Contactless microwave studies of weak localization in epitaxial graphene

A microwave detection method was applied to study weak localization in epitaxial graphene sheets grown on SiC substrates. Both coherence and scattering length values were obtained. The intervalley scattering length was found to be about one order of magnitude greater than the intravalley and warping lengths. The influence of width and amplitude of the signal on the decoherence rate was discussed. The temperature dependence and the angular dependence of the observed signal were presented. The decoherence rate was found to depend linearly on temperature, showing the electron-electron scattering mechanism, evidencing electron spin-flip scattering.

Figure 1

(a) The signal measured in an ESR spectrometer, proportional to $d\sigma$/$dB$ (black curve), at 2 K ($B$ perpendicular to graphene sheets). The green dashed line is the fitted linear background. Red curves show the numeric integral of the spectrum (solid line) and the background (dashed line), which represent the conductivity dependence on the magnetic field. (b) Fitted WL dependence of $d\sigma$/$dB$ with Eq. (2) after subtracting the linear background. The obtained coherence and scattering lengths are $L_{\varphi }$ $=$ 238 nm and $L_i$ $=$ 40 nm, $L_{\mathrm{lr}}$ $=$ 15 nm, respectively.

Figure 2

(a) P-P amplitude of the WL line as a function of the modulation amplitude (black circles) and square root of the microwave power (red open circles). The black line is the guide for the eye of linear dependence. (b) Normalized WL signal for a few chosen modulation amplitudes (0.05, 0.1, 0.5, and 1 mT), and for chosen microwave powers (0.047 43, 0.4743, 4.743, and 15 mW). As can be clearly seen, the shape of the WL signal does not depend on the measurement conditions.

Figure 3

(a) The temperature dependence of the decoherence rate ($B_{\varphi }$) and the coherence length ($L_{\varphi }$). Scattering rates and scattering lengths are $B_i$ $=$ 12.6 mT, $B_{\mathrm{lr}}$ $=$ 1 T, $L_i$ $=$ 115 nm, $L_{\mathrm{lr}}$ $=$ 13 nm. The dashed lines are fitted to $B_{\varphi }=B_s+AT$, with $B_s$ $=$ 1.9 mT and $A$ $=$ 0.178 mT/K. (b) P-P width ($W$) and inverse amplitude as a function of the decoherence rate. The line is fitted to $W=B_0+C{B}_{\varphi }$, with $B_0$ $=$ 4.8 mT, and $C$ $=$ 4.001. Large errors for the highest temperatures result from the low amplitude of the signal and small signal-to-noise ratio.

Figure 4

The angular dependence of the P-P amplitude and the P-P width. Black curves are the cosine (dashed line) and cosine square (solid line) dependence of the P-P amplitude. The red dashed line is 1/cos(Θ) dependence of the P-P width.