Near-Field Optical Mapping of Quantum Hall Edge States

We report on the mapping of quantum-Hall edge states by quasiresonant photovoltage measurements using a near-field scanning optical microscope. We have observed fine structures near sample edges that shift inward with an increase in magnetic field in accordance with the shift of the positions of the quantum-Hall edge states. We have found a transition from the weak disorder regime where compressible-incompressble strips are visible to the strong disorder regime where fluctuations smear out incompressible strips.

Figure 1

(a) Schematics of the measurement setup and the sample structure. The arrows indicate the direction of electrons in the edge states. (b) A cross-sectional profile of photovoltage across a Hall-bar structure at $B=1.98\mathrm{T}$ ($\nu =9.55$). (c) Mappings of PV near the edge of a Hall-bar structure and (d) their spatial derivative $dV/dx$ at $B=2.02\mathrm{T}$ ($\nu =9.36$), at the excitation energy of 1.5194 eV.

Figure 2

(a) Optical excitation energy dependence of cross-sectional intensity profiles of PV at $B=1.64\mathrm{T}$ ($\nu =11.5$) at the excitation energy of (i) 1.5083, (ii) 1.5120, (iii) 1.5139, (iv) 1.5176, and (v) 1.5194 eV. (b) Schematic illustration of optical transitions between valence and conduction bands. The shaded area in the conduction band indicates broadened Landau levels. (c) Optical excitation energy dependence of peak PV at $B=1.64\mathrm{T}$. (d) Optical excitation energy dependence of half-peak width at $B=1.64\mathrm{T}$.

Figure 3

Temperature dependence of cross-sectional intensity profiles of PV at $B=2.02\mathrm{T}$ ($\nu =9.36$) at temperatures of (i) 200 mK, (ii) 400 mK, (iii) 0.8 K, (iv) 1.5 K, (v) 2.0 K, (vi) 3.0 K, and (vii) 4.0 K, at the excitation energy of 1.5194 eV.

Figure 4

(a) Magnetic field dependence of mapping of the derivative of PV near the edge of a Hall-bar structure at the excitation energy of 1.5194 eV. The position of incompressible strip $x_k$ is shown by vertical lines. (b) Magnetic field dependence of mapping of PV at the excitation energy of 1.5120 eV. The scanning range was $2.1\mu \mathrm{m}\times 82\mathrm{nm}$.