A scanning single-electron transistor array microscope probes the Hall potential profile in the fractional quantum Hall regime

INTRO: In 1980 Klaus von Klitzing (Nobel prize in 1985) observed during low-temperature Hall measurements on two-dimensional electron systems hosted by MOSFETs, fixed values of the Hall resistances R_xy described with h/(ie^2) (i is integer) - nowadays denoted as integer quantum Hall effect (QHE). Since 1990 the QHE is used as a resistance standard and it played a key role in the redefinition of the Système Internationale d'unités (SI unit system), where from May 2019 the SI units are defined by fixing the values of fundamental physical constants as h, e, c and k_B. In 1982 Störmer, Tsui and Laughlin (Nobel prize in 1998) observed and discussed the fractional quantum Hall effect (FQHE) where further resistance plateaus are observable with R_xy=h/(ve^2) where v are special fractional numbers. The FQHE is currently understood on base of electron-electron interaction leading to quasi-particles with fractional effective charge. The main goal of this thesis was to use an one-dimensional single-electron transistor (SET) array as sensitive electrometer to locally probe Hall potential profiles in the fractional quantum Hall regime to determine where an externally biased current is distributed inside a two-dimensional electron system (2DES) hosted by an (Al,Ga)As heterostructure. MICROSCOPIC PICTURE: This thesis opens with an explanation of the microscopic picture of the integer quantum Hall effect where strong magnetic flux densities lead to the formation of Landau levels that are separated by an energy gap. This gap is responsible for the formation of electrically incompressible regions - with a well defined integer filling factor - within an otherwise compressible 2DES. A quantum Hall plateau shows two regimes: (1) the edge-dominated QH regime in the low magnetic field side and (2) the bulk-dominated QH regime in the high magnetic side of the plateau. SCANNING SET ARRAY MICROSCOPE: For experiments a scanning single-electron transistor (SET) array microscope with eight independent SETs on tips is used. The SET island sizes are about 155nm by 220nm, separated by 4µm. Single-electron charging energies up to 175µeV had been reached for these SETs. Measurements were performed at temperatures below 40mK in a 3He/4He dilution refrigerator with a 18T superconducting magnet, located in a highly vibrational reduced environment. MEASUREMENT PRINCIPLE: Electrostatic potential changes of the 2DES which result solely from an externally biased current are accessible via a two-step measurement technique probing calibrated Hall potential profiles. In this thesis a new method to extract and present local current density distributions from such Hall potential profiles is introduced. EXPERIMENTAL RESULTS: After systematic measurements of Hall potential profiles in the integer quantum Hall regime around filling factors v={3,2,1} the fractional quantum Hall regime with filling factors v=2/3 and v=3/5 is investigated for the first time with a scanning SET array microscope. Experimental results show a similar behavior for fractional and integer filling factors: (1) Hall potential profiles probed across the sample width evolve for varying magnetic flux densities in the same way, (2) the longitudinal resistance R_xx shows the same electrical breakdown behavior and (3) area scans in the fractional quantum Hall regime at fixed magnetic fields are spatially homogeneous. These similarities are seen for the first time and they contradict the widely used picture of a current transport along the edge. A final discussion at the end clarified, that the integer QH regime and the fractional QH regime have generally one thing in common: There is an evolution of the compressible/incompressible landscape within the 2DES which determines the current distribution in the 2DES. NEW SENSOR DEVELOPMENT: Additionally, a further milestone to the functionality of our scanning single-electron array microscope is developed: A free-standing Hall sensor tip which (i) allows another access to the current distribution inside the 2DES and (ii) makes also diamagnetic currents, which are already present in equilibrium, accessible. A calculation shows that this sensor can detect electrostatics and magnetic fields separately when a feedback loop is used