Quantum lifetime in ultrahigh quality GaAs quantum wells: Relationship to ${\mathrm{\Delta }}_{5/2}$ and impact of density fluctuations

We consider the quantum lifetime derived from low-field Shubnikov–de Haas oscillations as a metric of the quality of the two-dimensional electron gas in GaAs quantum wells that expresses large excitation gaps of the $\nu =\frac{5}{2}$ fractional quantum Hall state in the $N=1$ Landau level. In high quality samples, small density inhomogeneities dramatically impact the amplitude of Shubnikov–de Haas oscillations such that the canonical method [cf. Coleridge, Phys. Rev. B 44, 3793 (1991)] for determination of the quantum lifetime substantially underestimates ${\tau }_{\text{q}}$ unless density inhomogeneity is explicitly considered. We have developed a method that can be used to determine density inhomogeneity and extract the intrinsic ${\tau }_{\text{q}}$ by analyzing the Shubnikov–de Haas oscillations. However, even after accounting for inhomogeneity, ${\tau }_{\text{q}}$ does not correlate well with sample quality as measured by ${\mathrm{\Delta }}_{5/2}$, the excitation gap of the fractional quantum Hall state at 5/2 filling.

Figure 1

Impact of density inhomogeneity on low field transport measured at $T=0.3$ K. (a) Magnetoresistance as a function of $B$ after background subtraction for nominal 2DEG density $n_o=1.78\times {10}^{11}/{\text{cm}}^2$ and simulated trace with $\mathrm{\Delta }n/n_o\sim 0.23\%$ density inhomogeneity and ${\tau }_q=24.6$ ps (red). (b) Density spectrum obtained through a FFT of $\mathrm{\Delta }{R}_{xx}$ vs $1/B$. (c) Dingle plots from data (blue squares) and simulated trace (red line) shown in (a). The black line is a single-parameter least-squares fit of the data between 55 and 95 mT with the intercept fixed to 4; it clearly shows poor overlap with the data. (d) Two-dimensional plot of the fit error for various combinations of $\mathrm{\Delta }n$ and ${\tau }_{\text{q}}$ for data shown in (a). The fit error is minimized at $\mathrm{\Delta }n=4.15\times {10}^8{\text{cm}}^{-2}$ and ${\tau }_{\text{q}}=24.6$ ps.

Figure 2

Dingle plots from data taken at $T=0.3$ K and simulations. (a) Back gate 2DEG with nominal 2DEG density $n_o=1.75\times {10}^{11}/{\text{cm}}^2$. (b) A 91-nm-deep, single heterojunction 2DEG, $n_o=1.30\times {10}^{11}/{\text{cm}}^2$.

Figure 3

Low-field transport data of back-gated 2DEG with nominal 2DEG density $n_o=1.60\times {10}^{11}/{\text{cm}}^2$ taken at $T=10$ mK. (a) Magnetoresistance as a function of $B$ after background subtraction. (b) Dingle plots of experimental data and simulation.

Figure 4

Experimentally measured ${\tau }_{\text{q}}$ (red squares) and numerically calculated ${\tau }_{\text{q}}$ (solid lines) as a function of 2DEG density $n$. The black line shows the scattering time corresponding to scattering by donors in the doping well, while the blue line shows the scattering time corresponding to scattering by ionized impurities on the back gate. The red line shows the combined ${\tau }_{\text{q}}$ including both sources of scattering, and it indicates reasonable agreement with the experimental data.

Figure 5

Characteristic properties of the back-gated sample as a function of the electron density $n$. (a) Quantum lifetime ${\tau }_{\text{q}}$ extracted from Shubnikov–de Hass oscillations measured at $T=0.3$ K, both in dark (black) and after illumination (red). (b) The Landau level broadening based on the low-field quantum lifetime time with illumination, ${\mathrm{\Gamma }}_{\text{SdH}}=\hslash /2{\tau }_q$. (c) Gap energy for the $\nu =\frac{5}{2}$ FQHS with illumination.