A new Network Approach for the Integer Quantum Hall Effect

 

Introduction:


 So far the edge channel (EC) picture has been successfully used to explain the plateau values of the IQHE. In addition, for the case of EC back scattering, which is induced by a gate electrode in depletion mode, the EC picture on the basis of the Landauer-Büttiker (LB) formalism gives also correct values for the finite longitudinal resistance Rxx, provided that the involved ECs are either fully reflected or transmitted. This means that the bulk region below and outside the gate has to be insulating (plateau regime below and outside the gate). The situation gets more complicated if the bulk below and/or outside the gated region undergoes a QHE plateau-to-plateau transition. In this case edge and bulk currents occur simultaneously and the LB formula cannot be applied directly. The problem of a simultaneous occurrence of edge and dissipative bulk currents was recently successfully addressed by an alternative representation of back scattering in the EC-picture[1,2]. In this presentation we show, that this back scattering approach allows to calculate the resistance components Rxx and Rxy as well as the potential distribution in the bulk for actual sample geometry's. The key point is that this back scattering approach allows also to treat the coupling between EC-pairs like it occurs by tunneling between magnetically bound states at the saddle points of long range potential fluctuations in the bulk. Thus, it is possible to use this modified LB-like formalism also for describing the nodes of a Chalker-Coddington-like network representation of bulk transport in the high magnetic field limit. However, the essential difference to the Chalker-Coddington network is that we use the above mentioned back scattering formulation instead of a transfer matrix for describing the nodes[3]. The coupling (tunneling) at the nodes depends exponentially on the position of the Fermi level with respect to the saddle point energy. This finally means, that the coupling at the nodes of the network depends exponentially on the filling factor at the lateral location of the node. In this way we get a handle for including a lateral electron density profile within the network. This allows us to define realistic sample structures by shaping the lateral carrier density profile to e.g. a Hall bar with realistic contact arms and side depletion zones. Further more it is possible to model inhomogeneities like introduced e.g. by a gate electrode across the Hall bar.
 


Figure: Typical density profile, which is distributed over the nodes of the grid, which allows to calculate a local filling factor. At the same time it is shown, how a gate across the Hallbar can be created.
 

Each involved LL is represented by a complete network and all LLs are interconnected at the location of the metallic contacts (voltage probes and current contacts). The network is solved numerically and allows to study almost arbitrarily shaped samples including non-local and Corbino geometry. Besides the transport data, our network model delivers also the lateral potential distribution for each LL, which will be used for the animated examples.

[1] J.Oswald, G. Span, F. Kuchar, Phys. Rev. B58, 15401-15404 (1998)

[2] J.Oswald, "A New Model for the Transport Regime of the Integer Quantum Hall Effect: The Role of Bulk Transport in the Edge Channel Picture", Proc. 10th Int. Winterschool on New Developments in Solid State Physics, Mauterndorf, 23-27 Feb. 1998, Salzburg, Austria (invited),  Physica E3, 30-37 (1998)
 (for a preprint click here)

[3] J. Oswald, "A Novel networkmodel for the Hall Insulator", Proc. 4th Int. Symp. on New Phenomena in Mesoscopic Structures" Dec. 6-11 1998, Kauai, Hawaii, USA.  , in print (for a preprint click here)
 

Examples:

The arrows connecting the nodes of the network are colored according to the potential which they transmit. "Red" means a high potential, which is supplied to the current contact at the left and "green" corresponds to the low potential, which has been supplied to the current contact on the right. Any color other than green or red corresponds to an intermediate potential. Arrows of undefined potential appear in gray. According to this representation, edge channels can be identified as stripes of constant color. In this context the ohmic regime can be identified by a gradual change of color in the longitudinal direction. The shape of the sample is defined by distributing an appropriate lateral carrier density profile over the grid of the network like shown in the figure above.
2->1- Plateau transition:
The animation shows a magnetic field sweep as indicated in the line at the top. The sample represents a standard QHE-situation without a gate electrode. Starting at the n=2 plateau, the potential distribution indicates edge stripes in the 2nd Landau level (LL). Approaching the transition regime, the edge stripes get broader and finally fill the whole sample area, but still indicating a pure transverse potential profile. After this, the potential profile starts to turn in longitudinal direction and finally the potential distribution in the interior gets undefined, which indicates that the 2nd LL gets insulating. At the same time Rxy approaches the n=1 plateau. Since there is another completely filled LL below which tries to keep the edge potential constant, the longitudinal voltage drop between the longitudinal voltage probes does not completely follow the voltage drop in the 2nd LL.
Transition to the Hall Insulator (HI)
Same sample as shown in previous example. This animation demonstrates a field sweep to higher fields starting at the n=1 plateau. In this case just the 1st LL is involved. At the beginning the same happens as in the previous example, namely the stripes get wider and finally cover the whole sample area. However, as the potential profile turns to longitudinal, there no other lower LL which would like to keep the edge potential and therefore the potential drop between the longitudinal voltage probes can fully develop. This leads to a monotonous increase of  Rxx without peak behavior. It is important to realize, that in this regime Rxy tends to stay on the n=1 plateau while Rxx already steeply rises. This behavior is known as the "Quanten-Hall-Liquid - to - Insulator" transition. The reason for the deviation of Rxx at highest fields is the discrete numerical representation, which always contains some deviation from an ideal geometry, like in real samples.

Edge channel reflection at a gate electrode

A gate electrode is simulated like shown in the figure above. The carrier density below the gate is depleted to 50% of the carrierdensity outside the gate. In this situation a special condition occurs if the fillingfator outside the gate is n = 4 and n = 2 because in this cases the fillingfactor below the gate gets nG = 2 and  nG =1 respectively. This means for n = 4 two edge channels get transmitted through the gate and 2 edge channels get reflected. In the case n = 2 one edge channel gets transmitted and one reflected. The simulation shows the potential distribution in the second Landau level while the magnetic field sweeps in a way that the fillingfactor n changes from  n = 4 to n = 2. This means that at the beginning the 2nd Landau level should create a transmitted EC while at the end of the feld sweep it should create a reflected EC.
 
 

Supported by Österreichische Nationalbank - Jubiläumsfonds Project No. 6566
 

Back to the Homepage