826 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 3, JUNE 2004 The Width of AC Quantum Hall Plateaus Jürgen Schurr, Jürgen Melcher, Klaus Pierz, Günter Hein, and Franz-Josef Ahlers Abstract—The quantum Hall effect in GaAs heterostructures is investigated with alternating currents in the kilohertz frequency range. The width of the plateau shows no significant frequency dependence and is determined by an effective temperature. This effective temperature is higher than the bath temperature by an amount which depends on the current and the bath temperature. Whereas the electron temperature determined by other measure- ments of the dc conductivity shows a clear divergence toward the centre of the Landau levels, the effective temperature determined from the ac plateau width does not depend on the noninteger frac- tion of the filling factor [1], [2]. This unexpected result is now ex- plained as a matter of the definition of the effective temperature and not of unexpected physics. Index Terms—Characteristic length, effective temperature, low- frequency resistance, quantum Hall effect. I. INTRODUCTION A COAXIAL ac bridge is used to measure the relative dif- ference between the quantum Hall resistance and a reference resistor with the same nominal value. To obtain the shape of a quantum Hall plateau, the magnetic flux density is sweeped across the plateau and the bridge signal is monitored (see [2] for a more detailed description). For the determination of the temperature and current dependence of the plateau shape, the bath temperature was varied between 0.32 and 1.2 K, and the plateau shape is measured again. The ac current was typi- cally varied between 5 and 25 A. Sometimes the current was reduced down to a few nanoamperes to determine the plateau shape in the limit of zero current. The width of the plateau is read at an arbitrary, but fixed value of . This value is, of course, the same for all plateau to be compared, for example 2 10 . Such a small value was chosen because we wanted to investigate the width of the region where the resistivity is close to a quantized value, i.e., the width of the region which is essential for metrological applications. Our investigations relate therefore to the tails of the Landau levels. The width of the plateau is found to decrease with increasing current. It decreases also with increasing bath temperature so that a change of current and a change of bath temperature are equivalent ([2], Fig. 1). An effective temperature can, there- fore, be assigned to each current. To fix the absolute value of the effective temperature, it has been assumed that the effec- tive temperature in the limit of zero current is equal to the bath temperature. The effective temperature is found to increase linearly with the current , the rate being inversely proportional to the width of the Hall bar as shown in Fig. 2 ([1] and [2]). Such a de- Manuscript received February 5, 2003; revised January 8, 2004. The authors are with the Physikalisch-Technische Bundesanstalt (PTB), 38116 Braunschweig, Germany. Digital Object Identifier 10.1109/TIM.2004.827066 Fig. 1. Starting with a current of 3 A and a bath temperature of 0.32 K, the plateau shape changes in the same way when current or bath temperatures are increased. The frequency was 1 kHz. Note the break in the axis which cuts out the central plateau region. Fig. 2. (Left) Effective temperature increases linearly with current , shown for a sample with a width 2.7 mm and a frequency of 1 kHz. (Right) Corresponding increase rate measured with different samples is inversely proportional to the width of the Hall bar (measured at a frequency of 1 kHz). pendence of the effective temperature on the current density is valid also for dc hopping [3]. A theoretical model yields (1) where is the temperature of the bath, the quantum Hall resistance, and the Boltzmann constant. This equation describes the energy gain of the electrons in the electric Hall field along a mean free length . To find out if (1) and other theoretical scaling relations apply to our results of the plateau width, (1) is used just as a fit- ting function, the characteristic length being a free parameter which describes the rate of increase of the effective temperature with respect to the current . To determine the dependence of the characteristic length on the noninteger fraction of the filling factor , the plateau width was read at different values of from 10 up to 10 , and the whole analysis was repeated. Further, samples with different plateau widths (i.e., different electron mobilities) were investigated. It has been found that the parameter does not depend on how close the filling factor is to an integer value . Further, the parameter does not depend on the electron 0018-9456/04$20.00 © 2004 IEEE