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PHYSICAL REVIEW B 70, 144524 (2004)Hofstadter butterflies in a modulated magnetic field: Superconducting wire networkwith magnetic decorationYasuhiro Iye,* Eiichi Kuramochi, Masahiro Hara, Akira Endo, and Shingo KatsumotoInstitute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 227-8581, Japan(Received 1 April 2004; published 29 October 2004)Hofstadter butterfly spectra of tight-binding electron systems under spacially modulated magnetic fields arecalculated. The dependence of the spectrum on the symmetry and strength of the spatially varying magneticfield is elucidated. The Little-Parks oscillation of the superconducting network under a spatially modulatedmagnetic field produced by decoration with mesoscopic magnetic structure exhibits behavior reproducing theedge of the corresponding Hofstadter spectra.DOI: 10.1103/PhysRevB.70.144524PACS number(s): 74.78.w, 74.81.Fa, 75.75.a, 73.21.bI. INTRODUCTIONThe exquisite structure of the energy diagram of a tightbindingelectron system on a two-dimensional (2D) squarelattice subjected to a uniform perpendicular magnetic fieldwas first elucidated by the work of Hofstadter 1 and Wannier. 2The spectrum consists of q subbands, when the flux (in theunits of the flux quantum 0 =h/e) threading through eachplaquette is a rational number p/q. The spectra of this type,known as the Hofstadter butterfly, have since been calculatedfor a variety of lattices, 3–5 and they have been discussedin various contexts of condensed-matter physics includingthe quantum Hall effect 6,7 and the so-called flux phase in thet-J model. 8,9Observation of the butterfly spectra in real systems is aformidable task. For the ordinary crystalline lattices, the requiredmagnetic field exceeds 1000 T, which is beyond thereach of the present technology. It is therefore natural to turnto the artificial periodic structures, such as lateral superlatticesbased on semiconductor two-dimensional electron systems.Recent work by Albrecht et al. 10 on magnetotransportin an antidot lattice seems to capture a certain characteristicof the Hofstadter spectrum.Superconducting networks comprise another category ofsystems intimately related to the Hofstadter spectrum. Asdemonstrated by Pannetier et al., 11 the change in the transitiontemperature of a superconducting wire network with thefrustration parameter reproduces the fine structure of theupper edge of the Hofstadter butterfly spectrum. This correspondencearises from the fact that the linearized Ginzburg-Landau equation for a superconductor near its transition hasthe same form as the Schrödinger equation for the electronsystem. 12,13 Thus, with the advantage of macroscopic coherence,superconducting networks provide a convenient experimentalmodel of the Hofstadter butterfly, although admittedlythey can only probe the edge of the spectrum. This line ofstudy was subsequently extended to networks of various latticesymmetries, including triangular, honeycomb,Kagomé, 14 T 3 , 15 disordered, 16 quasiperiodic, 17 and fractal 18lattices.To the best of our knowledge, all of these previous studiesbasically employed a uniform external magnetic field. Itwould be interesting to extend the study to more generalcases of spatially varying magnetic field. This is the subjectof the present study. More specifically, we have calculatedthe Hofstadter spectrum for a square lattice under spaciallyvarying magnetic fields. We have also conducted experimentsusing a superconductor/ferromagnet (S/F) hybrid systemconsisting of a superconducting wire network and anarray of mesoscopic ferromagnets. An early version of ourstudy using an S/F hybrid system study has beenpublished. 