Electronic energy spectra and wave functions on the square Fibonacci tiling

Transcription

1 Phlosophcal Magazne, Vol. 86, Nos. 6 8, 21 February 11 March 2006, Electronc energy spectra and wave functons on the square Fbonacc tlng S. EVEN-DAR MANDEL and R. LIFSHITZ* School of Physcs and Astronomy, Raymond and Beverly Sackler Faculty of Exact Scences, Tel Avv Unversty, Tel Avv 69978, Israel (Receved 15 May 2005; n fnal form 19 July 2005) We study the electronc energy spectra and wave functons on the square Fbonacc tlng, usng an off-dagonal tght-bndng model, n order to determne the exact nature of the transtons between dfferent spectral behavours, as well as the scalng of the total bandwdth as t becomes fnte. The macroscopc degeneracy of certan energy values n the spectrum s nvoked as a possble mechansm for the emergence of extended electronc Bloch wave functons as the dmenson changes from one to two. 1. Background and motvaton We contnue our ntal studes [1] of the off-dagonal tght-bndng hamltonan on the square Fbonacc tlng [2], n order to gan a better quanttatve understandng of the nature of the transtons between dfferent spectral behavours n ths 2-dmensonal (2D) quascrystal. We also consder more carefully the transton of the spectrum from sngular-contnuous to absolutely contnuous, n gong from one to two dmensons, and the mplcatons of ths transton on the possble emergence of extended Bloch wave functons. The square Fbonacc tlng s constructed by takng two dentcal grds each consstng of an nfnte set of lnes whose nter-lne spacngs follow the well-known Fbonacc sequence of short (S ) and long (L) dstances and supermposng them at a90 angle. Ths constructon can be generalzed, of course, to any quasperodc sequence as well as to hgher dmensons. On ths tlng we defne an off-dagonal tght-bndng hamltonan wth zero on-ste energy, and hoppng ampltudes: t for vertces connected by short (S ) edges; 1 for vertces connected by long (L) edges; and zero for vertces that are not connected by edges. The 2-dmensonal Schro dnger equaton for ths model s gven by t nþ1 ðn þ 1, mþþt n ðn 1, mþþt mþ1 ðn, m þ 1Þþt m ðn, m 1Þ ¼E ðn, mþ, ð1þ *Correspondng author. Emal: ronlf@tau.ac.l Phlosophcal Magazne ISSN prnt/issn onlne # 2006 Taylor & Francs DOI: /

2 760 S. Even-Dar Mandel and R. Lfshtz where ðn, mþ s the value of a 2D egenfuncton on a vertex labelled by the two ntegers n and m, and E s the correspondng egenvalue. The hoppng ampltudes t j are equal to 1 or t accordng to the Fbonacc sequence as descrbed above, and we do not lose much by lmtng ourselves to the case t > 1, as the mportant features of the model depend on the rato of the two hoppng ampltudes. Wth no addtonal assumptons other than the absence of dagonal hoppng ths 2-dmensonal egenvalue problem, as well as ts hgher dmensonal versons, are all separable. Ths allows us to use the known solutons for the 1D problem [3, 4] n order to construct the solutons n two and hgher dmensons (as was done for smlar models n the past [5, 6]). Two-dmensonal egenfunctons can therefore be expressed as products jðn, mþ ¼ ðnþ j ðmþ, ð2þ wth egenvalues E j ¼ E þ E j, ð3þ where (n) and j (n) are two of the egenfunctons of the correspondng 1D egenvalue equaton on the Fbonacc chan, wth egenvalues E and E j. Consequently, the 2D densty of states (DOS) s gven by a convoluton of two 1D denstes of states. 2. Study of the 2D spectrum Although the spectrum of the 1D model s always a Cantor-lke set wth zero bandwdth and an nfnte number of bands, the 2D spectrum exhbts dfferent behavour for dfferent ranges of the parameter t. For t close to 1 the gaps n the 1D spectrum are small and dsappear as a result of the addton of spectra (3), producng an absolutely contnuous 2D spectrum. For t 1 the 2D spectrum retans the Cantor-lke structure of the 1D spectrum. For ntermedate values of t the number of bands tends to nfnty but the total bandwdth remans fnte and t s unclear whether the spectrum has an absolutely contnuous part. In order to study the exact structure of the 2D spectrum the total bandwdth and number of bands as a functon of t one has to perform a convoluton of two 1D spectra. The numercal study of the 1D spectrum s dffcult for low values of t (t 1) for whch the gaps are very small and for t 1 for whch the bands are very small. Further numercal dffcultes are encountered n the convoluton process tself. The desred 2D spectrum, consstng of the set of ponts for whch the 2D DOS s not zero, s gven by where ½a ðn Þ [ F N, j¼1 h a ðn Þ þ a ðn Þ j, b ðn Þ þ b ðn Þ j, ð4þ, b ðn Þ Š s the th nterval n the relevant 1D spectrum of the Nth approxmant and F N s the Nth Fbonacc number. Because F N N the number of ntervals n the unon (4) behaves lke 2N, and although many of the ntervals may overlap, any attempt to drectly study the band structure and the DOS requres a constructon

