In the framework of a tight-binding model, we study energy spectra and persistent currents in mesoscopic Fibonacci rings threaded by a magnetic flux. It is found that the flux-dependent electron eigenenergies E(Φ) in mesoscopic Fibonacci rings still form "bands" with respect to the flux Φ, but there is a scaling relation between the total "bandwidth" and the Fibonacci number. When the strength of the one-dimensional quasiperiodic potential increases, the persistent current decreases rapidly. Interestingly, for a generalized mixing model of mesoscopic Fibonacci rings, free-electron-like persistent current may appear if the number of electrons of the system takes a specific value.published_or_final_versio

Hu, A

Jiang, SS

Jin, GJ

Wang, ZD

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Title Persistent currents in mesoscopic Fibonacci ringsAuthor(s) Jin, GJ; Wang, ZD; Hu, A; Jiang, SSCitation Physical Review B - Condensed Matter And Materials Physics,1997, v. 55 n. 15, p. 9302-9305Issued Date 1997URL http://hdl.handle.net/10722/43200Rights Physical Review B (Condensed Matter). Copyright © AmericanPhysical Society.PHYSICAL REVIEW B 15 APRIL 1997-IVOLUME 55, NUMBER 15Persistent currents in mesoscopic Fibonacci ringsG. J. JinDepartment of Physics and National Laboratory of Solid State Microstructures,Nanjing University, Nanjing 210093, People’s Republic of ChinaZ. D. WangDepartment of Physics, The University of Hong Kong, Pokfulam Road, Hong KongA. Hu and S. S. JiangDepartment of Physics and National Laboratory of Solid State Microstructures,Nanjing University, Nanjing 210093, People’s Republic of China~Received 8 October 1996!In the framework of a tight-binding model, we study energy spectra and persistent currents in mesoscopicFibonacci rings threaded by a magnetic flux. It is found that the flux-dependent electron eigenenergiesE(F) in mesoscopic Fibonacci rings still form ‘‘bands’’ with respect to the flux F , but there is a scalingrelation between the total ‘‘bandwidth’’ and the Fibonacci number. When the strength of the one-dimensionalquasiperiodic potential increases, the persistent current decreases rapidly. Interestingly, for a generalized mix-ing model of mesoscopic Fibonacci rings, free-electron-like persistent current may appear if the number ofelectrons of the system takes a specific value. @S0163-1829~97!08816-4#In recent years, there has been considerable interest in thepersistent currents in one-dimensional metal or semiconduc-tor mesoscopic rings threaded by a magnetic flux.1–4 An es-pecially interesting fact is that there is a discrepancy aboutthe magnitude of the persistent currents between theories andexperiments.3,4 It is believed that disorder and interaction aretwo important factors that affect the amplitude of the current.Nevertheless, their influences have not been completelyclarified. On the other hand, great experimental and theoret-ical efforts have been devoted to physical properties in one-dimensional quasiperiodic structures.5–8 As is well known,the quasiperiodicity of Fibonacci structure has substantial ef-fects on physical properties of condensed matter, and mayprovide a profound understanding of the intermediate regionbetween periodic and random structures. Furthermore, a ra-tional approximation is often employed in theoretical treat-ments of quasiperiodic behavior in Fibonacci chains or su-perlattices. Fortunately, this rational condition can benaturally realized in mesoscopic rings with the structure ofFibonacci chain. Such rings may be fabricated artificially.Therefore, it is worthwile to explore the properties of persis-tent currents in mesoscopic Fibonacci rings. In this paper, wefirst derive the basic formulas for persistent currents in me-soscopic Fibonacci rings; then the numerical results as wellas relevant discussions are presented.A Fibonacci chain is the simplest one-dimensional quasi-periodic structure, with two building units denoted by A andB . Using these two units, a Fibonacci chain is formed ac-cording to the rule S j115$S j ,S j21%, S15A , S25AB . Forexample, S55ABAABABA . It is reasonable to consider Aand B as two kinds of site energies or hopping integrals.5 Amesoscopic Fibonacci ring is constructed by a finite Fi-bonacci chain with N sites. For convenience, we assume thatN5F j , where j is a generation number, and F j is a Fi-bonacci number obeying the recursion relation F j5F j221F j21 with F15F051. For a mesoscopic ring, it may bereasonable to consider F j;103.550163-1829/97/55~15!