–8]. In 2001, Abrecht et al. haveprepared a two-dimensional square lattice using GaAsand observed the Hofstader butterfly in energy spectraby applying a perpendicular magnetic field [9]. Amongthese various lattice systems kagome lattice structure oc-cupies a very special position because of its fascinatingproperty of having a ferromagnetic ground state at zerotemperature where the single particle energy spectra havea complete dispersionless flat band in the tight-bindingapproximation.In 1993, Mielke and Tasaki [10, 11] have investigatedthat the Coulomb interaction between the degenerateelectronic states induces ferromagnetism at zero temper-ature, when the flat band is half-filled with electrons.In 2002, Kimura et al. [12] have shown that the flatband is completely destroyed by the application of a per-pendicular magnetic field, and by calculating the Drudeweight (D) in a closed system with Hubbard interac-tion they have predicted that the magnetic field caninduce a metal-insulator as well as a ferromagnetic-to- ∗ Electronic address: [email protected] paramagnetic transition. Ishii and Nakayama in 2004have shown that the exitonic binding energy of a kagomelattice is larger than other two-dimensional (2D) or evenone-dimensional (1D) lattices because of the macroscopicdegree of degeneracy and the localized nature of the flatband states [13], in contrast to the concept that exitonicbinding energy in spatially higher dimension is smallerthan those of spatially lower dimension. To elucidate thetransport properties of such unique flat band electronicstates, Ishii and Nakayama again in 2006, have stud-ied electron transport through a kagome lattice chain, in
SOURCE KAGOME LATTICE NANORIBBON DRAINB
FIG. 1: (Color online). Schematic view of a Kagome nanorib-bon attached to finite width leads, viz, source and drain, inpresence of a perpendicular magnetic field. presence of an in-plane electric field [14]. They have ob-served a large current peak arising from electronic trans-mission through the flat band states when the electricfield is applied in perpendicular direction to the kagomechain, while no current is observed when the field is ap-plied along the chain. These anisotropic features are theoutcome of the itinerant and localized characteristics ofthe flat band like states which are originated due to quan-tum interference effect and are largely sensitive to the ex-ternal perturbations like magnetic field or electric field.These are the motivation of our present work wherewe investigate two-terminal quantum transport througha finite size kagome lattice ribbon in presence of a per-pendicular magnetic field using Green’s function tech-nique [15, 16] within Landauer-B¨uttiker formalism [17].Following an analytical description, in presence of themagnetic field, of a kagome lattice nanoribbon we studynumerically the transmittance spectra together with thedensity of states (DOS) and current-voltage ( I - V ) char-acteristics. Most interestingly we notice that the current-voltage characteristics reflect the feature of insulator-to-metal transition, when the equilibrium Fermi level E F isfixed at the flat band i.e., E F = − t where t being thenearest-neighbor hopping strength and the magnetic fluxis switched from zero to a non-zero one. Again, a simi-lar but reverse transition takes place when the electronfilling is set at 2 t i.e., E F = 2 t .The paper is organized as follows. With a brief in-troduction and motivation (Section I), in Section II, wedescribe the model and theoretical formulation to de-termine the transmission probability, DOS and currentthrough the nanostructure. The analytical and numeri-cal results are illustrated in Section III. Finally, in SectionIV, we summarize our results. II. THEORETICAL FORMULATIONA. Model and Hamiltonian