3 Electronc energy spectra and wave functons 761 of the whole set. The amount of memory requred to store such sets makes exact calculaton mplausble for hgh values of N regardless of the value of t. Luckly, the total bandwdth of the 2D spectrum can be calculated wthout full knowledge of the DOS. Ths s done by examnaton of a set of unformly dstrbuted energes n the relevant range and testng for each energy whether t les wthn one of the above ntervals. For a suffcently large set of energes ths procedure yelds the measure of the spectrum after proper normalzaton. For gven values of N and t the 1D spectrum s a symmetrc set of F N ntervals wthn ½ r N ðtþ, r N ðtþš, where r N (t) s the maxmal egenenergy of the Nth 1D approxmant. The 2D spectrum s therefore a subset of the nterval ½ 2r N ðtþ,2r N ðtþš, whch gves an upper bound of 4r N ðtþ on ts total bandwdth. Fgure 1 shows the total bandwdth dvded by 4r N ðtþ as a functon of t for a sequence of approxmants N ¼ 6,...,17. For values of t up to 2 the spectrum conssts of a sngle band. For t s above 2 the normalzed bandwdth becomes dependent on N. Usng fntesze scalng (as suggested by Ashraff et al. [6]) we can obtan an estmate for the crtcal value t c at the transton between the ntermedate and large t regmes, where the total bandwdth vanshes. Fgure 2 shows the total bandwdth as a functon of 1/N for dfferent values of t > 2. Although some oscllatory behavour s observed, the behavour seems qute lnear and extrapolaton to 1=N! 0 yelds a predcton for the total bandwdth n the nfnte N lmt. By plottng these estmates as a functon of t we obtan fgure 3 whch agan exhbts an overall lnear behavour. Extrapolaton yelds an estmate for a crtcal value of t c 5:375 for the transton between the ntermedate and large t regmes. It s worth notng that as t s ncreased and the gaps grow, the degeneracy of energy levels decreases. At the asymptotc lmt each nterval n the 2D spectrum should be exactly doubly degenerate (neglectng a sngle nterval around E ¼ 0) N=6 N=7 N=8 N=9 N=10 N=11 N=12 N=13 N=14 N=15 N=16 N= t Fgure 1. Total bandwdth of the 2D spectrum dvded by 4r N (t). For values of t up to 2 the spectrum conssts of a sngle band regardless of N. At hgher values the bandwdth clearly depends on N.

4 762 S. Even-Dar Mandel and R. Lfshtz 2d bandwdth t=3 t=3.2 t=5.2 t=3.4 t=3.6 t=3.8 t=4 t=4.2 t=4.4 t=4.6 t=4.8 t= Fgure 2. Total bandwdth as a functon of 1/N for dfferent values of t. Lnear extrapolaton to 1/N! 0 gves a predcton for the total bandwdth at the nfnte N lmt. 1/N 6 5 projected bandwdth t= t Fgure 3. Extrapolaton of the predcted nfnte lmt bandwdth to fnd the crtcal value t c, above whch the total bandwdth tends to zero. When ths lmt s obtaned the 2D bandwdth should be approxmately equal to F N B 1D ðt, N Þ, where B 1D ðt, N Þ s the bandwdth of the correspondng 1D spectrum. The numercal data shows that ths lmt s to be obtaned well beyond t c. It s hence evdent that the large t regme of the 2D spectrum s further dvded nto two

