Direct extraction of electron parameters from magnetoconductance analysis in mesoscopic ring array structures

We report an approach for examining electron properties using information about the shape and size of a nanostructure as a measurement reference. This approach quantifies the spin precession angles per unit length directly by considering the time-reversal interferences on chaotic return trajectories within mesoscopic ring arrays (MRAs). Experimentally, we fabricated MRAs using nanolithography in InGaAs quantum wells which had a gate-controllable spin-orbit interaction (SOI). As a result, we observed an Onsager symmetry related to relativistic magnetic fields, which provided us with indispensable information for the semiclassical billiard ball simulation. Our simulations, developed based on the real-space formalism of the weak localization/antilocalization effect including the degree of freedom for electronic spin, reproduced the experimental magnetoconductivity (MC) curves with high fidelity. The values of five distinct electron parameters (Fermi wavelength, spin precession angles per unit length for two different SOIs, impurity scattering length, and phase coherence length) were thereby extracted from a single MC curve. The methodology developed here is applicable to wide ranges of nanomaterials and devices, providing a diagnostic tool for exotic properties of two-dimensional electron systems.

Figure 1

MRA sample fabricated by nanolithography. (a) An example of chaotic billiard ball trajectory within the MRA that returns to the initial point indicated by red filled circle (CG). (b) Scanning electron micrograph of the fabricated MRA. Colors were added to the original image for clarity. The reddish-colored region contributes to the electron transport, whereas the bluish-colored regions are isolated areas that do not contribute to the transport. The dark lines and curves are the ditches created by reactive ion etching, where squares drawn inside the circles are just to adjust the proximity effect in the electron beam lithography. (c) Illustration showing the arrangement of the MRA in a Hall bar device. (d) Cross-sectional illustration of the MRA device along the white line segment indicated in (b) and the MOCVD-grown epitaxial layer structure. The doping densities ($n_1$, $n_2$) are ($1.5\times {10}^{24}{\mathrm{m}}^{-3}$, $2.5\times {10}^{24}{\mathrm{m}}^{-3}$) and ($2.0\times {10}^{24}{\mathrm{m}}^{-3}$, $2.0\times {10}^{24}{\mathrm{m}}^{-3}$) for the first (Sec. 2b) and second (Sec. 4d) samples in the main text, whose sample names are KH3-1 and KH1-3, respectively.

Figure 2

Gate dependence of the MC curves of MRA. (a)–(c) (Colored curves) Experimental MC curves ($\mathrm{\Delta }\sigma$) at selected gate voltages ($V_{\mathrm{g}}$) corresponding to the horizontal lines $A$–$C$, $B^{\prime }$, and $C^{\prime }$ in (d). (Black dashed curves) $\mathrm{\Delta }\sigma$ generated by the “chaos only” model [see (e)] maximizing the $r^2$ values with the corresponding experimental MC curves (see Sec. 3d). (d)–(f) Image plots of $\mathrm{\Delta }\sigma$ from (d) the experiment at a dilution refrigeration temperature, (e) the “chaos only” model, and (f) the “chaos+impurity” model. $k_{\alpha }$ values in (a)–(c), (e), and (f) are related to $V_{\mathrm{g}}$ by $k_{\alpha }=-0.9614\times (V_{\mathrm{g}}-0.8547)\left(\mathrm{rad}/\mu \mathrm{m}\right)$.

Figure 3

$r^2$ values (see Sec. 3d) calculated between the measured and simulated MC curves at $V_{\mathrm{g}}=0.85$ V, where $k_{\alpha }=0\mathrm{rad}/\mu \mathrm{m}$, as a function of $k_{\mathrm{c}}$ and $L$. The magnetic field ranges used in the calculation of $r^2$ are $\left|B\right|\le 0.32$ mT and $\left|B\right|\le 1.42$ mT in the “chaos only” and “chaos+impurity” models, respectively. (a) Result in the chaos only model (${\ell }_{\mathrm{ex}}=10\mu \mathrm{m}$). (b) Result in the chaos+impurity model (${\ell }_{\mathrm{ex}}=1\mu \mathrm{m}$). Only the regions corresponding to $r^2\ge 0.95$ and $r^2\ge 0.98$ with positive correlation are color scaled in (a) and (b), respectively.

Figure 4

Parameter extraction from the “chaos+impurity” model. (a) (Colored curves) Replot of the experimental $\mathrm{\Delta }\sigma \mathrm{s}$ shown in Figs. 2. Each curve was shifted vertically for clarity. (Black dotted curves) $\mathrm{\Delta }\sigma \mathrm{s}$ generated by the chaos+impurity model maximizing the $r^2$ values with the corresponding experimental MC curves. (b)–(d) Various parameter values extracted from the chaos+impurity model as a function the gate voltage $V_{\mathrm{g}}$. These parameters are (b) the Fermi wavelength ${\lambda }_{\mathrm{F}}$, (c) electron phase coherent length $L_{\varphi }$, and (d) extrinsic scattering length ${\ell }_{\mathrm{ex}}$. The dotted curve in (b) was obtained from Eq. (4) for the pristine bulk 2DES using the $k_{\alpha }$ vs $V_{\mathrm{g}}$ relation extracted in Sec. 4a.

Figure 5

MC curve of MRA in a wide magnetic field range. (a), (b), (d), (e) Red curves represent the same experimental $\mathrm{\Delta }\sigma$ data of MRA taken at $T_{\mathrm{el}}\sim 500\mathrm{mK}$, where $k_{\alpha }$ is tuned to be zero by $V_{\mathrm{g}}$. (b) and (e) are the magnifications of (a) and (d), respectively, for a narrower magnetic field range. Black dotted curves in (a) and (b) are the $r^2$ maximized $\mathrm{\Delta }\sigma$ in the “chaos+impurity” model using only a partial set of the closed-loop trajectories that fall in the areal range $S/S_0\le 0.5$ [the hatched region in (c)]. Black dotted curves in (d) and (e) are the $r^2$ maximized $\mathrm{\Delta }\sigma$ in the chaos+impurity model using all the closed-loop trajectories [the hatched region in (f)]. (c), (f) Black solid and dashed curves are the areal distributions of the nominal and weighted return counts, respectively [the latter with $\exp (-L_i/L_{\varphi })$], using $\mathrm{\Delta }S=0.05{\mu \mathrm{m}}^2$ for the histogram bin. Parameter values used in this simulation are $k_{\mathrm{c}}=0.121\mathrm{rad}/\mu \mathrm{m}$, ${\ell }_{\mathrm{ex}}=0.94\mu \mathrm{m}$, and $L_{\varphi }=75\mu \mathrm{m}$.

Figure 6

Temperature dependence of the MC curve in MRA. (a) MC curves at several different $T$'s. Experimental results are indicated by colored solid curves and the $r^2$-maximized simulated results from the “chaos+impurity” model are indicated by black dotted curves. (b) Temperature dependence of the phase coherence times extracted from the chaos+impurity model (colored solid circles). ${\tau }_{\mathrm{ee}}$, ${\tau }_{\mathrm{AAK}}$, and ${\tau }_{\varphi }^{\mathrm{ring}}$ [Eqs. (5), (7), and (8)] are plotted by the blue dashed, green dotted, and green dashed lines, respectively.