Analyzing the Effects of Capacitances-to-Shield in Sample Probes on AC Quantized Hall Resistance Measurements

We analyze the effects of the large capacitances-to-shields existing in all sample probes on measurements of the ac quantized Hall resistance R(H). The object of this analysis is to investigate how these capacitances affect the observed frequency dependence of R(H). Our goal is to see if there is some way to eliminate or minimize this significant frequency dependence, and thereby realize an intrinsic ac quantized Hall resistance standard. Equivalent electrical circuits are used in this analysis, with circuit components consisting of: capacitances and leakage resistances to the sample probe shields; inductances and resistances of the sample probe leads; quantized Hall resistances, longitudinal resistances, and voltage generators within the quantum Hall effect device; and multiple connections to the device. We derive exact algebraic equations for the measured R(H) values expressed in terms of the circuit components. Only two circuits (with single-series “offset” and quadruple-series connections) appear to meet our desired goals of measuring both R(H) and the longitudinal resistance R(x) in the same cool-down for both ac and dc currents with a one-standard-deviation uncertainty of 10(−8) R(H) or less. These two circuits will be further considered in a future paper in which the effects of wire-to-wire capacitances are also included in the analysis.