Applied Superconductivity - Walther Meißner Institut - Bayerische ...

174 R. GROSS AND A. MARX Chapter 4I I ID D DW W WFigure 4.9: Sketch of various geometries for planar dc-SQUIDs with washer geometry.has the inductanceL ≃ 1.25 µ 0 D (4.1.52)in the limit W ≫ D. That is, the loop inductance scales with the inner diameter D. In contrast, theeffective area of the SQUID loop is 31A eff ∝ D ·W , (4.1.53)that is, it scales with the outer dimension of the washer. Of course, one cannot increase W arbitrarily,since for large W one starts to trap magnetic flux quanta in the washer area during cool down even insmall magnetic fields. The thermally activated motion of these flux quanta generate disturbing 1/ f -noise.An experimental example for the flux focusing effect is shown in Fig. 4.10 for a geometry shown in theinset. Here, A eff /D 2 is plotted versus W 2 /D 2 on a double logarithmic scale. In such plot a straight linewith slope 1/2 is expected according to (4.1.53) in good agreement with the data.We briefly address the inductance and the coupling of a spiral input coil that can be put on top of thewasher as shown in Fig. 4.11. Neglecting the parasitic inductance associated with the Josephson junctions,the following expressions for the inductance L i of the spiral input coil, the mutual inductance M iand the coupling coefficient α 2 between the spiral and the SQUID loop are found:L i ≃ n 2 L + L s (4.1.54)M i ≃ √ n 2 L · L = nL (4.1.55)α 2 ≃11 + L s /n 2 L . (4.1.56)Here, L s is the stripline inductance of the spiral coil, n the number of turns of the spiral input coil andn 2 L the geometric self-inductance of the input coil. For the estimate of M i we have assumed that the fluxdue to a current flowing in the spiral input coil is perfectly coupled in the SQUID hole. The couplingcoefficient α is obtained from the expression α = M i / √ L i L = nL/ √ (n 2 L + L s )L. As an example, forD = 20 µm we obtain L ≃ 30 pH. For a 50 turn input coil we obtain L i ≃ 75 nH 32 and M i ≃ 1.5 nH. The31 M.B. Ketchen, W.J. Gallagher, A.W. Kleinsasser, S. Murphy, J.R. Clem, in SQUID’85, H.D. Hahlbohm and H. Lübbigeds., Walther de Gruyter, Berlin (1985), p. 865.32 The stripline inductance is usually negligible for a 50 turn coil.© Walther-Meißner-Institut