Hall effect under null current conditions

Hall effect under null current conditions R. G. Mani and K. von Klitzing Max Planck Institut fiir Festkiirperforschung, Heisenbergstrasse 1, 070569 Stuttgart, Germany

(Received 9 July 1993; accepted for publication 22 December 1993) In a double boundary geometry, it is shown that dual independent Hall effects may be realized simultaneously by injecting two currents into the device, and the Hall effect on each boundary reflects only the current injected via the same boundary. A study of the magnetoresistance shows, however, that branch dissipation may be reduced through the optimal orientation of the injected currents. The results help clarify the signature edge currents in experiment by demonstrating that a Hall effect may be generated under null (netj current conditions.

The classical Hall effect1 is widelv understood in terms of the van der Pauw theorem,’ which asserts that the Hall effect may be measured without a knowledge of the current distribution provided that small contacts lie at the circumference of a singly connected specimen. Yet, earlier work of Juretschke et aZ.,3 suggests that the Hall constant remains unaffected by isolated, nonconducting “cylinders parallel to magnetic field and perpendicular to applied electric field.“3 The apparent difference between van der Pauw” and Juretschke et aL3 over the role of sample topology in the Hall effect, might be partially reconciled by asserting that simple connectedness was invoked because the complexity of the proof of (van der Pauw’s) theorem’ was reduced under these conditions, and that it does not necessarily imply that the Hall effect is manifested only in singly connected specimens.4*5Yet, it is also important to note that Ref. 3 does not define possible measurements of the Hall effect in a multiply connected medium. It asserts that a measurement of a sample containing holes, in the “usual experimental Hall setup,” would reveal an unmodified Hall constant. Even this latter prediction is not as general as one might presume. For example, studies of small system8---some including a small hole in a small cross7-and arrays of antidots in a macroscopic Hall bar,8 have shown that microscopic holes can modify the Hall constant, although naively one might not expect such effects based on Ref. 3. Such reasoning suggests that there remain unresolved paradoxes in the understanding. of the Hall effect. Here, we suggest that compatibility may be achieved between Refs. 2 and 3, at least in macroscopic systems, through the identification of inversion symmetry (within a sign change), and superposition in the Hall effect. We show that the existence of these properties allows for the possibility of several independent Hall effects or resistances in multiply connected, macroscopic, homogeneous systems utilizing multiple boundary-injected currents.9 For the special case of a doubly connected sample, we demonstrate dual classical Hall effects (or resistances) for dual injected currents even when the net current vanishes in both branches of the doubly connected structure. The results show that it is possible to measure a classical Hall effect under null (net) current conditions. The Hall effect is usually examined in the Hall configuration [see Fig. l(a)], where the measurement is defined unambiguously in terms of Hall’s experiment.’ Upon inserting a hole in the sample [Fig. l(b)], the measurement becomes 1262

Appl. Phys. Lett. 64 (IO), 7 March 1994

less well defined until one observes that the hole could, in principle, be made vanishingly small. Then, it is apparent that the geometry should be viewed as a Hall bar-with-a-hole [Fig. l(b)], where the Hall resistance, R,, ought to be defined in terms of the injected current, IA,B, and the external Hall effect, Vo,, , i.e., R,=Vo,,/I,,, , as in the simply connected specimen [Fig. l(a)]. Given the doubly connected specimen, one might also measure the Hall-like signals on the left branch, i.e., V,., in Fig. l(bj, and that on the right branch, i.e., V6,F in Fig. l(b), since one expects the sum Defining the Hall resistance in terms of vD,4+v6,F=vD,F* the interboundary signals requires, however, a knowledge of

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EJIG.1. (a) Hall effect measurement in the Hall bar configuration. (b) A Hall bar with a hole. The measurement shown allows for a determination of the Hall resistance from the injected current and the measured Hall voltage, i.e., %y=VD,FIrA,B ’ A determination of the Hall resistance from interboundary voltage measurements requires a knowledge of the current distribution. (c) Application of an inversion transformation [see (b)] results in the anti-Hall bar configuration where the Hall resistance may be determined from the interior injected current and the interior Hall voltage, i.e., V4,,/l,,,=R,. (d) superimposing configurations (b) and (c) produces an anti-Hall bar inside a Hall bar with two boundary-injected currents, which results in two welldefined Hall resistances V,,,I'Z,,~ and V4.6/11,2in a single specimen.

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8 1994 American Institute of Physics

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the branch current. This point indicates that van der Pauw’s theorem” is technically correct because, in multiply connected systems, there are additional Hall signals, such as I’D,4 and V6,F, which do not allow an evaluation of R, without a determination of the current distribution. An inversion transformation applied to the Hall barwith-a-hole [compare Figs. l(b) and l(c)], shifts the current and Hall voltage-contacts into the sample interior, and by synmetry this configuration must generate a classical Hall effect within the interior boundary such that the quotient of the interior Hall effect divided by the interior injected current equals the Hall resistance, i.e., R, = V4,6/11,2 and, therefore, one might also determine the Hall resistance in such a configuration without a knowledge of the current distribution.” As the interior arrangement [Fig. l(c)] is the complement of the exterior configuration [Fig. l(a)], one might identify this class of Hall effect measurements as Hall measurements in the “anti-Hall bar” configuration. Such reasoning indicates that, in general, measurements of the Hall resistance may be carried out without a knowledge of the current distribution even in multiply connected systems, so long as current and voltage contacts associated with a particular measurement all lie on the same boundary, regardless of whether the boundary lies in the interior or the exterior of the sample. One might examine superposition and, therefore, include the anti-Hall configuration within the standard Hall setup, and simultaneously inject two currents into the sample [see Fig. l(d)]. In the following, we describe the results of such experiments.’ Lock-in based measurements were carried out using a double current technique with electrically separated supplies in double boundary devices fabricated from GaAs/AlGaAs heterostructures, and alloyed Au-Ge/Ni contacts provided electrical access to the interior and exterior boundaries of the two-dimensional electronic system [see Fig. l(d)]. A simple analysis indicated that the sample was characterized by nil15 K)=2.5X101t cmb2 and ~(115 K)=7XlO” cm”/V s. Thus ,uB