Dynamical Scaling of the Quantum Hall Plateau Transition

Using different experimental techniques, we examine the dynamical scaling of the quantum Hall plateau transition in a frequency range $f=0.1–55\mathrm{G}\mathrm{H}\mathrm{z}$. We present a scheme that allows for a simultaneous scaling analysis of these experiments and all other data in literature. We observe a universal scaling function with an exponent $\kappa =0.5\pm 0.1$, yielding a dynamical exponent $z=0.9\pm 0.2$.

Figure 1

(a) Conductivity of sample 1 measured in the coaxial setup. Here shown are the conductivity peaks at the $\nu =4\to 3$ and $\nu =3\to 2$ plateau transition. (b) Conductivity of sample 2 at the $\nu =2\to 1$ transition, measured in the waveguide reflection setup for very high frequencies and in a classical Hall setup for dc. (c) Frequency dependence of the width of the QH plateau transitions, measured for $\nu >2$ as full width and for $\nu =2\to 1$ as half width at half maximum [24, 26]. For the waveguide experiment, different symbols denote different samples from the same wafer or different cooldown cycles with slightly different carrier densities.

Figure 2

Comparison of the transition width vs frequency for all experiments [10, 19, 20, 21, 22, 23, 24, 25]. Multiple symbols for the same experiment denote different samples or cooldowns (waveguide), transitions (coaxial, Lewis), or temperatures (Engel, Lewis, Shahar) as presented in Table .

Figure 3

(a) Normalized plateau transition width $y=\Delta B(f,T)/\Delta B(f\approx 0,T)$ vs $x=hf/k_{B}T$ for all data presented in Table  and Fig. 2. The solid line results from a fit of $F(x)=[1+(\alpha x{)}^2{]}^{\kappa /2}$ [Eq. (5)] to all data except those for $f>20\mathrm{G}\mathrm{H}\mathrm{z}$ and the Balaban experiment ($\alpha \approx 2$, $\kappa =0.5\pm 0.1$). The dashed line depicts a reduced $\kappa =0.4$. (b) Rescaling of $y=\Delta \nu (f,T)/\Delta \nu (f\approx 0,T)$ by the scaling function $F(x)$ allows one to judge the quality of scaling.