Calculating the effects of longitudinal resistance in multi-series-connected quantum Hall effect devices

Abstract

Many ac quantized Hall resistance experiments have measured significant values of ac longitudinal resistances under temperature and magnetic field conditions in which the dc longitudinal resistance values were negligible. We investigate the effect of non-vanishing ac longitudinal resistances on measurements of the quantized Hall resistances by analyzing equivalent circuits of quantized Hall effect resistors. These circuits are based on ones reported previously for dc quantized Hall resistors, but use additional resistors to represent longitudinal resistances. For simplification, no capacitances or inductances are included in the circuits. The analysis is performed for many combinations of multi-series connections to quantum Hall effect devices. The exact algebraic solutions for the quantized Hall resistances under these conditions of finite ac longitudinal resistances provide corrections to the measured quantized Hall resistances, but these corrections do not account for the frequency dependences of the ac quantized Hall resistances reported in the literature.

Uncertainty evaluation in a two-terminal cryogenic current comparator

“…The relative change in the quantized Hall resistance using a tripleseries connection in dc can be calculated from a mathematical model of the QHR [4]. In that reference an offset of less than 0.001 µΩ/Ω was found with typical leads and longitudinal resistance.…”

“…This improves the sensitivity [2] but the winding resistance must be measured to correct the value of each resistor. Since this bridge measures two-terminal resistors, we use the tripleseries connection technique to reduce errors in measurements of the QHR, as described in [4]. In a condition of balance, the bridge equations are: (1) .…”

Analyzing the effects of capacitances-to-shield in sample probes on AC quantized Hall resistance measurements

“…Under these current and magnetic field conditions, the drain contact pad D ′ and the potential probe contact pads 1 ′ , 3 ′ , and 5 ′ at the device periphery are at higher potentials than contact pads S ′ , 2 ′ , 4 ′ , and 6 ′ . These current and flux density directions are chosen to be consistent with those we have used in earlier calculations [ 19 , 22 – 23 ].…”

“…Diamond-shaped voltage generator arrays of Ricketts and Kemeny [ 18 ] are employed in the equivalent circuit of the QHE device, rather than the ring-shaped voltage generator arrays introduced later by Delahaye [ 24 ] and then subsequently used by Jeffery, Elmquist, and Cage [ 25 ]. Although both arrays give essentially identical results [ 22 ], the calculations are much simpler with the diamond arrays when longitudinal resistances are included in the circuits [ 22 ]. We therefore use diamond arrays.…”

Derivation of an electronic equivalent of QHE devices

“…Very detailed analytical studies of QHE circuits have been reported in the literature [4], nevertheless, if the measurement circuit is not oversimplified, the equations become very complex [5]. Numerical techniques are useful [6] in the study of such a problem, allowing quick tests of solutions corresponding to different values of the circuit parameters.…”