19,20 An S/F hybrid system of a different type fromthe present one was studied by Nozaki et al. 21 The Hofstadterspectrum for a T 3 lattice under a modulated magnetic field istheoretically studied by Oh. 22The remainder of this paper is organized as follows. Calculationsof the Hofstadter spectra for a square network undervarious patterns of a spatially varying magnetic field arepresented in the next section. Experiments using asuperconductor/ferromagnet hybrid system are described inSec. III, and discussed in comparison with the calculatedspectra. Finally, Sec. IV gives a summary of the presentwork and some remarks.II. CALCULATION OF THE HOFSTADTER SPECTRAA. The original Hofstadter problemThe Hofstadter problem in its original form, i.e., the tightbindingelectron spectrum on a 2D square lattice with latticeconstant a under a uniform magnetic field H, is obtained bysolving the following Schrödinger equation: n,m = t n−1,m + t n+1,m + te −2in n,m−1 + te 2in n,m+1 ,1where n,m represents the wave function at the latticesite n,m, t is the nearest-neighbor transfer-matrix element,and / 0 =eHa 2 /h is an avarage magnetic flux threadinga unit square. The parameter is often called a frustrationparameter. The above equation can be reduced byputting n,m = e ik x na e ik y ma nto the following one-dimensional Harper equation:21098-0121/2004/70(14)/144524(9)/$22.50 70 144524-1©2004 The American Physical Society

IYE et al. PHYSICAL REVIEW B 70, 144524 (2004) n = te −ik x a n−1 + te ik x a n+1 +2t cosk y a −2n n .3When is a rational number p/q, p and q beingmutually prime, the size of the magnetic unit cell becomesqa,a, namely, x+qa,y=x,y and x,y+a=x,y.Figure 1 shows the tight-binding square lattice withthe choice of gauge appropriate to = p/q. The energyeigenvalues are obtained by diagonalizing the followingmatrix:2t cosk y a −2 te ik x a 0 ¯ 0 e −ik x ate −ik x a 2t cosk y a −4 te ik x a 0 ¯ 00 4 0 00 ¯ 0 te −ik x a 2t cosk y a −2q −1 te ik x ate ik x a 0 ¯ 0 te −ik x a 2t cosk y a −2q.The result is the well-known butterfly spectrum, shown inthe topmost panel of Fig. 3.B. Checkerboard magnetic fieldWe consider a spatially varying magnetic field whichconsists of a uniform component and a component varyingin a checkerboard pattern, as shown in Fig. 2. Here, denotes the flux per plaquette of the uniform componentof the magnetic field, and denotes the flux per plaquettewhich alternates in sign in the checkerboard pattern. Theassignment of the Peierls phase factor for this flux patternis shown in the figure. For = p/q, the system is invariantunder translation 2qa,0 or a,a. The relevant Schrödingerequation reads n,m = t n−1,m + t n+1,m + te −2in−m+1 n,m−1+ te 2in−m e 2i n,m+1 n − m odd, 5 n,m = t n−1,m + t n+1,m + te −2in−m+1 e 2i n,m−1+ te 2in−m n,m+1 n − m even.Spectra obtained by diagonalization of the corresponding2q2q matrix are shown in Fig. 3. The spectra are symmetricwith respect to transformations, →1±, so that calculationover the range =0– 1 2suffices. Five panels in Fig. 3correspond to =0, 1 8 , 1 4 , 3 8 , and 1 2, respectively. The topmostpanel =0 is the original Hofstadter butterfly spectrum.Introduction of nonzero deforms the spectrum in such away that, for instance, the spectral weight at the band center(van Hove singularity at =0) for =0 is smeared, and aquasigap develops there with increasing . The bottommostpanel = 1 2is identical to the topmost one except that thespectrum is shifted by 1 2along the horizontal axis. Thatthis should be so can be readily understood by recalling thefollowing: At = 1 2 , two adjacent cells enclose + 1 2 and − 1 2flux, respectively. Addition of a uniform flux = 1 2to thesystem changes them to 1 and 0, which is equivalent to the=0,=0 configuration. The same relation (shift by halfperiod)holds between the spectra for = 3 8[panel (d)] andfor = 1 8 [panel (b)].At= 1 4[panel (c)], the periodicity in becomes half the original one. In other words, the states at=integer and at =half-integer become equivalent for= 1 4. Again, this can be easily understood by recalling thatthe flux configuration of two adjacent cells is +34 ,+1 4at= 1 4 ,= 1 2, which is equivalent to −14 ,+1 4, and hence to +14 ,−1 4at = 1 4 ,=0 .FIG. 1. Tight-binding square lattice with assignment of thePeierls phase factor to each bond, for a uniform external magneticflux = p/q.FIG. 2. Square lattice subjected to a spatially varying magneticfield in a checkerboard pattern and a uniform field .The assignment of the Peierls phase factor is indicated bythe arrows.144524-2

