Hofstadter butterflies in a modulated magnetic field - APS Link ...
Hofstadter butterflies in a modulated magnetic field - APS Link ...
Hofstadter butterflies in a modulated magnetic field - APS Link ...
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IYE et al. PHYSICAL REVIEW B 70, 144524 (2004)FIG. 10. Structure of the samples. (a) Sample A for the checkerboard<strong>field</strong>. (b) Sample B for the stripe <strong>field</strong>. The external <strong>field</strong>parallel to the network plane controls the magnetization of the ferro<strong>magnetic</strong>array. The amplitude of the spatially <strong>modulated</strong> <strong>magnetic</strong><strong>field</strong> can be changed by the azimuthal angle of the parallel<strong>field</strong>.III. EXPERIMENT WITH SUPERCONDUCTINGWIRE NETWORKSA. Experimental methodThe samples used <strong>in</strong> the present study were fabricated onsilicon substrates by the follow<strong>in</strong>g steps.(i) Electrode pads were first formed by electron beam lithographyand gold evaporation.(ii) The network pattern (square lattice) was def<strong>in</strong>ed byelectron beam lithography and the niobium wire networkwas formed by ion-beam sputter<strong>in</strong>g deposition and the liftoffprocess.(iii) A protect<strong>in</strong>g 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 ferro<strong>magnetic</strong>material 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 <strong>field</strong>). 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 po<strong>in</strong>t <strong>in</strong> the fabrication was to achievegood positional and angular registration between thesuperconduct<strong>in</strong>g network and the overlaid ferromagnetarray.Two samples (A and B) were <strong>in</strong>tensively studied. Thesesamples represent the checkerboard <strong>field</strong> case (sample A)and the stripe <strong>field</strong> 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 wires<strong>in</strong> the y direction, as shown <strong>in</strong> Fig. 10(a). For sample B,250-nm-wide strips of 60-nm-thick nickel were placedon top of every other l<strong>in</strong>es <strong>in</strong> the y direction as shown <strong>in</strong>Fig. 10(b).Measurements of the superconduct<strong>in</strong>g properties wereconducted by use of a cross-coil superconduct<strong>in</strong>g magnetsystem, consist<strong>in</strong>g of a 6 T Helmholtz coil and a 1 Tsolenoid. The horizontal <strong>field</strong> was used to fix the magnetizationof the ferro<strong>magnetic</strong> array and thereby controlthe strength of the spatially vary<strong>in</strong>g <strong>field</strong> (parameter or ).The vertical <strong>field</strong> supplied the uniform <strong>field</strong> (parameter )for the network. The four-term<strong>in</strong>al resistance of the networkwas measured by a standard ac lock-<strong>in</strong> technique. Cryogeniccontrol was achieved by a variable temperature <strong>in</strong>sert Dewar144524-6