Weak localization in high-quality two-dimensional systems

We investigate five different methods of modeling the correction to the magnetoconductivity due to the weak localization effect in two-dimensional (2D) systems. The phase breaking rate is extracted using each method by fitting experimental magnetoconductivity data of high-quality 2D GaAs hole systems over the range of carrier densities and temperatures that weak localization is observed. We find that despite corrections to the magnetoconductivity differing by more than $100\%$ between different methods valid beyond the diffusion approximation, the phase breaking rate extracted is approximately the same. We also find that if diffusive transport is incorrectly assumed in high-quality systems, then values of the phase breaking rate approximately 2.5 times too high are extracted. We demonstrate the regime in which the diffusive transport approximation holds and explain previous discrepancies in the literature where phase breaking rates much higher than expected from Fermi-liquid theory have been obtained. We find good agreement of the phase breaking rate with Fermi-liquid theory until $k_{F}l$ begins to approach 1.

Figure 1

(a), (b), and (c) show the conductivity correction due to the weak localization, $\Delta {\sigma }_{\text{wl}}$, generated by the five methods with for $z=0.01$, 0.1, and 0.5, respectively. $▵$, Hikami; $엯$, Kawabata; $◻$, Wittmann; $\times$, Zduniak; $∎$, Dmitriev. Note the varying $y$-axis scale on each plot. The arrow on the right shows the effect of increasing temperature or, equivalently, increasing the sample quality at a fixed temperature.

Figure 2

In (a), (b), and (c) we have replotted the conductivity correction due to the weak localization, $\Delta {\sigma }_{\text{wl}}$, on a linear scale. $\Delta {\sigma }_{\text{wl}}$ is generated by each method with $z=0.01$, 0.05, and 0.1, respectively. $▵$, Hikami; $엯$, Kawabata; $◻$, Wittmann; $\times$, Zduniak; $∎$, Dmitriev. Note the different $y$-axis scale on each plot. The arrow on the right shows the effect of increasing temperature or, equivalently, increasing the sample quality at a fixed temperature.

Figure 3

(a) Typical low field ${\rho }_{xx}$ and ${\rho }_{xy}$ data, (b) the derived ${\sigma }_{xx}$ and the sum of the Drude and hole-hole interaction terms (dashed line), and (c) the difference of the ${\sigma }_{xx}$ data and the fitted Drude and hole-hole terms. These data come from sample $\text{A1433}∕37$ with $p=4.50\times {10}^{10}{\text{cm}}^{-2}$.

Figure 4

(a) Comparison of fitted points (symbols) and experimental data for each of the five methods using the two-stage procedure fit. The experimental data (solid line) come from $\text{A1433}∕37$ with $p=4.50\times {10}^{10}{\text{cm}}^{-2}$; symbols give best fit according to method; $▵$: Hikami; $엯$, Kawabata; $◻$, Wittmann; $\times$, Zduniak; $∎$, Dmitriev. Each method is offset with associated experimental data by $0.2e^2∕h$ for clarity. (b) Simultaneous three-parameter fit results, again from sample $\text{A1433}∕37$ with $p=4.50\times {10}^{10}{\text{cm}}^{-2}$, each method is offset by $0.2e^2∕h$ for clarity. Note that the experimental data appears different in each fit in (b) because the scaling of the reduced magnetic field depends on $\tau$.

Figure 5

(a) Activation plots for three different sample densities for $\text{A1433}∕37$. The lines are least-squares fits to the five highest-temperature points. (b) $T_{\text{hole}}$ vs $T_{\text{lattice}}$ plots for each sample at the extremes of the density range studied.

Figure 6

(a)–(c) Magnetoconductivity data for sample $A1433∕37$ at the density and temperatures indicated, and (d)–(f) ${\tau }_{\varphi }^{-1}$ computed by each method of generating $▵{\sigma }_{\text{wl}}$. $\Delta$, Hikami; $엯$, Kawabata; $◻$, Wittmann; $\times$, Zuduniak; and $∎$, Dmitriev. The solid and dashed lines show predictions of Fermi-liquid theory according to Eq. (16) with $F_0^{\sigma }=-0.3$ and 0, respectively.

Figure 7

(a)–(c) Magnetoconductivity data for sample $T402∕5$ at the density and temperatures indicated and (d)–(f) ${\tau }_{\varphi }^{-1}$ computed each method of generating $▵{\sigma }_{\text{wl}}$. $\Delta$, Hikami; $엯$, Kawabata; $◻$, Wittmann; $\times$, Zuduniak; and $∎$, Dmitriev. The solid and dashed lines show predictions of Fermi-liquid theory according to Eq. (16) with $F_0^{\sigma }=-0.3$ and 0, respectively. For (a) this produces a negative prediction of ${\tau }_{\varphi }$ in the $F_0^{\varphi }=-0.3$ case as discussed in the text.

Figure 8

(a) $P_n$ generated by the techniques described in text for $z=0.01$ and $z=0.5$ with $b=0.01$, (b) $P_n$ generated for $z=0.01$ and $z=0.5$ at $b=0.01$ or $10$ for the two groups of curves as indicated.

Figure 9

$\Delta {\sigma }_{\text{wl}}$ generated with a truncated series expansion for (a) Zduniak and (b) Dmitriev methods. Each expansion is truncated at $n=10$, ${10}^2$, ${10}^3$, ${10}^4$, or ${10}^5$ terms as indicated.

Figure 10

Diagram of computation method used to calculate $P_n$. Beyond $n={10}^5$ $P_n$ is calculated using Eq. (B2) for all $b$. The small, medium, and large boundaries $n_S$, $n_M$, and $n_L$ are marked at $b=0.01$.