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VOLUME 83, NUMBER 5 PHYSICAL REVIEW LETTERS 2A UGUST 1999Comment on “Fractal Conductance Fluctuations in Fig. 1(b)] which track the curves and reveal the samedeviation from linearity at largeDB.in a Soft-Wall Stadium and a Sinai Billiard”2Since the theory formulates FCF in terms ofDG [3],Sachrajda et al. [1] claim their investigations of semi- this is an obvious method for their detection. A box count-conductor billiards provide the first observation of fractal ing parameterN is also used to detect FCF in [1]. The frac-2dconductance fluctuations (FCF) over 2 orders of magni- tal dimension d is calculated using N DB , wheretude in magnetic field B. Here we reanalyze the data for 1, d, 2. Figure 1(d) shows log N vs log DB for10 10the Sinai billiard (SB) in [1] plus previously reported data the traces of Fig. 1(a). For the nonfractal model curve[2] for this billiard and show that the FCF are limited to (bottom), the gradient must be21 (corresponding todone order. 1) at smallDB, which is the gradient of the line shown. ForIf conductanceG vsB curves exhibit FCF, the variance DB. 0.3 mT, the periodicity in our model curve causes2of G increments scales with B increments as DG the associated log plot to deviate from linearity. Under-422DDB , where the fractal dimensionD obeys1, D, standing how the periodicity in the experimental GB2 [3]. The top three traces in Fig. 1(a) show experimen- curves affects the shapes of the log N vs log DB ...

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VOLUME83, NUMBER5
P H Y S I C A L R E V I E W L E T T E R S
Comment on “Fractal Conductance Fluctuations in a Soft-Wall Stadium and a Sinai Billiard”
Sachrajdaet al.[1] claim their investigations of semi-conductor billiards provide the first observation of fractal conductance fluctuations (FCF) over 2 orders of magni-tude in magnetic fieldB. Here we reanalyze the data for the Sinai billiard (SB) in [1] plus previously reported data [2] for this billiard and show that the FCF are limited to oneorder. If conductanceGvsBcurves exhibit FCF, the variance ofGincrements scales withBincrements asDG2  DB422D, where the fractal dimensionDobeys1,D, 2[3]. The top three traces in Fig. 1(a) show experimen-talGBplots for the SB, with biasesVI(from the top) of10.7,20.1, and22.9V applied to the circular gate [see Fig. 1(a) inset]. The20.1V trace is the same data as Fig. 1(b) of [1], except thatGis plotted (as appropri-ate for the FCF analysis) rather than resistance. The10.7 and22.9 AsV traces are from [2].VIis made more nega-tive, the empty billiard (top trace) is transformed into the SB [2]. The bottom trace in Fig. 1(a) shows a model os-cillatory function constructed from a cosine superimposed on a smooth background similar to that of the experi-mental data. Figure 1 also shows plots of log10DG2 [Fig. 1(b)] and its derivative [Fig. 1(c)] against log10DB, for the curves in Fig. 1(a). Over the range that FCF ex-ist, the traces in Fig. 1(c) will be horizontal lines with or-dinates422Dsatisfying1,D,2 is clear from. It Fig. 1(c) that the data traces exhibit FCF over one order of magnitude or less inDB expected, the model curve is. As not fractal: at smallDBthe horizontal line givesD1, but at largerDBthe analysis breaks into oscillations as the underlying period is detected. Similar oscillations in the data traces of Fig. 1(c) again indicate a dominant period at largeDBby the inspection of the raw is confirmed . This data in Fig. 1(a): a clear period develops as the circle is in-troduced. Comparisons of Figs. 1(b) and 1(c) demonstrate the importance of a derivative analysis. Using plots simi-lar to Fig. 1(b), Sachrajdaet al.claim to observe FCF over twoorders inDB[arrows in Fig. 1(c) indicate this range]. Our derivative analysis shows that FCF occur only over one order. The curves in Fig. 1(b) are calculated using overlappingDB (statisticallyintervals. Nonoverlapping independent)DBintervals giveDG2values [triangles
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in Fig. 1(b)] which track the curves and reveal thesame deviation from linearity at largeDB. Since the theory formulates FCF in terms ofDG2[3], this is an obvious method for their detection. A box count-ing parameterNis also used to detect FCF in [1]. frac- The tal dimensiondis calculated usingN DB2d, where 1,d,2. Figure 1(d) shows log10Nvs log10DBfor the traces of Fig. 1(a). For the nonfractal model curve (bottom), the gradientmustbe21(corresponding tod1) at smallDB, which is the gradient of the line shown. For DB.0.3mT, the periodicity in our model curve causes the associated log plot to deviate from linearity. Under-standing how the periodicity in the experimentalGBcurves affects the shapes of the log10Nvs log10DBplots is thereforeessentialfor accurate assessment of theDB range for which FCF occur. It is insufficient to perform a linear fit without considering the limits imposed by this periodicity. The experimental traces of Fig. 1(d) all reveal similar deviations caused by the periodicity inGBwhich prevents the observation of FCF over two orders. The de-viations from the FCF behavior revealed in Figs. 1(b) and 1(d) occur at identicalDBvalues but are less pronounced in Fig. 1(d), indicating that the box counting method is less sensitive to the emergence of periodicity. Using this technique, Sachrajdaet al.again claim to see FCF over two orders, while Fig. 1(d), together with the more sensi-tiveDG2confirms that FCF are limited to onemethod, order. Sachrajdaet al.state that “almost any smooth func-tion can be satisfactorily fitted by a power law for just one order of magnitude.” Applying their criterion to Fig. 1, the SB data are only suggestive of FCF. R. P. Taylor,1A. P. Micolich,1T. M. Fromhold,2 and R. Newbury1 1School of Physics, UNSW, Sydney, 2052, Australia 2School of Physics and Astronomy, Nottingham University Nottingham NG7 2RD, United Kingdom Received 5 June 1998 PACS numbers: 73.23.Ad, 05.40. – a, 05.45. – a, 72.80.Ey
[1] A. S. Sachrajdaet al.,Phys. Rev. Lett.80, 1948 (1998). [2] R. P. Tayloret al.,Phys. Rev. Lett.78, 1952 (1997); Phys. Rev. B56 (1997). 733, R12 [3] R. Ketzmerick, Phys. Rev. B54, 10 (1996). 841
FIG. 1. The traces are vertically offset. Intercepts are as follows: (a) (top) 264, 218, 151, and 152mS; (b)22.20,22.52, 22.91, and23.28; (c) 1.28, 1.53, 1.72, and 2; (d) 5.91, 6.16, 6.06, and 5.78. In (b) and (d) the dashed lines show theDB values above which departure from FCF behavior occurs. In (d) the gradient lines correspond tod1.20(top), 1.15, 1.10, and 1.00. The inset in (a) is a schematic diagram of the SB in [1].
© 1999 The American Physical Society