/9302~4!/$10.00In the tight-binding approximation, the Schro¨dinger equa-tion for an electron in a one-dimensional mesoscopic Fi-bonacci ring readst l11c l111v lc l1t lc l215Ec l , ~1!where l is the site index, t l the hopping integral, and v l theon-site energy. For simplicity and without loss of generality,the spin freedom is not taken into account in the presentpaper. There are two simple types of one-dimensional quasi-periodic models. One is a transfer model, in which v l istaken to be constant, but t l is chosen to be two values tA andtB in a Fibonacci sequence. The other is an on-site model, inwhich t l is constant, but v l is set to be vA and vB in theFibonacci sequence. In addition, there is a mixing model, inwhich both v l and t l take two values. For concreteness, weconsider the on-site model first, in which two parametersvA and vB represent the quasiperiodic potential experiencedby an electron in a mesoscopic Fibonacci ring.Since t l is a constant, we can take its absolute value as anenergy unit, or simply define t l521. In the on-site model,Eq. ~1! can be transformed into a matrix form asS c l11c l D 5Tl11,lS c lc l21D , ~2!whereTl11,l5S 2~E2v l! 211 0 D . ~3!When the ring is threaded by a magnetic flux F , which leadsto the twisted boundary conditions for the wave functions ofthe electrons in the ring, 1 the equation for the global transfermatrix can be written as9302 © 1997 The American Physical Society55 9303BRIEF REPORTSS cN11cN D 5M jS c1c0D 5ei2pF/F0S c1c0D , ~4!where F05hc/e is the flux quantum, and M j5) l51N Tl11,1 . Tl11,l and M j are all unimodular. All physicalquantities are periodic in F with the period F0.Denoting x j5(1/2)TrM j , the flux-dependent energyspectra of an electron in the mesoscopic Fibonacci ring canbe obtained from the equation asx j5cos~2pF/F0!. ~5!The persistent current contributed from the nth level isIn~F!52c]En~F!]F52pcF0sin~2pF/F0!]x j /]En. ~6!Here x j and ]x j /]E can be evaluated recursively. The use-ful recursion relations arex j52x j21x j222x j23 ~7!and]x j]E 52S x j21 ]x j22]E 1 ]x j21]E x j22D2 ]x j23]E , ~8!with the initial conditionsx152~E2vA!/2, x052~E2vB!/2, x2151,]x1 /]E52 12 , ]x0 /]E52 12 , ]x21 /]E50.To calculate the charge stiffness later, we need also to evalu-ate the second-order derivatives]2E]~F/F0!2524p2S ]x j]E D23FcosS 2p FF0D S ]x j]E D21sin2S 2p FF0D ]2x j]E2 G . ~9!The corresponding recursion relation is found to be]2x j]E2 52S x j21 ]2x j22]E2 1]2x j21]E2 x j2212]x j21]E]x j22]E D2]2x j23]E2 , ~10!with the initial conditions]2x1/]E2 50, ]2x0/]E2 50, ]2x21/]E2 50.The first nonzero term of the second derivatives is]2x2 /]E251. Equations ~5!–~10! are our main analyticalresults, which will be used in numerical calculations. Theseequations are also suitable for the transfer model or even forthe mixing model, provided the initial conditions are modi-fied.In the following numerical calculations, we setvA52vB5v . Obviously, the parameter v represents thestrength of the quasiperiodicity. Once the Fibonacci genera-tion number is chosen to be j , the total number of energylevels is F j for any fixed flux. At zero temperature, electronswill occupy the levels one by one from the lowest level. Ifthe number of electrons of the system is Ne , which is equalto the highest occupied-level index m , the energy of the sys-tem is given byE~F!5 (n51mEn~F!, ~11!and the total persistent current of the system isI~F!5 (n51mIn~F!. ~12!Since the persistent current in a mesoscopic ring is com-pletely determined by flux-dependent energy spectra of thesystem, we address it at first. Figure 1 shows the numericalresults of the energy spectra for j510 and v50.5. At a fixedflux, for example F50.25F0, there are F10589 eigenener-gies. It can be seen clearly that they form a Cantor set. Ac-tually, all En;F curves form three subband structures,which have been shown in inset ~a! of Fig. 1. This charac-teristics is quite similar to those for the electrons, phonons,and spin waves in Fibonacci chain or superlattices. 