Direct Observation of Valley Hybridization and Universal Symmetry of Graphene with Mesoscopic Conductance Fluctuations

In graphene, the valleys represent spinlike quantities and can act as a physical resource in valley-based electronics to produce novel quantum computation schemes. Here we demonstrate a direct route to tune and read the valley quantum states of disordered graphene by measuring the mesoscopic conductance fluctuations. We show that the conductance fluctuations in graphene at low temperatures are reduced by a factor of 4 when valley triplet states are gapped in the presence of short-range potential scatterers at high carrier densities. We also show that this implies a gate tunable universal symmetry class that outlines a fundamental feature arising from graphene’s unique crystal structure.

Figure 1

(a) Resistance ($R$) vs gate voltage ($V_{\mathrm{BG}}$) characteristics for a single-layer device (dev. I) at $T=10\mathrm{mK}$. Inset shows the schematic along with the false color scanning electron microscope image of a typical graphene transistor. (b) Evolution of magnetoconductance with increasing electron density measured at $T=4.5\mathrm{K}$. Solid lines are fit to the weak localization theory [24, 25]. (c) Conductance fluctuations as a function of gate voltage for dev. I, shown for various temperatures. Curves at different temperatures are shifted for clarity. (d) Variance in conductance, calculated from the conductance fluctuations in gate voltage range $-50$ to $-46\mathrm{V}$ for dev. I, plotted as a function of temperature.

Figure 2

Magnetic field dependence of the magnitude of conductance fluctuations for dev. III at $T=10\mathrm{mK}$ at various gate voltages: (a) 42.5 V, (b) 12.5 V, and (c) $-34\mathrm{V}$. For comparison, we have plotted the ratio $N(B)/N(B=0)$, with $N(B)=⟨\delta G^2⟩/⟨G{⟩}^2$ at the magnetic field $B$. Inset shows the $R\mathrm{\text{−}}{V}_{\mathrm{BG}}$ characteristics of dev. III at $T=10\mathrm{mK}$ with arrows indicating the three measured regions.

Figure 3

(a) Variance in conductance ($⟨\delta G^2⟩$) vs gate voltage ($V_{\mathrm{BG}}$) at 10 mK for dev. I. (b) Gate voltage dependence of $L_{\varphi }$ and $L_i$ at $T=10\mathrm{mK}$, extracted from the weak localization fits, are shown for dev. I. (c) The variance in conductance in a phase-coherent box of area $L_{\varphi }^2$, $⟨\delta G_{\varphi }^2⟩$, extracted from (a) and (b), is plotted with gate voltage for dev. I. The factor of 4 is indicated by the dashed lines. (d) Conductance ($G$) vs gate voltage ($V_{\mathrm{BG}}$) for the device at $T=10\mathrm{mK}$. The solid line indicates the linear region in both electron and hole sides. The vertical dotted lines in (c) and (d) indicate the densities where the short-range scattering dominate. The shaded region indicates the inhomogeneous region near the Dirac point.

Figure 4

[(a) and (b)] Conductance ($G$) vs gate voltage ($V_{\mathrm{BG}}$) at 10 mK for dev. II and dev. III, respectively. The solid line indicates the linear region in both electron and hole sides. [(c) and (d)] The variance from conductance fluctuations in a phase-coherent box of area $L_{\varphi }^2$ is plotted with gate voltage for devs. II and III; see details in Table . The factor of 4 is indicated by the dashed lines. The vertical dotted lines indicate the densities beyond which the short-range scattering dominates. (e) Schematic describing the crossover from orthogonal to symplectic universality class depending on the presence of short-range scattering, which breaks the effective valley-isospin rotational symmetry. The red and purple arrows in the center represent valley isospin states that hybridized into singlet and triplets in different regimes of intervalley scattering. $⟨\delta G_0^2⟩$ is the variance in conductance in a single phase-coherent box for an ordinary disordered metal.

Figure 5

[(a–d)] Gate voltage dependence of the ratio $⟨\delta G_{\varphi }^2⟩/⟨\delta G_0^2⟩$ for dev. III, plotted for various temperatures at $B=100\mathrm{mT}$, where $⟨\delta G_0^2⟩$ is the variance in the conductance for a single phase-coherent box at $V_{\mathrm{BG}}=-68\mathrm{V}$.