HOFSTADTER BUTTERFLIES IN A MODULATED… PHYSICAL REVIEW B 70, 144524 (2004)FIG. 3. Hofstadter spectra for a square lattice subjected to a checkerboard field and a uniform field . Five panels shows spectrafor different amplitudes of the checkerboard field: (a) =0, (b) = 1 8 , (c) = 1 4 , (d) = 3 8 , and (e) = 1 2 , respectively.C. Stripe magnetic fieldWe next consider the case of a stripe pattern of fieldmodulation, as shown in Fig. 4. Here, denotes the flux perplaquette, which alternates in sign for every other column.The assignment of the Peierls phase factor for this fluxpattern is shown in the figure. The magnetic unit cell for= p/q is 2qa,a, i.e., the system is invariant under translationby 2qa,0 or 0,a. The relevant Schrödinger equationreads n,m = t n−1,m + t n+1,m + te −2in e 2i n,m−1+ te 2in e 2i n,m+1 n odd, 6 n,m = t n−1,m + t n+1,m + te −2in n,m−1+ te 2in n,m+1 n even.The spectra obtained by diagonalization of the corresponding2q2q matrix are shown in Fig. 5. Only the spectrafor = 1 8 and 1 4are shown, since the spectra for =0and = 1 2 are identical to those for =0 and = 1 2in Fig. 3,and the spectrum for = 3 8is none other than thatfor = 1 8 shifted by 1 2along the horizontal axis. Wecan immediately note the differences between the spectrafor a stripe field and those for a checkerboard field. Introductionof nonzero (stripe field) causes lifting of degeneracy144524-3

IYE et al. PHYSICAL REVIEW B 70, 144524 (2004)FIG. 4. Square lattice subjected to a spatially varying magneticfield in a stripe pattern and a uniform field . The assignmentof the Peierls phase factor is indicated by the arrows.at =0 for = 1 2, while the splitting occurs at integer for the checkerboard field, as seen earlier. It is evidentthat the spectra for stripe fields are considerably “darker”(more bands and fewer gaps) as compared with thosefor checkerboard fields. This is in line with a generaltrend that the Hofstadter spectrum becomes “darker”as one introduces rectangular anisotropy in the squarelattice. 9D. Comparison with a honeycomb latticeAlthough it is a kind of excursion to a side road,it is interesting to compare the spectra for the checkerboardFIG.FIG. 5. Hofstadter spectra for different values of , the strengthof the stripe pattern field. (a) = 1 8 and (b) = 1 4. The spectra for=0 and = 1 2are identical with Figs. 3(a) and 3(e), respectively.The spectrum for = 3 8 is the same as the one for = 1 8except it isshifted by 1 2along the horizontal axis.6. (a) A square lattice with different transfer integralst and t. When t=0 [panel (b)], it is topologically equivalentto a honeycomb lattice [panel (c)]. Panel (d) shows unit “brick” ofthis lattice and the one for a square lattice under checkerboard field(Fig. 2).field shown in Fig. 3 with that for a honeycomb latticeunder a uniform magnetic field. Let us consider a squarelattice shown in Fig. 6(a), in which the transfer integralfor every other vertical bond t (dotted lines) can take adifferent value from that for the rest of the bonds t (solidlines).The relevant Schrödinger equation reads n,m = t n−1,m + t n+1,m + te −2in n,m−1 + te 2in n,m+1 .7Setting t=t reduces the system to a simple square lattice.On the other hand, setting t=0 [Fig. 6(b)] makes it topologicallyequivalent to a honeycomb lattice [Fig. 