We start with Fig 1, where a finite width kagome lat-tice ribbon, subject to a perpendicular magnetic field ~B ,is attached to two semi-infinite multi-channel leads, com-monly known as source and drain. These leads are char-acterized by the electrochemical potentials µ S and µ D ,respectively, under the non-equilibrium condition whenan external bias voltage is applied. Both the leads havealmost the same cross section as the sample to reducethe effect of the scattering induced by wide-to-narrowgeometry at the sample-lead interface. The whole sys-tem is described within a single electron picture by asimple tight-binding Hamiltonian with nearest-neighborhopping approximation.The Hamiltonian representing the entire system can bewritten as a sum of three terms, H = H kagome + H leads + H tun . (1)The first term represents the Hamiltonian for the kagomelattice ribbon which is coupled to two electron reser-voirs through conducting leads i.e., source and drain.In Wannier basis the Hamiltonian of the ribbon in non-interacting picture reads as, H kagome = X i ǫ c † i c i + X h ij i h ˜ t ij c † i c j + h.c. i (2)where, ǫ refers to the site energy of an electron at each siteof the kagome ribbon and ˜ t ij corresponds to the nearest-neighbor hopping integral between the sites in presenceof a perpendicular magnetic field. The effect of magneticfield ~B (= ~ ∇ × ~A ) is incorporated in the hopping term ˜ t ij through the Peierl’s phase factor and for a chosen gaugefield it becomes, ˜ t ij = t e − i πφ ~rj R ~ri ~A.~dl (3)where, t gives the nearest-neighbor hopping integral inthe absence of magnetic filed and φ (= ch/e ) is the ele-mentary flux quantum. The specific choice of the gaugefield in this case and the exact calculation of the Peierl’s ζ
74 6Unit Cell Finite size Kagome LatticeNx Ny = 2= 4 ; B n=3n=2n=1n=0
FIG. 2: (Color online). Schematic view of a Kagome latticein presence of a perpendicular magnetic field where the unitcell configuration (dashed region) and the co-ordinate axesare shown. phase factor has been discussed in the sub-sequent sec-tion. c † i and c i correspond to the creation and annihila-tion operators, respectively, of an electron at the site i ofthe nanoribbon.The second and third terms of Eq. 1 describe the T-B Hamiltonians for the multi-channel semi-infinite leadsand sample-to-lead coupling. In Wannier basis they canbe written as follows. H leads = H S + H D = X α = S,D (X n ǫ l c † n c n + X mn t l (cid:2) c † m c n + h.c. (cid:3)) , (4)and, H tun = X S,D t c h c † i c m + c † m c i i . (5)Here, ǫ l and t l stand for the site energy and nearest-neighbor hopping integral in the leads. c † n and c n are thecreation and annihilation operators, respectively, of anelectron at the site n of the leads. The hopping integralbetween the boundary sites of the lead and the sample isparametrized by t c . B. Calculation of the Peierl’s phase factor
Let us now evaluate ˜ t ij incorporating the Peierl’s phasefactor.We choose the gauge for the vector potential ~A associ-ated with the magnetic field ~B (= B ˆ z ), perpendicular tothe lattice plane, in the form, ~A = − By ˆ x + By √ y = (cid:18) − , √ , (cid:19) By. (6)This specific choice is followed from a literature [18], andthe purpose of doing so is solely due to the simplificationof the factor R ~A.~dl along various paths of the ribbon.With this particular choice of gauge, we determine˜ t ij for the three different types of paths of the ribbonthrough which an electron can hop in the following ways. • Case 1:
First, we consider the hopping along the ζ axis(see Fig. 2). In this case, our choice of gauge ensuresthat the component of ~A along ζ axis is zero i.e., A ζ = 0. Downward pointing triangleUpward pointing triangle ijk ij k
FIG. 3: (Color online). Upward and downward pointing tri-angles labeled with proper site indices.
So, R ~A.~dl = 0. Therefore, ˜ t ij = t when an electron hopsalong the ζ axis, either along the + ve , − ve or its paralleldirection. • Case 2:
Here, we consider the hopping along the X (+ve for forward hopping and − ve for backward hopping)direction only. In this case R ~A.~dl = R ( − By ) dl = − Bya . a is the lattice spacing. Now, for n = 0 line, y = 0,and therefore, ˜ t ij = t . For the line n = 1, y = √ a ,and hence, R ~A.~dl = −√ Ba . If we set φ = √ Ba ,the flux enclosed by the smallest triangle (as shown inFig. 3), then for the line n = 1, ˜ t ij = t e i πφφ . Therefore,in general, we can write the hopping term for n -th lineas, ˜ t ij = t e i πnφφ (7)where, n = 0 , , , . . . (2 N y − • Case 3:
