Dual ordinary, integral quantum, and fractional quantum Hall effects in

PHYSICAL REVIEW B

VOLUME 55, NUMBER 23

15 JUNE 1997-I

Dual ordinary, integral quantum, and fractional quantum Hall effects in partially gated doubly connected GaAs/Alx Ga12x As heterostructure devices R. G. Mani* Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany ~Received 28 October 1996! Partially gated doubly connected quasi-two-dimensional GaAs/Alx Ga12x As heterostructure devices, which include striplike, bimodal electron density, n, distributions are used to demonstrate that the Hall resistance determined from a fixed set of voltage contacts can vary strongly with the off-diagonal resistivity r xy of the material connecting the source and the drain. The results are also used to deduce the response of homogeneous carrier density Hall configurations subject to similarly patterned inhomogeneous magnetic fields. @S0163-1829~97!04323-3#

In the laboratory, the Hall effect is often used to characterize the material properties of semiconducting and metallic systems because the technique is relatively straightforward and the relation between the Hall voltage V H , the applied current I, and the material off-diagonal resistivity r xy is especially simple in ‘‘homogeneous’’ specimens (R xy 5 r xy ) such as those based on the two-dimensional electron gas ~2DEG! in GaAs/Alx Ga12x As heterostructure devices, where V H 5IR xy and R xy is the Hall resistance.1,2 The simplicity and robustness of this very old effect have also led to the annual manufacture of millions of Hall-effect devices for tasks ranging from position sensing and contactless switching to magnetometry and metrology.1,3–10 Yet, in such realworld applications, Hall sensors are often subject to inhomogeneous magnetic fields, which invalidates the use of the R xy 5 r xy relation. Under such conditions, it becomes useful to know how the Hall resistance (R xy ) reflects an inhomogeneous magnetic field, if one wishes to optimize device performance.6 In addition, even in the laboratory, where one is able to control the homogeneity of the magnetic field, one often does not know a priori whether a specimen under investigation satisfies the necessary homogeneity criterion so far as the carrier density is concerned. Then, one sometimes wonders: what is the Hall response of a specimen with an inhomogeneous carrier density? In order to address such questions, we have investigated transport in specimens including striplike, bimodal electron density distributions within the 2DEG device. The results indicate that the Hall effect at a fixed set of contacts ~in the inhomogeneous n specimen subject to a homogeneous magnetic field! can vary strongly with the off-diagonal resistivity of the material connecting the source and the drain. The samples used in this study were fabricated from molecular-beam-epitaxy- ~MBE-! grown GaAs/Alx Ga12x As heterostructures, which were characterized by 2 31011 cm22