527 Aspointed out by Bu¨ttiker and co-workers,1 there is a corre-spondence between an electron in a small normal one-dimensional ring and an electron in a periodic potential, soflux-dependent electronic eigenenergies form ‘‘band’’ struc-tures with respect to F . In the mesoscopic Fibonacci rings,this conclusion is still correct. To see the En(F) curvesclearly, the five lowest En(F) curves are plotted in inset ~b!of Fig. 1. Every En(F) curve changes smoothly with flux.However, the quasiperiodicity affects the energy spectra of amesoscopic Fibonacci ring significantly. One example isthat the total ‘‘bandwidth’’ B @5(nuEn(F/F050)2En(F/F05 12)u] diminishes rapidly with increasing Fi-bonacci generation number j . There exists a scaling relationbetween B and Fj , i.e., B;F j2a, as shown in Fig. 2. TheFIG. 1. The energy eigenvalues of a mesoscopic Fibonacci ring for theon-site model, where j510 (N589), v50.5, and F/F050.25. The inset~a! shows the flux-dependent energy spectra for 21/2,F/F0,1/2. Theinset ~b! shows the five lowest-energy ‘‘bands.’’ All energies are in unit ofut lu.9304 55BRIEF REPORTSpower index a depends on v . For v50.4, 0.8, and 1.2,a50.22, 0.49, and 0.75, respectively. Although the scalingindex a may be quite different, this scaling behavior is simi-lar to those in other quasiperiodic systems,9 as the commonorigin of the scaling comes from the quasiperiodicity of thestructure.The persistent current in a mesoscopic ring is obtainedfrom the energy spectra in terms of Eqs. ~6! and ~12!. Usu-ally, the narrower the ‘‘bandwidth’’ is, the smaller the maxi-mum persistent current. Here let us consider a case with evennumber of electrons and near half-filled, where j512(N5F j5233) and Ne5116. When v50, the relationshipbetween the persistent current and the magnetic flux is free-electron behavior. As v increases, I(F) is suppressed, asshown in Fig. 3, where the maximum persistentcurrent of the periodic tight-binding model I05(4pc/NF0)sin(Nep/N) is taken to be the current unit. To see theFIG. 2. Log-log plot of the total ‘‘bandwidth’’ B and the Fibonaccinumber F j . B is also in unit of ut lu.FIG. 3. The persistent current I vs flux F for various on-site energies,where j512 (N5233), Ne5116. The inset gives the dependence of I on theon-site energy v at F/F050.25.suppression clearly, we plot I against v at a fixed F in theinset of Fig. 3. Here the overall behavior of I(F) is some-what similar to that in a disordered mesoscopic ring. 2 But,unlike disordered systems, the mesoscopic Fibonacci ring isdeterministic and no average procedure needs to be taken. InFig. 4, we show the results for an odd Ne case. One can findthat I;F curves shift F/F05 12 along the flux axis whencompared with that for even Ne in Fig. 3, as in the periodicsystems.10 Also, we calculate the charge stiffness, which rep-resents the response of the persistent current to the appliedflux, and is defined as11D5F j4p2]2E~F!]~F/F0!2. ~13!By combining Eqs. ~9! and ~10! with Eq. ~13!, the numericalresults of D for F/F050, 0.3, 0.35, and 0.4 are plotted, asshown in the inset of Fig. 4. We can see that, as v increases,the response of the persistent current to the applied flux ismore and more difficult for F,0.35; while for 0.35,F/F0,0.5, the situation becomes a little bit complicated.When v is significantly large, uDu vanishes for all F . Whenv!0, D approaches 0.636, which can be analytically de-rived from the energy spectra formulas of the tight-bindingperiodic mesoscopic ring. Similar behaviors can also befound for even-number electrons, but with F being shiftedby F0/2.The transfer model can be treated in a similar way. Infact, the main results are almost the same. Here we wish toaddress further an issue of higher transmission and extendedstates in the mixing model. As in a generalized Fibonaccichain,8 we need to take into account three reduced energyparameters: the on-site energy v , the hopping integral t , andthe electron energy E , where t5tAA /tAB with tAA and tAB asthe hopping integrals between corresponding sites (tAB istaken as the energy unit!. In this model, there may existextended states, as in other quasiperiodic and disorderedFIG. 