IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 2, APRIL2007 431 Phenomenological Model for Frequency-Related Dissipation in the Quantized Hall Resistance Blaise Jeanneret and Frédéric Overney Abstract—Recently, ac measurements of the quantized Hall resistance have shown a linear relationship between the deviation of the Hall resistance from the perfectly quantized value ΔR H and the dissipation in the system represented by the longitudinal resistivity ρ ac xx . In this paper, we present a phenomenological model based on electrodynamic arguments that model this rela- tion. The dissipation is due to the displacement current flowing in the system. All the microscopic features are included in a complex tensorial dielectric susceptibility which remains to be investigated if more physical insight in the ac transport properties of the 2-D electron gas is desired. Index Terms—AC quantum Hall effect (QHE), AC resistance, dielectric losses, dissipation, frequency dependence, metrology. I. I NTRODUCTION S INCE 1990, the dc quantum Hall effect (QHE) has been successfully used as a representation of the unit of re- sistance: the ohm. The universal nature of transport in a 2-D electron gas (2DEG) at low temperature and high magnetic field provides a practical realization of a primary standard of resistance (see [1] for the latest review). Technical guidelines have also been written to ensure the highest possible accuracy in the measurements [2]. The situation for ac measurements of the QHE is not as well settled, although an important progress has been accomplished lately [3]. The major problem encountered with the ac mea- surements is a systematic dependence of the Hall resistance on current and frequency. These dependencies are related to losses either within the sample itself or losses due to current leaking to (or from) the sample’s surroundings. The losses can be compensated by the use of external gates [4], [5]. From our point of view, however, it is preferable to remove all the possible sources of leakage around the sample [5]. This approach leads to a strong reduction of the losses and allows the study of the physical processes taking place in the QHE device itself. In such a configuration, it was recently shown (see Fig. 1) that the deviation of the Hall resistance R H from its perfectly quan- tized value was proportional to the dissipation in the sample represented by the longitudinal ac resistivity ρ ac xx [3], [6] ΔR H = s ′ ρ ac xx (1) where s ′ is the proportionality factor. A linear relationship was already observed some 20 years ago for dc measurements [7], Manuscript received July 11, 2006; revised October 27, 2006. The authors are with the Swiss Federal Office of Metrology (METAS), 3003 Bern-Wabern, Switzerland (e-mail: blaise.jeanneret@metas.ch). Digital Object Identifier 10.1109/TIM.2007.891162 Fig. 1. Relative deviation of the Hall resistance R H , measured at the center of the i =2 plateau, plotted as a function of ρxx/R H for a GaAs/AlGaAs sample at a temperature of 0.3 K. The current was varied between 10 and 100 μA for each frequency range. A linear fit to the data gives a slope of s ′ =1.6. where this behavior was related to variable range hopping and thermal activation. However, at audio frequencies, additional physical mechanisms give rise to the dissipation of a different nature. By introducing a complex tensorial electric suscepti- bility, the linear relation of (1) can be deduced. Unfortunately, the calculation of the ac susceptibility is clearly beyond the scope of such a phenomenological approach. II. MODEL At dc, the sheet current density -→ J and the electric field - → E are related by the conductivity tensor σ. The conductivity σ and resistivity ρ =(σ) -1 tensors have their usual form σ =  σ xx σ xy -σ xy σ xx  (2) ρ = 1 σ 2 xx + σ 2 xy  σ xx -σ xy σ xy σ xx  . (3) Under full quantization, ρ xy = σ -1 xy = R H = R K /i and ρ xx = σ xx =0, where i is the Hall plateau index and R K ≡ h/e 2 . At finite frequencies, one must take the displacement field - → D into account - → J (t)= σ -→ E (t)+ d ∂ -→ D (t) ∂t . (4) The last term in (4) represents the sheet density of the displacement current -→ J D . The coefficient d is the thickness of 0018-9456/$25.00 © 2007 IEEE