6(c)].Therefore, by changing the parameter t from t to 0, thesystem evolves continuously from a square to a honeycomblattice.Figure 7 shows calculated spectra for (a) t=0.5t and (b)t=0 (honeycomb lattice). It is seen that they exhibit a resemblancein overall shape with the spectra for = 1 8 and 1 4for the checkerboard field [panels (b) and (c) of Fig. 3]. Theresemblance actually runs over the whole range of the twomodels, i.e., between the checkerboard field [Eq. (6)] with changing from 0 to 1 2and the square-to-honeycomb evolution[Eq. (7)] with t changing from t through 0 to −t. Thespectra at both ends are indeed identical: =0⇔t=t and=1/2⇔t=−t.In retrospect, this resemblance is something one shouldhave anticipated, because in both cases the structural unitis a rectangle consisting of two adjacent unit squares, asshown in Fig. 6(d). Both Fig. 2 and Fig. 6(a) are thenconstructed by bricklaying these unit rectangles in the144524-4

HOFSTADTER BUTTERFLIES IN A MODULATED… PHYSICAL REVIEW B 70, 144524 (2004)FIG. 7. Spectra for the model defined in Fig. 6. The panels (a)and (b) are for t=0.5t and 0 (honeycomb lattice), respectively. Thespectra for t=t and t=−t are the same as the panels (a) and (e) ofFig. 3, respectively.pattern shown in Fig. 6(b). It is also interesting to notethat the spectra shown in Fig. 3 resemble those obtained fora lattice fermion model in the context of two-dimensionald-wave superconductivity. 23E. Relation to superconducting networksThe transition temperature of a superconducting wire networkunder a uniform magnetic field H is related with theeigenvalue max corresponding to the upper edge of the energyspectrum for the tight-binding model by the followingrelation: 11–13T c HT c0= 2 0a 2arccos 2 maxz. 8Here, z is the number of nearest-neighbor nodes. Typically,the relative change in the transition temperature is smallT c T c0 , and T c is linearly related with max .Figure 8 shows the evolution of the lower edgeof the Hofstadter spectra as a function of the amplitude (a) of the checkerboard field and (b) of the stripe field.(Since the upper and the lower edge are identical, weshow the latter for the sake of ease of visual comparisonwith the experimental data to be shown later.) The curveon the front-left face ( or =0) represents the loweredge of the original Hofstadter butterfly spectrum. In Fig. 9,the same set of data are presented as a gray-scale plot.A notable difference between the checkerboard and stripeFIG. 8. Stereo plot of the minimum energy as a function of and (or ). Namely, these figures show evolution of the bottomedge of the Hofstadter spectrum with the amplitude of the spatiallyvarying magnetic field of (a) the checkerboard pattern and (b) thestripe pattern.field case is manifest upon comparison of the curves on thefront-right face =integer. The curve in Fig. 8(a) has acusp at = 1 2 , while that in Fig. 8(b) is rounded at = 1 2 .Correspondingly, the latter rises more steeply near the originthan the former. These features are to be compared with theexperimental results in the next section. An interesting featureof the energy landscape shown in Fig. 8(a) is that theenergy at = 1 4 and 3 4 is constant =−2 2 irrespective of thevalue of , which can be seen more clearly in the contourplot of Fig. 9(a).FIG. 9. Contour plot of the same set of data as Fig. 8.144524-5