Finally, we consider the case where electronshop along the direction from site 2 to site 3 and all its parallel directions. In this case, ~r j Z ~r i ~A.~dl = B √ (cid:0) y j − y i (cid:1) = √ B (cid:0) ζ j − ζ i (cid:1) (8)It can be shown by straightforward calculation that for anupward pointing triangle (shown in Fig. 3) the modifiedhopping strengths are given by,˜ t i → j = t e − i πφφ ( n + ) , and , ˜ t j → i = t e i πφφ ( n + ) . (9)The value of n belongs to the base line of the triangle.Similarly, for a downward pointing triangle (shown inFig. 3) the modified hopping integrals can be written as,˜ t i → j = t e − i πφφ ( n + ) , and , ˜ t j → i = t e i πφφ ( n ′ − ) . (10)Here, n ′ = n + 1, as for a downward pointing triangle thesites i and j do not belong to the same value of n . C. Evaluation of the transmission probability andcurrent by Green’s function technique
To obtain the transmission probability of an electronthrough such a bridge system, we use Green’s functionformalism. Within the regime of coherent transport andin the absence of Coulomb interaction this technique iswell applied.The single particle Green’s function operator repre-senting the entire system for an electron with energy E is defined as, G = ( E − H + iη ) − (11)where, η → + .Following the matrix form of H and G the problemof finding G in the full Hilbert space H can be mappedexactly to a Green’s function G effkagome corresponding toan effective Hamiltonian in the reduced Hilbert space ofthe ribbon itself and we have, G = G effkagome = (cid:16) E − H kagome − Σ S − Σ D (cid:17) − , (12)where, Σ S ( D ) = H † tun G S ( D ) H tun . (13)These Σ S and Σ D are the contact self-energies intro-duced to incorporate the effect of coupling of the kagomeribbon to the source and drain. It is evident from Eq. 13that the form of the self-energies are independent of theconductor itself through which transmission is studied.Following Lee and Fisher’s expression for the transmis-sion probability of an electron from the source to drainwe can write, T SD = Tr[ Γ S G r Γ D G a ] . (14)Γ α ’s ( α = S and D ) are the coupling matrices represent-ing the coupling between the ribbon and the leads andthey are mathematically defined by the relation, Γ α = i (cid:2) Σ rα − Σ aα (cid:3) . (15)Here, Σ rα and Σ aα are the retarded and advanced self-energies associated with the α -th lead, respectively.It is shown in literature by Datta et al. [15, 16] thatthe self-energy can be expressed as a linear combinationof a real and an imaginary part in the form, Σ rα = Λ α − i ∆ α . (16)The real part of self-energy describes the shift of the en-ergy levels and the imaginary part corresponds to thebroadening of the levels. The finite imaginary part ap-pears due to incorporation of the semi-infinite leads hav-ing continuous energy spectrum. Therefore, the couplingmatrices can easily be obtained from the self-energy ex-pression and is expressed as, Γ α = − Σ α ) . (17)Considering linear transport regime, at absolute zerotemperature the linear conductance g is obtained usingtwo-terminal Landauer conductance formula, g = 2 e h T ( E F ) . (18)With the knowledge of the effective transmission proba-bility we compute the current-voltage ( I - V ) characteris-tics by the standard formalism based on quantum scat-tering theory. I ( V ) = 2 eh ∞ Z −∞ T [ f S ( E ) − f D ( E )] dE. (19)Here, f S ( D ) ( E ) = (cid:20) e E − µS ( D ) kBT (cid:21) − is the Fermi func-tion corresponding to the source and drain. At absolutezero temperature the above equation boils down to thefollowing expression. I ( V ) = 2 eh E F + eV Z E F − eV T ( E ) dE. (20)In our present work we assume that the potential droptakes place only at the boundary of the conductor. D. Evaluation of the self-energy
Finally, the problem comes to the point of evaluat-ing self-energy for the finite-width, multi-channel 2Dleads [19, 20]. Now, for the semi-infinite leads (source and drain) as the translational invariance is preservedin X direction only, the wave function amplitude atany arbitrary site m of the leads can be written as, χ m ∝ e ik x m x a sin( k y m y a ), with energy E = 2 t L [cos( k x a ) + cos( k y a )] . (21)In Eq. 21, k x is continuous, while k y has discrete valuesgiven by, k y ( n ) = nπ ( m + 1) a (22)where, n = 1 , , . . . m . m is the total number of trans-verse channels in the leads. In our case m = 2 N y .The self-energy matrices have non-zero elements onlyfor the sites on the edge layer of the sample coupled tothe leads and it is given by,Σ rS ( D ) ( m, n ) = 2 n y + 1 X k y sin( k y m y a )Σ r ( k y ) sin( k y n y a )(23)where, Σ r ( k y ) is the self-energy of the each transversechannel with a specific value of k y . It is expressed in thefollowing from:Σ r ( k y ) = t c t L (cid:20) ( E − ǫ ( k y )) − i q t L − ( E − ǫ ( k y )) (cid:21) (24)with ǫ ( k y ) = 2 t L cos( k y a ), when the energy lies withinthe band i.e., | E − ǫ ( k y ) | < t L ; and,Σ r ( k y ) = t c t L (cid:20) ( E − ǫ ( k y )) ∓ q ( E − ǫ ( k y )) − t L (cid:21) (25)when the energy lies outside the band. The − ve signcomes when E > ǫ ( k y )+2 | t L | , while the + ve sign appearswhen E < ǫ ( k y ) − | t L | . III. RESULTS AND DISCUSSION
