Scaling flow diagram in the fractional quantum Hall regime of $\mathrm{GaAs}∕{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructures

The temperature driven flow lines of the Hall and dissipative magnetoconductance data $({\sigma }_{xy},{\sigma }_{xx})$ are studied in the fractional quantum Hall regime for a two-dimensional (2D) electron system in $\mathrm{GaAs}∕{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructures. The flow lines are rather well-described by a recent unified scaling theory developed for both the integer and the fractional quantum Hall effect in a totally spin-polarized 2D electron system which predicts that one $({\sigma }_{xy},{\sigma }_{xx})$ point determines a complete flow line.

Figure 1

The scaling flow diagram showing the coupled evolution of the diagonal $\left({\sigma }_{xx}\right)$ and Hall $\left({\sigma }_{xy}\right)$ conductivity components with increasing coherence length for a totally spin-polarized 2D electron system in the range $0<{\sigma }_{xy}<1$. Solid lines represent the semicircle separatrixes and dotted lines give examples of some other flow lines.

Figure 2

The diagonal $\left({\sigma }_{xx}\right)$ and Hall $\left({\sigma }_{xy}\right)$ conductivities as a function of the gate voltage for sample two at a 6.0 T magnetic field for different temperatures below 0.3 K. The arrows indicate the gate voltage positions of the ${\sigma }_{xx}$ minima for fractional QHE states $2∕3$ and $1∕3$.

Figure 3

The flow diagram of the temperature-dependent $[{\sigma }_{xy}\left(T\right),{\sigma }_{xx}\left(T\right)]$ data points (a) for sample 1 from 0.37 down to 0.05 K (for $1∕3<{\sigma }_{xy}<2∕3$) and down to 0.11 K (for ${\sigma }_{xy}<1∕3$) at different magnetic fields in the range 8.5–21 T, and (b) for sample two from 0.3 down to 0.035 K at different gate voltages for $B=2.5$ (diamonds), 2.8 (squares, a few data), and 6 T (circles). Different fillings of the symbols are used for different magnetic fields (data of sample 1) and for different gate voltages (data of sample 2). Solid lines represent the theoretical vertical and semicircle separatrixes. Dotted lines show flow lines close to experimental data.

Figure 4

A part of the flow diagram of Fig. 3b in the vicinity of the theoretical critical point (0.3, 0.1). Experimental points (diamonds for $B=2.5$, squares for 2.8, and circles for 6 T) are connected by solid lines and the direction of their shift with lowering temperature are shown by arrows. The theoretical semicircle separatrixes are presented by dashed and dotted lines. The dash-dotted line is obtained by a slight deformation of the dotted theoretical line.