IYE et al. PHYSICAL REVIEW B 70, 144524 (2004)FIG. 10. Structure of the samples. (a) Sample A for the checkerboardfield. (b) Sample B for the stripe field. The external fieldparallel to the network plane controls the magnetization of the ferromagneticarray. The amplitude of the spatially modulated magneticfield can be changed by the azimuthal angle of the parallelfield.III. EXPERIMENT WITH SUPERCONDUCTINGWIRE NETWORKSA. Experimental methodThe samples used in the present study were fabricated onsilicon substrates by the following steps.(i) Electrode pads were first formed by electron beam lithographyand gold evaporation.(ii) The network pattern (square lattice) was defined byelectron beam lithography and the niobium wire networkwas formed by ion-beam sputtering deposition and the liftoffprocess.(iii) A protecting layer of germanium was deposited ontop of the wire network, so as to prevent oxidation of niobiumand to keep it from direct contact with the ferromagneticmaterial to be deposited next.FIG. 11. Magnetoresistance traces at different temperatures forsample A. These data were taken at =0, i.e., =0.FIG. 12. Evolution of the Little-Parks oscillation for sample Awith the value of (checkerboard field). The traces are verticallyoffset for clarity.(iv) An array of mesoscopic ferromagnets (cobalt ornickel) was placed on top by electron beam lithography, ionbeamsputter deposition, and liftoff.The crucial point in the fabrication was to achievegood positional and angular registration between thesuperconducting network and the overlaid ferromagnetarray.Two samples (A and B) were intensively studied. Thesesamples represent the checkerboard field case (sample A)and the stripe field case (sample B), respectively. Thesuperconductor part of the sample consisted of a squarenetwork of 100100 unit cells, made of niobium wire150 nm wide and 40 nm thick. The lattice period was500 nm for sample A and 750 nm for sample B. For sampleA, 150200 nm 2 rectangular dots of 80-nm-thick cobaltwere placed on top of the center of every other bond wiresin the y direction, as shown in Fig. 10(a). For sample B,250-nm-wide strips of 60-nm-thick nickel were placedon top of every other lines in the y direction as shown inFig. 10(b).Measurements of the superconducting properties wereconducted by use of a cross-coil superconducting magnetsystem, consisting of a 6 T Helmholtz coil and a 1 Tsolenoid. The horizontal field was used to fix the magnetizationof the ferromagnetic array and thereby controlthe strength of the spatially varying field (parameter or ).The vertical field supplied the uniform field (parameter )for the network. The four-terminal resistance of the networkwas measured by a standard ac lock-in technique. Cryogeniccontrol was achieved by a variable temperature insert Dewar144524-6

HOFSTADTER BUTTERFLIES IN A MODULATED… PHYSICAL REVIEW B 70, 144524 (2004)FIG. 13. Evolution of the Little-Parks oscillation for sample Bwith the value of (stripe field). The traces are vertically offset forclarity.and a feedback circuit for heater power. Use of a sampleholder with rotating stage enabled us to precisely align theorientation of the network sample with respect to the magneticfield.The plane of the wire network was adjusted so as to makeit parallel to the horizontal magnetic field. The horizontalmagnetic field was typically set at 1 T, which was strongenough to fully saturate the magnetization of the ferromagnet.The value of for the checkerboard field (or for stripe field) was varied by changing the azimuthalangle of the horizontal field. As is clear from the configurationshown in Fig. 10, when =0 the total flux dueto the stray field from the mesoscopic magnets adds upto zero for each plaquette (provided that the lithographicalsymmetry is perfect), so this corresponds to =0. Asthe horizontal field is rotated from =0, the flux piercingeach plaquette grows in magnitude, and the sign alternatesbetween adjacent cells. The value of could thus becontrolled in a contiunuous manner (roughly proportionalto for small ). A typical set of measurements consistedof sweeping the vertical magnetic field for different settingsof .B. Results and discussionFigure 11 shows a series of magnetoresistance tracesfor sample A under =0 at different temperatures belowT c . Little-Parks oscillations with period 7.7 mT, whichagrees well with the unit-cell area 0.25 m 2 , are clearlyseen with additional dips at half-integers. The small shiftFIG. 