We begin by referring the values of different parametersused for our calculations. Throughout our presentation,we set ǫ = ǫ l = 0 and fix all the hopping integrals ( t , t l and t c ) at −
1. We measure the energy scale in unit of t and choose the units where c = e = h = 1. The magneticflux φ is measured in unit of the elementary flux quantum φ = ch/e . A. Analytical description of energy dispersionrelation in presence of magnetic field
To make this present communication a self containedstudy let us start with the energy band structure of afinite width kagome lattice nano-ribbon. We obtain itanalytically.Here we follow a general approach to evaluate theband structure of a quasi one-dimensional kagome latticenanoribbon. We establish an effective difference equationsimilar to the case of an 1D infinite chain and this canbe done by proper choice of a unit cell from the ribbon.The schematic view of a unit cell is shown by the dashedregion of Fig. 2. We consider the ribbon to be made upof the unit cells consisting of N atomic sites. Within H a L Φ= Π Π Π Π- -
202 Wavevector H k x L E n er gy H E L H b L Φ= Π Π Π Π- -
202 Wavevector H k x L E n er gy H E L H c L Φ= Π Π Π Π -
202 Wavevector H k x L E n er gy H E L FIG. 4: (Color online). Energy levels as a function of k x for a finite width kagome nanoribbon considering N y = 5 fordifferent values of magnetic flux φ . the tight-binding approximation the effective differenceequation for the n -th unit cell reads as,( E − ˜ ǫ ) ψ n = ˜ τ ψ n +1 + ˜ τ † ψ n − . (26)Here, ψ n is a column vector with N elements represent-ing the wave function at the n -th unit cell. ˜ ǫ is a N × N dimensional matrix and it represents the site-energy ma-trix of the unit cell. ˜ τ corresponds to the hopping in-tegral between two neighboring unit cells with identicaldimension of the site-energy matrix.Since the system is translationally invariant along the X direction, the vector ψ n can be written as, ψ n = ˜ A e ik x λ n . (27)Here, ˜ A = A A ... A N . (28)Substituting the value of ψ n in Eq. 26 we have,( E − ˜ ǫ ) = τ e ik x λ + τ † e − ik x λ (29)The non-trivial solutions of Eq. 29 are obtained from thefollowing condition. (cid:12)(cid:12) E − ˜ ǫ − τ e ik x λ − τ † e − ik x λ (cid:12)(cid:12) = 0 (30)Thus, simplifying Eq. 30, we get the desired energy-dispersion relation ( E vs. k x curve) for the finite widthkagome nanoribbon.As illustrative example, in Fig. 4 we show the varia-tion of energy levels as a function of wave vector k x for afinite width kagome nanoribbon considering N y = 5 for - - H Φ L E n er gy H E L FIG. 5: (Color online). Energy levels as a function of flux φ for a finite size kagome ribbon considering N x = 6 and N y = 2. The circular region is re-plotted in the inset to showthe removal of the degeneracy with magnetic flux. different values of magnetic flux φ . Most interestinglywe see that, when φ = 0, a highly degenerate and dis-persionless flat band appears at E = − t (blue line ofFig. 4(a)) along with the dispersive energy levels. But,as long as the magnetic flux is switched on, the degener-acy of the energy levels at the typical energy E = − t isbroken and the flatness of these levels gets reduced i.e.,they start to be dispersive. This feature is clearly ob-served from Fig. 4(b). Finally, when the flux φ is set at φ /
2, the flat band again re-appears in the spectrum andit situates at the bottom of the energy spectrum i.e., at E = 2 t (red line of Fig. 4(c)), which is exactly oppositeto that of the case when φ = 0. This feature provides thecentral idea for the exhibition of magnetic field inducedmetal-insulator transition in a kagome lattice ribbon. B. Energy-flux characteristics
