Quantum interference in a macroscopic van der Waals conductor

Quantum corrections to charge transport can give rise to an oscillatory magnetoconductance, typically observed in mesoscopic samples with a length shorter than or comparable to the phase coherence length. Here, we report the observation of magnetoconductance oscillations periodic in magnetic field with an amplitude of the order of $e^2/h$ in macroscopic samples of highly oriented pyrolytic graphite (HOPG). The observed effect emerges when all carriers are confined to their lowest Landau levels. We argue that this quantum interference phenomenon can be explained by invoking moiré superlattices with a discrete distribution in periodicity. According to our results, when the magnetic length ${\ell }_B$, the Fermi wavelength ${\lambda }_F$, and the length scale of fluctuations in local chemical potential are comparable in a layered conductor, quantum corrections can be detected over centimetric length scales.

Figure 1

(a) Longitudinal $c$-axis magnetoresistivity ${\rho }_{zz}$ for a HOPG sample (sample E) as a function of magnetic field $B$ up to 37 T. The inset shows the low-field quantum oscillations as a function of $B^{-1}$. Above 7.5 T, both holes and electrons are confined in the lowest Landau levels ($n=0$). (b) Longitudinal resistance $R_{zz}$ in the magnetic field range of 10–30 T. Above 10 T, unexpected additional resistance oscillations are observed, which become more visible after a smooth background subtraction as shown in (c). While the low-field quantum oscillations are only weakly temperature dependent, these oscillations quickly fade out with temperature.

Figure 2

(a) Oscillating part $\delta G$ of the conductance in units of $e^2/h$ as a function of magnetic field measured on different HOPG samples at $T=0.4$ K in both transverse (sample A) and longitudinal (samples B, D, and E) configurations. The curves are shifted for clarity. (b) Fast Fourier transform (FFT) spectra of $\delta G$ for the same samples. Despite differences in the oscillation patterns, all spectra display the same characteristic frequencies that differ only in their FFT amplitude from one sample to another.

Figure 3

Size dependence of the magnetoresistance oscillations $\delta R_{xx}$: (a) Comparison of the resistance oscillations $\delta R_{xx}$ measured on a sample with a voltage lead contact length ranging from 1.4 to 12.3 mm at $T=150$ mK. (b) Oscillation amplitude $\delta R_{xx}$ as a function of the resistance $R_{xx}$ at two different magnetic fields ($B=9.8$ and 13.6 T) deduced from (a).

Figure 4

(a) Temperature dependence of the magnetoconductance oscillations $\delta G_{zz}$ as a function of magnetic field for temperatures from 0.35 to 4.3 K. (b) Temperature dependence of $\delta G_{zz}$ at $B=26$, 29, and 34 T. In the field-induced state ($B=34$ T) we observe that the amplitude of the peak does not change considerably, while it strongly increases with decreasing temperature below the field-induced state ($B=26$ and 29 T).