14. Comparison of T c with the edge of the Hofstadterspectra for (a) checkerboard and (b) stripe field. The symbols represent−T c obtained from the resistance change as a function of or . The curves represent the bottom edge of the correspondingHofstadter spectra scaled to fit with the experimental data. The solidsymbols and the solid curve correspond to =0. The open symbolsand the dashed curve correspond to = 1 2 .of zero magnetic field is due to residual misalignmentbetween the horizontal magnetic field and the film plane.The fine structures at rational values of other than12 (and weaker features at 1 3 and 2 3) were not clearly observedin this sample. This is probably attributable to lithographicalimperfection, especially nonperfect registry betweenthe superconducting wire network and the mesoscopicmagnet array, which tends to smear the higher-orderstructures.Then the temperature was fixed at a point where thesample resistance was about 0.01R n , and a series of magnetoresistancetraces were taken by sweeping the vertical fieldfor different settings of . Figure 12 shows the traces thusobtained. Here, successive traces correspond to incrementsof by 1 degree, and they are vertically offset for clarity.The horizontal axis is converted to , with a proper correctionfor the above-mentioned zero-field shift.The trace marked A corresponds to =0. With incrementsof , the dips at half-integers deepen and those at integersbecome shallow, until they become equal for the traceB. Here, the period of oscillation becomes half the originalone. This corresponds to = 1 4. With a further increase144524-7

IYE et al. PHYSICAL REVIEW B 70, 144524 (2004)of , the dips at half-integer and integer ’s change overin depth. The trace C, which corresponds to = 1 2, is shiftedby a half-period from the original one (trace A). Theevolution with thus goes on and completes one cycle atthe trace E.Figure 13 shows a similar set of traces for sample B,representing the stripe field case. 24 Although the overall evolutionof the trace with is similar to Fig. 12, there are twodistinct features to be noted. First, comparing the traces B inthe two figures, the relative amplitude of oscillation at thepoint of half-periodicity for sample B = 1 4is muchsmaller than the corresponding one for sample A = 1 4. Secondly,although difficult to see directly from Figs. 12 and 13,there is a substantial difference in the way the depth of integerdips (and that of the half-integer dips) evolves with or, between the two cases. This is more clearly seen in Fig.14, which shows the shift in the transition temperature with (or ) for the two cases. The values of T c were obtainedfrom the experimentally measured changes in resistance byconverting them with use of the resistive transition curveRT. The amplitude T c is consistent with Eq. (8) with030 nm. The difference in T c by a factor of 2 betweensamples A (checkerboard) and B (stripe) is chiefly due to thedifference in the unit-cell area a 2 . The curves in the figurerepresent the shape of the bottom edge of the correspondingHofstadter spectra, seen on the front-right facet of the stereoplot in Fig. 8. The solid symbols and the solid curve correspondto =0. The open symbols and the dashed curve correspondto = 1 2. The agreement between the experimentaldata and the calculated curves indicates that the present systemof a superconducting network with magnetic decorationcaptures some of the characteristic features of the Hofstadterspectra under the spatially modulated magnetic field discussedin this paper.IV. CONCLUSIONWe have calculated the Hofstadter butterfly spectra for a2D tight-binding square lattice subjected to a spatially modulatedmagnetic field. The corresponding experimental