Measuring the Hall resistance of a two-dimensional electron system (2DES) in magnetic fields of typically several Tesla leads to the observation of Hall resistance plateaus as a function of magnetic field (Quantum Hall effect, QHE [1]). The values are well described by *R*_{H}=*h*/i*e*^{2}, where h is the Planck constant, e the elementary charge and i an integer value. Within a plateau, the longitudinal resistance, i.e., a voltage drop *V*_{x} along the 2DES edge, vanishes. As the Hall resistance value is reproduced to high accuracy in different material systems, since 1990 the QHE is used as resistance standard. Today, the QHE opens the way for the redefinition of the SI units in terms of fundamental constants.

For several years we have performed scanning probe measurement at 1.4K on 2DES embedded in narrow (typ. 10µm wide) Hall bars made from (Al,Ga)As heterostructures. By interpreting the evolution of Hall potential profiles with increasing magnetic field, we could obtain a detailed microscopic picture of the current distribution inside the 2DES [2]. An important ingredient is the presence of incompressible regions inside the 2DES, which are characterized by the fact that locally the Fermi level lies between two Landau levels, i.e., the states of the lower Landau level are completely filled, those of the next higher level are empty. A Hall voltage drop over such an incompressible region causes a drift of all states, leading to a local current density *j*=i*e*^{2}/*hE*_{H}, perpendicular to the direction of the Hall field *E*_{H}. The integer number i is the number of filled Landau levels of the incompressible region. If the Hall voltage drop, induced by the biased current I, happens in a cross section of the 2DES only over incompressible regions of same filling factor i, the Hall voltage *V*_{H}=*h*/i*e*^{2} I is measured between the two edges of this cross section.

It turned out that under high magnetic fields incompressible strips can be present in the electrostatic depletion region along the edges of the 2DES while the bulk is compressible. The innermost incompressible strips at both edges of a cross section carry the current through the sample, leading to a quantized Hall resistance. They evolve with increasing magnetic field, i.e., they get wider and shift with its center position towards the bulk. The incompressible strips coming from opposite edges of the 2DES merge when we expect that the Landau level filling factor in the bulk approaches an integer value. As the 2DES contains slight variations in the electron concentration due to disorder, the bulk is then mostly incompressible with embedded compressible droplets. The biased current is widely distributed in this incompressible bulk flowing around the compressible droplets. In summary, the lower magnetic field side of a Hall resistance plateau is governed by the presence of incompressible strips at the edges of the 2DES whereas the extension of the upper plateau side is determined by the strength of disorder in the bulk.

In recent time, we have concentrated on investigating by our scanning probe technique in parallel to magneto-transport measurements the evolution of the Hall potential landscape with increasing bias. As expected from our microscopic picture we could identify basically two different behaviors towards QHE breakdown, where a finite voltage drop *V*_{x}>0 becomes measurable along the Hall bar, indicating electrical dissipation.

**Fig. 1:**(a) Applying a voltage bias V

_{DS}between source (S) and drain (D) contact to a 2DES − embedded in a Hall bar − under quantum Hall conditions, the longitudinal voltage drop

*V*

_{x}is zero. However, with rather large voltage bias a longitudinal voltage drop becomes detectable indicating the breakdown of the quantum Hall effect. The lower and upper magnetic field side of a Hall resistance plateau show different

*V*

_{x}vs.

*V*

_{DS}behavior. This is also seen in the evolution of Hall potential profiles with increasing

*V*

_{DS}: (b) For the low magnetic field side, Hall voltage drops appear for small

*V*

_{DS}symmetrically at both edges, whereas with increasing V

_{DS}the distribution becomes asymmetric. The smooth evolution is nicely seen by plotting the Hall potential profiles in color-scale versus

*V*

_{DS}(see (d)). (c) For the high magnetic field side, the Hall voltage drop happens widely distributed over the bulk, and the profile is not changing to high

*V*

_{DS}, till with a small further increase a sudden profile change is observed (see also (e)).

On the lower magnetic field side of a plateau, a small current is equally distributed between the opposite edges in a Hall bar cross section. With increasing bias, the Hall voltage drop over these incompressible strips becomes asymmetric until the current flows almost completely close to one edge (**Fig. 1(b),(d)**). This behavior is naturally explained in our microscopic picture [3]: The width of an incompressible strip at the edge depends on the electrostatic potential drop over it. The Hall voltage drop acts on one edge with, on the other edge against the intrinsic potential drop which is given in magnitude by the gap between Landau levels and turns up towards the edge. Therefore, on one edge the width is increased enhancing the Hall voltage drop here, whereas the incompressible strip vanishes on the other edge, limiting the Hall voltage drop there. As almost the whole Hall voltage drop appears over a small incompressible region in the cross section, the Landau level bending is huge there, which might allow for inter Landau level tunneling as breakdown mechanism or strong local heating due to a small, but finite longitudinal resistivity, caused by scattering of thermally excited electrons inside the incompressible strip. On the other edge where the incompressible strip has lost its width, electrons can relax from higher to lower Landau levels by photon emission. Consistently with the microscopic picture, where local electric fields gradually increase with increasing bias, the evolution of a longitudinal voltage drop *V*_{x} is gradually appearing with increasing bias. We denote this regime the edge-dominated QHE breakdown.

**Fig. 2:**Scans of the Hall potential landscape over an area of the Hall bar for different bias voltages V

_{DS }and a magnetic field near the high-

*B*edge of a Hall resistance plateau. Obviously, the Hall potential drop happens in the bulk. From (a) to (f) the bias voltage was increased from 20mV to 120mV in steps of 20mV. For low bias, the Hall potential profiles are similar in the different cross sections. At high bias, the profiles change strongly along the sample. Small regions of enhanced electrical fields are visible.

In contrast, on the upper magnetic field side of a Hall resistance plateau, where the current is widely distributed inside the 2DES bulk, *V*_{x}>0 appears rather abruptly at large bias (**Fig. 1(c),(e)**). We call this regime the bulk-dominated breakdown. We found that the Hall potential landscape remains stable up to high bias − the local Hall voltage drops are moderate, until a small further increase in bias dramatically changes the Hall potential landscape. Such an abrupt change creates abruptly locally enhanced electric fields that enhance locally inter Landau level transitions or heating, and give rise to an abrupt increase of the longitudinal voltage drop (see also **Fig. 2**). Self-consistent simulations which do not include breakdown mechanisms also show that in the bulk-dominated regime where disorder in the bulk is important the current distribution is changing with bias. A small change in the sample bias can lead to the formation of a dominant incompressible segment in a cross section, carrying most of the current.

In summary, our recent results on the evolution of Hall potential profiles with increasing bias − obtained by an almost non-invasive scanning probe technique − clarifies the breakdown of QHE in such samples, and supports our microscopic understanding of the quantum Hall effect, which contradicts the QHE model of current carrying edge states, commonly cited in the context of topological insulators.