The behavior of single-particle energy levels as a func-tion of flux φ for a finite size kagome lattice is shownin Fig. 5, where the energy eigenvalues are obtained bydiagonalizing the Hamiltonian matrix. Here we choose H a L N Y = N X = Φ= - - T H E L H b L N Y = N X = Φ= - - T H E L H c L N Y = N X = Φ= - - T H E L FIG. 6: (Color online). Transmission probability as a functionof energy for three different lengths of the kagome nanoribbonwhen N y is fixed at a particular value ( N y = 2) in the absenceof magnetic flux φ . N x = 6 and N y = 2. This E - φ spectrum is sometimescalled as Hofstader butterfly spectrum. From the spec-trum it is clearly observed that a highly degenerate en-ergy level is present at E = − t when φ is set at zero,and, the application of a very small non-zero flux helps tobreak the degeneracy which is clearly shown in the insetof Fig. 5. The presence of even a very small magnetic fluxaffects the phase of the electronic wave functions and thusdestroys the quantum interference originated flat band.The flat band again comes back because of the quan-tum interference effect at the energy E = 2 t when φ isswitched to φ /
2. The E - φ spectrum is periodic in φ withperiodicity φ (= 1 in our chosen unit system) and it is mirror symmetric about φ = φ /
2. For an infinitely largesystem when φ = n/ m , n and m are two arbitrary in-tegers, 3 m magnetic mini-bands appear in the spectrum,and, for m = 1 i.e., φ = n/
8, the number of mini-bandsis smallest and the gap becomes widest. But in our case,we do not observe such gaps, as they are smeared outbecause of the decoherence of the wave functions.
C. Length dependence on transmission probability
As illustrative example, in Fig. 6, we show the varia-tion of transmission probability as a function of inject-ing electron energy for a finite size kagome ribbon inthe absence of magnetic flux φ . Here we fix the width( N y = 2) and vary the length N x of the ribbon. Forthree different lengths the results are shown in (a), (b)and (c), respectively. Sharp resonant peaks are observed N y g H E L FIG. 7: (Color online). Conductance as a function of ribbonwidth N y for a particular energy E = 0 .
75 in the absence ofmagnetic flux φ . in the T - E spectrum associated with the eigenenergiesof the ribbon, and therefore, it can be predicted thatthe transmission spectrum manifests itself the electronicstructure of the ribbon. In our numerical calculations,though all the hopping parameters are set to the samevalue ( t = t L = t c = − Φ= H a L - - Ρ H E L Φ= H b L - - Ρ H E L Φ= H c L - - Ρ H E L H a L N Y = N X = Φ= - - T H E L H b L N Y = N X = Φ= - - T H E L H c L N Y = N X = Φ= - - T H E L FIG. 8: (Color online). Transmission probability (right column) and density of states (left column) as a function of energy fora finite size kagome ribbon ( N x = 6 and N y = 2) for different values of flux φ . more dense, but the quantized nature and values of T ( E )remains unchanged as the ribbon width is kept fixed ata certain value. D. Conductance quantization
In order to understand the dependence of conductance g on the ribbon width N y , in Fig. 7, we show the vari-ation of two-terminal conductance as a function of N y in the absence of magnetic flux when N x is fixed at 30.The conductance is directly proportional to the numberof transverse modes in the leads which is defined by thefactor m = 2 N y . Following Landauer formula the con-ductance at T = 0 k is related to the transmission proba-bility T ( E F ) as given in Eq. 18. It gives the total trans-mission probability of an electron through the ribbon byadding the net contributions of all the channels. Thuswe can write, T ( E F ) = m X i,j =1 T ij ( E F ) (31) where, T ij ( E F ) corresponds to the transmission probabil-ity between the i and j -th modes of the two leads. Withthe increase of N y , the number of propagating transversemodes ( m ) also increases, and therefore, the enhance-ment in the magnitude of conductance is achieved whichis clearly shown in Fig. 7. E. Effect of magnetic flux on transport properties
Now we focus our attention on the effect of magneticflux on electronic transport through a finite size kagomeribbon. We illustrate it by presenting transmission prob-abilities and current-voltage characteristics.