situationsare realized by fabricating a superconducting wire networkdecorated with a mesoscopic ferromagnet array. Theevolution of Little-Parks oscillation with the amplitude of themodulated magnetic field in these samples exhibits characteristicfeatures of the two different types of modulation, thatis, checkerboard and stripe pattern. These results establishthe consistency of the calculated Hofstadter spectra with thechange in T c , although there is a fundamental limitation thatonly the edge of the spectrum can be probed by the superconductingsystem.Control of flux distribution in the superconducting networkby mesoscopic magnetic decoration may prove to be auseful technique in the development of devices using superconductingflux quanta. It should be of much interest to explorethe vortex dynamics in these artificial potential landscapes.More generally, the mesoscopic ferromagnet/superconductor hybrid system may provide a wider arena forthe exploration of vortex physics and possible device applications.ACKNOWLEDGMENTSWe thank Professor D. Yoshioka, Professor H. Aoki, ProfessorH. Fukuyama, Professor S. Maekawa, Professor F.Nori, and Dr. M. Koshino for helpful conversations. Thiswork has been supported by a Grant-in-Aid for COE Research“Quantum Dot and Its Application” (No. 12CE2004),and by a Grant-in-Aid for Scientific Research (No.13304025), from the Ministry of Education, Culture, Sports,Science and Technology (MEXT), Japan.*Electronic address: iye@issp.u-tokyo.ac.jp1 D.R. Hofstadter, Phys. Rev. B 14, 2239 (1976).2 G.H. Wannier, Phys. Status Solidi B 88, 757 (1978).3 F.H. Claro and G.H. Wannier, Phys. Rev. B 19, 6068 (1979).4 Y. Hatsugai and M. Kohmoto, Phys. Rev. B 42, 8282 (1990).5 Ch. Kreft and R. Seiler, J. Math. Phys. 37, 5207 (1996).6 D.J. Thouless, M. Kohmoto, M.P. Nightingale, and M. den Nijs,Phys. Rev. Lett. 49, 405 (1982).7 A.H. MacDonald, Phys. Rev. B 28, 6713 (1983).8 P. Lederer, D. Poilblanc, and T.M. Rice, Phys. Rev. Lett. 63,1519 (1989).9 Y. Hasegawa, Y. Hatsugai, M. Kohmoto, and G. Montambaux,Phys. Rev. B 41, 9174 (1990).10 C. Albrecht, J.H. Smet, K. von Klitzing, D. Weiss, V. Umansky,and H. Schweizer, Phys. Rev. Lett. 86, 147 (2001).11 B. Pannetier, J. Chaussy, R. Rammal, and J.C. Villegier, Phys.Rev. Lett. 53, 1845 (1985); see also B. Pennetier, in QuantumCoherence in Mesoscopic Systems, edited by B. Kramer, NATOAdvanced Study Institute Series Vol. 254 (Plenum Press, NewYork, 1991), p. 457.12 S. Alexander, Phys. Rev. B 27, 1541 (1983).13 R. Rammal, T.C. Lubensky, and G. Toulouse, Phys. Rev. B 27,2820 (1983).14 M.J. Higgins, Y. Xiao, S. Bhattacharya, P.M. Chaikin, S. Sethuraman,R. Bojko, and D. Spencer, Phys. Rev. B 61, R894 (2000).15 C.C. Abilio, P. Butaud, Th. Fournier, B. Pannetier, J. Vidal, S.Tedesco, and B. Dalzotto, Phys. Rev. Lett. 83, 5102 (1999).16 M.A. Itzler, A.M. Behrooz, C.W. Wilks, R. Bojko, and P.M.Chaikin, Phys. Rev. B 42, 8319 (1990).17 M.A. Itzler, R. Bojko, and P.M. Chaikin, Phys. Rev. B 47, 14 165(1993).18 B. Doucot, W. Wang, J. Chaussy, B. Pannetier, R. Rammal, A.Vareille, and D. Henry, Phys. Rev. Lett. 57, 1235 (1986).19 S. Ito, M. Ando, S. Katsumoto, and Y. Iye, J. 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HOFSTADTER BUTTERFLIES IN A MODULATED… PHYSICAL REVIEW B 70, 144524 (2004)23 Y. Morita and Y. Hatsugai, Phys. Rev. Lett. 86, 151 (2001).24 The reason why the range of for Fig. 13 is wider than for Fig.12 is simply due to the difference in the magnetic decoration.First, the ferromagnetic material for sample B was nickel incontrast to cobalt for sample A. Secondly, the distance betweenthe ferromagnet array and the superconducting network wasabout five times larger for sample B than sample A. These twofactors made the value of required for the same amplitude ofmagnetic field modulation much larger for sample B than forsample A.144524-9