5 regmes, one (at extremely large t) n whch the energy levels are no-more than doubly degenerate, and another (at moderately large t) n whch energy levels may have a hgher degeneracy. 3. Emergence of extended wave functons Electronc energy spectra and wave functons 763 Snce the egenfunctons of the 1D hamltonan are crtcally localzed t s clear that the 2D egenfunctons (2) should also be crtcally localzed (fgure 4a c). However, the exstence of a regme for t close to 1 n whch the spectrum, or at least a part of t, s absolutely contnuous hnts at the exstence of extended wave functons. We proposed n the past [1] that the degeneracy of the 2D spectrum, at values of t close to 1, may provde an explanaton for the emergence of extended wave functons as the dmensonalty ncreases. To llustrate the possble role of degeneracy n the emergence of extended wave functons we analyse the E ¼ 0 egenfuncton. These functons are macroscopcallydegenerate for any value of t. A set of 1D egenfunctons can be found by drectly (a) (b) (c) (d) Fgure 4. Wave functons on a sngle D approxmant. (a) A crtcal wave functon n, m; (b) a symmetrzed crtcal wave functon 2 1=2 n, m þ m, n ; (c) a zero energy wave functon n, n; and (d) a lnear combnaton of zero energy wave functons producng an extended Bloch wave functon along the dagonal, clearly showng the underlyng Fbonacc modulaton.

6 764 Electronc energy spectra and wave functons dagonalzng the 1D hamltonan for a sngle approxmant wth perodc boundary condtons. The F N egenfunctons obtaned n ths way generate FN 2 egenfunctons for the 2D model, F N of whch have E ¼ 0. These egenfunctons, gven by ðn, mþ ¼ð 1Þ m ðnþ ðmþ, are each crtcally localzed, but we can demand that a lnear combnaton P N ¼1 a ðn, mþ of them have certan values on a gven subset of stes by solvng the equatons for the coeffcents a. Settng the values of a lnear combnaton to be dentcal on half the ponts along the dagonal results n egenfunctons of the form shown n fgure 4d. Ths wave-functon s an extended Bloch functon along the dagonal clearly showng the underlyng Fbonacc modulaton but s strongly localzed n the perpendcular drecton. Whether the degeneracy of other energy values s suffcently hgh to allow for a smlar process to be carred out s not clear, but the chances for success should strongly mprove as the tunnellng ampltude t approaches 1. Ths also corresponds to the regme n whch the spectrum s (at least partly) absolutely contnuous. For large t, where the spectrum s sngular contnuous, the degeneracy of levels other than E ¼ 0 s very small and no extended wave functons are expected to be found. Note that n 3 dmensons all the energes of the orgnal 1D spectrum are macroscopcally degenerate. Ths means that extended wave-functons, smlar to that of fgure 4d, can be constructed for a wder range of energes. Thus, as the dmensonalty further ncreases so does the degeneracy of energy levels, and wth t the ablty to construct Bloch wave functons. Acknowledgments Ths research s supported by the Israel Scence Foundaton through Grant No. 278/00. References [1] R. Ilan, E. Lberty, S. Even-Dar Mandel, et al., Ferroelectrcs (2004). [2] R. Lfshtz. J. Alloys and Compounds (2002). [3] M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev. Lett (1983); S. Ostlund, R. Pandt, D. Rand, et al., Phys. Rev. Lett (1983); M. Kohmoto and J.R. Banavar, Phys. Rev. B (1986); M. Kohmoto, B. Sutherland and C. Tang, Phys. Rev. B (1987). [4] T. Janssen, n The Mathematcs of Long-Range Aperodc Order, edted by R.V. Moody (Kluwer, Dordrecht, 1997), p. 269; T. Fujwara, n Physcal Propertes of Quascrystals, edted by Z.M. Stadnk (Sprnger, Berln, 1999), chap. 6; J. Hafner and M. Krajcı, bd., chap. 7; D. Damank, n Drectons n Mathematcal Quascrystals, edted by M. Baake and R.V. Moody (AMS, Provdence, 2000), p [5] K. Ueda and H. Tsunetsugu, Phys. Rev. Lett (1987); W.A. Schwalm and M.K. Schwalm, Phys. Rev. B (1988); J.X. Zhong and R. Mosser, J. Phys.: Condens. Matter (1995); S. Roche and D. Mayou, Phys. Rev. Lett (1997); Yu.Kh. Veklov, I.A. Gordeev and E.I. Isaev, JETP (1999); Yu.Kh. Veklov, E.I. Isaev and I.A. Gordeev, Mater. Sc. Eng (2000). [6] J.A. Ashraff, J.-M. Luck and R.B. Stnchcombe, Phys. Rev. B (1990).