1. Transmission probability and DOS spectra
In Fig. (8), we present the variation of transmissionprobability (right column) along with the nature of den-sity of states (DOS) profile (left column) as a function ofenergy for a finite size kagome ribbon considering threedifferent values of magnetic flux φ . In the top of theleft column, DOS is shown when φ is set at 0. A sharppeak is observed at the right edge of the energy banddue to localized states. These states are highly degener-ate and are generally pinned at E = − t . The existenceof these localized states is a characteristic feature of thiskind of topology due to quantum interference effect be-tween the electronic wave functions. Correspondingly, inthe transmittance spectrum (top of the right column) weobtain transmission peaks at the positions of extendedeigenstates but no peak is observed exactly at E = − t referring to the localized states.The presence of an external magnetic field disturbsthe phases of the wave functions resulting into annihi-lation of the localized states and thus a continuum of E F = H a L - - - .1 - .050.05.1 Voltage H V L C u rre n t H I L Φ= Φ= E F =- H b L - - - .1 - .050.05.1 Voltage H V L C u rre n t H I L Φ= Φ= FIG. 9: (Color online). Current-voltage characteristics fora finite size kagome ribbon ( N x = 6 and N y = 2). Thepossibility of magnetic field induced metal-insulator transitionfor two different values of E F is shown. extended eigenstates is observed as shown in the mid-dle panel of the left column where φ is set to an arbi-trary value 0 .
3. Accordingly, the transmission peaks areobserved and notably finite transmission probability isobtained at E = − t , which was zero in the φ = 0 case.This feature bears the crucial importance for showing theproperty of insulator-to-metal transition.Finally, when the magnetic flux is set equal to the halfflux quantum i.e., φ = φ /
2, again the quantum inter-ference effect takes place in such a way that the sharppeak associated with the localized states re-appears inthe DOS spectrum (bottom of the left column), but nowat the left edge ( E = 2 t ) of the energy band. Simi-larly, for this energy transmission probability vanishes and peaks in the transmittance spectrum are observedwhere the extended states are situated (bottom of theright column).
2. Current-voltage characteristics
Here we explore the possibility of magnetic field in-duced metal-insulator (MI) transition by investigatingthe current-voltage ( I - V ) characteristics through a finitesize kagome ribbon. It is shown in Fig. 9, where twodifferent values of E F are chosen. The current throughthe nano-structure is obtained via Landauer-B¨uttiker for-malism by integrating over the transmission curve (seeEq. 20). When the equilibrium Fermi level E F is fixedat − t , almost zero current is obtained for small biasvoltage in the absence of any magnetic field (blue line inFig. 9(a)) since the localized states, placed at E = − t ,do not contribute anything in electronic conduction. Asthe magnetic field is switched on, localization gets de-stroyed. Therefore, setting E F at the particular value − t , sufficiently large current is obtained (red line inFig. 9(a)) compared to the previous case where no mag-netic field is given which yields the metallic behavior.From the careful observation we see that when E F = − t ,the best current magnification is achieved for φ = φ / t , instead of − t . When φ = 0, a large currentresponse (metallic state) is observed due to the presenceof extended energy eigenstates around E F . On the otherhand, for φ = φ /
2, highly degenerate localized statesare formed at E = 2 t , and thus, a very low current isachieved in response to the applied bias voltage whichleads to the insulating phase. IV. CLOSING REMARKS
To summarize, in the present paper we investigate elec-tron transport properties through a finite size kagomelattice nanoribbon attached to two finite width leads byusing Green’s function technique within the frameworkof Landauer-B¨uttiker formalism. The model quantumsystem is described by a simple tight-binding Hamilto-nian. Following the analytical description of energy dis-persion relation of a finite width kagome nanoribbon inpresence of magnetic field, we compute numerically thetransmittance-energy spectra together with the densityof states and current-voltage characteristics. From theenergy dispersion curve we see quite interestingly that,at φ = 0, a highly degenerate and dispersionless flat bandappears at E = − t . But as long as the magnetic fieldis switched on the degeneracy at this typical energy isbroken and the flatness of these levels gets reduced i.e.,they start to be dispersive. Finally, when the flux φ is set at φ /
2, the flat band again re-appears and liesat the bottom of the spectrum i.e., at E = 2 t . Thisphenomenon provides the central idea for the exhibitionof magnetic field induced metal-insulator transition in akagome lattice nanoribbon and it is justified by study-ing the current-voltage characteristics for two differentchoices of the equilibrium Fermi energy E F . Our analysiscan be utilized in designing nano-scale switching devices.The results presented in this communication are worked out for absolute zero temperature. However, theyshould remain valid even in a certain range of finite tem-peratures ( ∼
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