4. The persistent current I vs flux F for various on-site energies,where j512, and Ne5117. The inset shows the variation of the chargestiffness with the on-site energy v at F/F050, 0.30, 0.35, and 0.40, re-spectively.55 9305BRIEF REPORTSsystems,12 which correspond to higher transmission coeffi-cients. If the three energy parameters satisfy a relation8En5v @~11t2!/~12t2!# , ~14!we can show that the global transfer matrix of the systemM j5RCN, whereRC5S 2t21~En2v ! 2tt21 0 D . ~15!FIG. 5. The persistent current I as a function of F in the mixing model,where t52, v50.6. ~a! Ne585. ~b! Ne586.In terms of Chebyshev polynomials of the second order, wehaveM j51sinu S sin@~N11 !u# 2t sin~Nu!t21sin~Nu! 2sin@~N21 !u# D , ~16!where cosu[2(1/2t)(En2v). From Eq. ~16!, it is straight-forward to show that x j5cos(Nu). Thus, from Eq. ~5!, weare able to have u5(2p/N)(n1F/F0). If En is the highestenergy level ~i.e., n5m), Eq. ~6! leads toIm~F!54pcN sinF2pN Sm1 FF0D G ,which is just the result of a periodic tight-binding model.Since the Im(F) is the dominant term in I(F),13 a free-electron-like persistent current is naturally expected. There-fore, if the electrons are appropriately accommodated in theenergy levels such that Em satisfies Eq. ~14! approximately,the persistent current is very close to the free-electron case@see the numerical results calculated from Eqs. ~6! and ~12!in Fig. 5#, and the transmission coefficients can be quitelarge or even close to 1. For example, when t52, v50.6,and E521 from Eq. ~14!, the transmission coefficientt50.9968.14 For j512, the energies of the 85th and 86thenergy bands are very close to 21. The deviation of theenergy from En521 is less than 5% as F changes.In summary, we have investigated the persistent currentsin mesoscopic Fibonacci rings. The effects of quasiperiodic-ity on the energy spectra as well as on the persistent currentshave been elucidated.This work was supported by Hong Kong RGC Grant No.HKU262/95P, the National Natural Science Foundation ofChina, and the Provincial Natural Science Foundation ofJiangsu.1M. Bu¨ttiker et al., Phys. Lett. 96A, 365 ~1983!; R. Landauer andM. Bu¨ttiker, Phys. Rev. Lett. 54, 2049 ~1985!.2H. F. Cheung et al., Phys. Rev. B 37, 6050 ~1988!.3L. P. Le´vy et al., Phys. Rev. Lett. 64, 2074 ~1990!; V. Chan-drasekhar et al., ibid. 67, 3578 ~1991!; D. Mailly et al., ibid. 70,2020 ~1993!.4V. Ambegaokar and U. Eckern, Phys. Rev. Lett. 65, 381 ~1990!;F. von Oppen and E. K. Riedel, ibid. 66, 84 ~1991!; B. L. Alt-shuler et al., ibid. 66, 88 ~1991!; M. Abraham and R. Berkovits,ibid. 70, 1509 ~1993!; G. Bouzerar et al., Phys. Rev. B 49, 8258~1994!.5M. Kohmoto et al., Phys. Rev. Lett. 50, 1870 ~1983!; Phys. Rev.B 35, 1020 ~1987!.6R. Merlin et al., Phys. Rev. Lett. 55, 1768 ~1985!; P. Hawrylakand J. J. Quinn, ibid. 57, 380 ~1986!; B. L. Johnson and R. E.Camley, Phys. Rev. B 44, 1225 ~1991!.7 J. W. Feng et al., Phys. Rev. B 52, 15 312 ~1995!; G. J. Jin et al.,ibid. 54, 1883 ~1996!.8V. Kumar, J. Phys. Condens. Matter 2, 1349 ~1990!.9M. Kohmoto, Phys. Rev. Lett. 51, 1198 ~1983!; P. Hawrylak andJ. J. Quinn, ibid. 57, 380 ~1986!; T. Janssen and M. Kohmoto,Phys. Rev. B 38, 5811 ~1988!.10Z. D. Wang and Jian-Xin Zhu, Phys. Rev. B 52, 5275 ~1995!;Jian-Xin Zhu et al., ibid. 52, 14 505 ~1995!.11G. Bouzerar et al., Phys. Rev. B 49, 8258 ~1994!; T. Giamarchiand B. S. Shastry, ibid. 51, 10 915 ~1995!.12V. Kumar and G. Ananthakrishna, Phys. Rev. Lett. 59, 1476~1987!; D. H. Dunlap et al., ibid. 65, 88 ~1990!.13For various quasiperiodic strengths v , we have calculated the cur-rent of the highest occupied level Im(F) and the total currentI(F), and find that Im(F) is dominant in the I(F), which is dueto the fact that the contributions from the levels of n and n11(n,m and m@1) cancel each other, as in the periodic mesos-copic ring.14 If we consider E , v , and t as three independent parameters, anddenote the global transfer matrix as M (N) with matrix elementsmi , js (i , j51,2), where mi , j is a function of the three energyparameters, the transmission coefficient can be obtained ast~N!542E2@t212t121~t222t11!E/2#21~t221t11!2~12E2/4!.

Persistent currents in mesoscopic Fibonacci rings

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Persistent currents in mesoscopic Fibonacci rings

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