Fig. 4: Comparison of inner and outer EC interference and couplings across transition. | Nature Communications

Fig. 4: Comparison of inner and outer EC interference and couplings across transition.

From: Strongly coupled edge states in a graphene quantum Hall interferometer

Fig. 4

a Conductance \({G}_{{{\rm{D}}}}\) oscillations on the inner EC (\({T}_{{{\rm{QPC}}}1}={T}_{{{\rm{QPC}}}2}=1.5\)) with \({V}_{{{\rm{PG}}}}\) and \(B\), for \({V}_{{{\rm{MG}}}}=1.2{{\rm{V}}}\). Dotted black lines highlight conductance maxima. Left inset: illustration of inner EC interference configuration. b 2D FFT of the \({G}_{{{\rm{D}}}}\) oscillations in (a) showing peak \({{{\boldsymbol{f}}}}_{{{\rm{i}}}}\) (vector corresponding to blue arrows) and its harmonics, where \({\Phi }_{0}\equiv h/e\). c \({G}_{{{\rm{D}}}}\) oscillations on the outer EC (\({T}_{{{\rm{QPC}}}1}={T}_{{{\rm{QPC}}}2}=0.5\)) at the same density set by \({V}_{{{\rm{MG}}}}\). Dotted black lines with identical slope to (a) highlight phase jumps. Left inset: illustration of outer EC interference configuration. d 2D FFT of oscillations in (c) showing the peaks \({{{\boldsymbol{f}}}}_{{{\rm{o}}}}\) (red arrows), \({{{\boldsymbol{f}}}}_{{{\rm{o}}}+{{\rm{i}}}}\), and \({{{\boldsymbol{f}}}}_{{{\rm{o}}}-{{\rm{i}}}}\) and their harmonics. e Magnitude of the phase jump on the outer EC as a function of \({V}_{{{\rm{MG}}}}\). Each data point is averaged over a \(\sim 0.25{{\rm{V}}}\) range in \({V}_{{{\rm{MG}}}}\); error bars indicate \(\pm 1\) standard deviation in this range. Unfilled data points represent zero observable phase jumps over the range, hence we infer a magnitude of \(0\) or \(-1\). \({G}_{{{\rm{xy}}}}\) of the device taken in an identical measurement to Fig. 2a, reflecting the expected filling \({\nu }_{{{\rm{MG}}}}\), is plotted for reference. Top inset: cartoon of the outer and inner EC evolution with increasing \({V}_{{{\rm{MG}}}}\). f Magnitudes \({I}_{{{\rm{o}}}}\), \({I}_{{{\rm{o}}}+{{\rm{i}}}}\), and \({I}_{{{\rm{o}}}-{{\rm{i}}}}\) of the respective peaks \({{{\boldsymbol{f}}}}_{{{\rm{o}}}}\), \({{{\boldsymbol{f}}}}_{{{\rm{o}}}+{{\rm{i}}}}\), and \({{{\boldsymbol{f}}}}_{{{\rm{o}}}-{{\rm{i}}}}\) as a function of \({V}_{{{\rm{MG}}}}\). \({I}_{{{\rm{o}}}}\), \({I}_{{{\rm{o}}}+{{\rm{i}}}}\), and \({I}_{{{\rm{o}}}-{{\rm{i}}}}\) are normalized by the sum \({I}_{{{\rm{o}}}}+{I}_{{{\rm{o}}}+{{\rm{i}}}}+{I}_{{{\rm{o}}}-{{\rm{i}}}}\) to show their relative contributions. Each data point is extracted from a 2D FFT (Supplementary Fig. 7). g Magnetic field frequency multiplied by \({\Phi }_{0}\) for peaks \({{{\boldsymbol{f}}}}_{{{\rm{o}}}}\), \({{{\boldsymbol{f}}}}_{{{\rm{i}}}}\), \({{{\boldsymbol{f}}}}_{{{\rm{o}}}+{{\rm{i}}}}\), and \({{{\boldsymbol{f}}}}_{{{\rm{o}}}-{{\rm{i}}}}\) tracked through the transition. Note that \({{{\boldsymbol{f}}}}_{{{\rm{i}}}}\) is measured from a separate measurement of interference on the inner EC (Supplementary Fig. 8). h Same as (g) but for plunger gate frequency. Horizontal dashed lines in (g, h) indicate the corresponding \({{{\boldsymbol{f}}}}_{{{\rm{o}}}}\) and \({2{{\boldsymbol{f}}}}_{{{\rm{o}}}}\) values before the transition. Black (red) dots show calculated \({{{\boldsymbol{f}}}}_{{{\rm{o}}}}\pm {{{\boldsymbol{f}}}}_{{{\rm{i}}}}\) from outer and inner EC data, which match the peaks identified as \({{{\boldsymbol{f}}}}_{{{\rm{o}}}+{{\rm{i}}}}\) and \({{{\boldsymbol{f}}}}_{{{\rm{o}}}-{{\rm{i}}}}\), respectively.

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