Phase coherent transport in graphene nanoribbons and graphene nanoribbon arrays

We have experimentally investigated quantum interference corrections to the conductivity of graphene nanoribbons at temperatures down to 20 mK studying both weak localization (WL) and universal conductance fluctuations (UCFs). Since in individual nanoribbons at milli-Kelvin temperatures the UCFs strongly mask the weak localization feature we employ both gate averaging and ensemble averaging to suppress the UCFs. This allows us to extract the phase coherence length from both WL and UCF at all temperatures. Above 1 K the phase coherence length is suppressed due to Nyquist scattering, whereas at low temperatures we observe a saturation of the phase coherence length at a few hundred nanometers, which exceeds the ribbon width, but stays below values typically found in bulk graphene. To better describe the experiments at elevated temperatures, we extend the formula for one-dimensional (1D) weak localization in graphene, which was derived in the limit of strong intervalley scattering, to include all elastic scattering rates.

Figure 1

(a) Scanning electron microscope image of a typical sample from this work. The length of the GNRs is 1 $\mu$m, the width 70 nm. Two palladium contacts are visible. (b) Array of GNRs with a GNR length of 1 $\mu$m in between two palladium contacts. (c) Sample C: Two-terminal resistance as a function of $V_{bg}$ for different temperatures at zero magnetic field. (d) Zoom-in of the GNR array, every GNR has a width of 40 nm and a spacing of 30 nm to the next, the zoom-in area of panel (b) is marked in white.

Figure 2

The conductance $G$ as a function of magnetic field $B$ of an individual GNR shows quantum interference phenomena for temperatures from $T=1.7$ to 48 K (a) and $T=50$ to 900 mK (b).

Figure 3

Sample A: Conductance $G$ as a function of magnetic field $B$ at (a) $T=1.7$ K and (b) $T=48$ K. The weak localization feature is fitted by using Eqs. (1) (orange) and (2) (blue dashed line). Both fit formulas reproduce the data well at low temperatures (a), but at higher temperature Eq. (2) is more appropriate.

Figure 4

Sample B, gate averaging: The magnetic field dependence of the conductance was measured at different gate voltages and the arithmetic mean was calculated. The average conductance $G$ of the 40 nm GNR at $T=20$ mK clearly shows the weak localization feature. Fitting the conductance dip by Eqs. (1) and (2) yields a phase coherence length of 100 nm.

Figure 5

Magnetotransport data of a GNR array. In comparison to the data of individual ribbons Fig. 2, the measurements of the arrays clearly show a suppression of the universal conductance fluctuations for all temperatures (a) and (b), whereas the weak localization feature is not affected. Blue dashed lines are best-fit curves to Eq. (2).

Figure 6

Phase coherence lengths determined via different methods for (a) sample A, (b) sample C, (c) sample B, and (d) sample D. The phase coherence length $L_{\varphi }$ was determined by different methods like weak localization (WL1 and WL2), the amplitude of the UCFs (RMS), and the autocorrelation function (AUT). The data points obtained by fitting the WL feature to the simple formula [Eq. (1)] are plotted as black, open squares (WL1) and by fitting the full formula [Eq. (2)] as black, filled squares (WL2), respectively. The orange, dashed lines represent the thermal length $L_T$.