Density dependence of the $\nu =\frac{5}{2}$ energy gap: Experiment and theory

A density-tunable GaAs-AlGaAs heterostructure is used to study the density dependence of the filling factor $\nu =\frac{5}{2}$ and other fractional and reentrant integer quantum-Hall states in the second Landau level. The activation energy at $\nu =\frac{5}{2}$ can be determined for densities between 1.3 and $2.7\times {10}^{11}/{\text{cm}}^2$ and reaches up to 310 mK. The 5/2 energy gap is calculated numerically, including finite width and Landau-level-mixing corrections, both as a function of electron density. The discrepancy between theory and experiment increases moderately with density and reaches about 1.5 K at the highest density. We argue that the activation energy is strongly influenced by disorder from ionized donors and attribute this to the surprisingly large size of the 5/2 quasiparticles. We find that the quasiparticles have a diameter of at least 12 times the magnetic length or 150 nm at a magnetic field of 4 T. Implications for heterostructure design are discussed.

Figure 1

Characteristic properties of our sample as a function of the electron density $n$. (a) Electron mobility. (b) Quantum lifetime extracted from Shubnikov-de-Haas oscillations.

Figure 2

(Color) Evolution of the longitudinal and Hall resistances with density and temperature, shown between $\nu =2$ and 3. (a) Density dependence of $R_{xx}$ and $R_{xy}$ (densities in units of ${10}^{11}/{\text{cm}}^2$). All traces are taken at 15 mK. Filling 5/2 is marked by arrows. Other filling factors can be identified by comparing with panel (c). $R_{xx}$ traces are offset for clarity. (b) Temperature dependence of $R_{xy}$ and (c) $R_{xx}$ for $n=2.5\times {10}^{11}/{\text{cm}}^2$.

Figure 3

(Color) Arrhenius plots for (a) the 5/2 state and (b) the reentrant integer quantum-Hall state at filling factor 2.57. Different symbols/colors correspond to different densities (examples given in units of ${10}^{11}/{\text{cm}}^2$). Activation energies are calculated from linear fits in the activated regime. (c) Density dependence of the activation energies for the fractional quantum-Hall states listed in the legend and for the $\nu =2.57$ reentrant integer quantum-Hall state.

Figure 4

(Color) Comparison of our measured 5/2 gaps to literature values (year of publication in parentheses). Mobilities $\mu$ of the used samples are given in units of ${10}^6{\text{cm}}^2/\text{Vs}$.

Figure 5

(Color) (a) Comparison of our measured 5/2 gaps to theory. See text for details. (b) Reduction in dimensionless gap parameter due to the normalized finite width $w/l_B$ of the electronic wave function, obtained from exact diagonalization of up to 12 and 20 electrons. (c) Normalized finite width as a function of electron density. Left dashed line and inset: actual finite width. Right dashed line and inset: effective width due to Landau-level mixing. Solid red line: combined effect of FW and LL mixing.

Figure 6

To determine the size of quasiholes for $\nu =\frac{5}{2}$ and 1/3 we put two quasiholes on a sphere occupied by $N=20$ and 12 electrons, respectively (left inset). The missing charge associated with the quasiholes is obtained from integration along the polar angle $\theta$. For 1/3 it levels at $2l_B$ from the poles (right inset). For 5/2 no plateau is observed even for 20 electrons. The quasihole diameter is thus $\gtrsim 12l_B$, which is about 150 nm at $B=4\text{T}$.