Imaging the Breakdown of Ohmic Transport in Graphene

Ohm’s law describes the proportionality of the current density and electric field. In solid-state conductors, Ohm’s law emerges due to electron scattering processes that relax the electrical current. Here, we use nitrogen-vacancy center magnetometry to directly image the local breakdown of Ohm’s law in a narrow constriction fabricated in a high mobility graphene monolayer. Ohmic flow is visible at room temperature as current concentration on the constriction edges, with flow profiles entirely determined by sample geometry. However, as the temperature is lowered below 200 K, the current concentrates near the constriction center. The change in the flow pattern is consistent with a crossover from diffusive to viscous electron transport dominated by electron-electron scattering processes that do not relax current.

Figure 1

Room temperature current imaging at the charge neutrality point. (a) Experimental setup. A diamond probe containing a 30 nm-deep, single NV center is scanned over a $h$-BN-encapsulated monolayer graphene device while current flows through an etched constriction. The stray magnetic field produced by the flowing current is measured via shifts in the NV magnetic resonance spectrum, and the measured field is used to reconstruct the underlying current distribution. (b) Optical image of the graphene device. (c) Scanning NV magnetometry signal inside the area indicated by the white dashed line in (b) at room temperature and at the charge neutrality point. The image boundaries, where the field varies slowly, are measured with sparser sampling, and linear interpolation is used to fill in these regions (see the Supplemental Material [18]). The dotted white line shows the edges of the device as indicated by the NV fluorescence rate. (d) Reconstructed current density magnitude $|j|$ at 298 K and carrier density near the CNP ($n<0.06\times {10}^{12}{\mathrm{cm}}^{-2}$). The current reconstruction is performed over the entire area enclosed by the white dashed box in (b), and the image areas in (c) and (d) are cropped relative to the full image by $1.8\mu \mathrm{m}$ in each direction. (e) A linecut of $|j|$ taken along the dashed white line in (d). The light blue band corresponds to the uncertainty in the reconstructed current density. The dashed red line plots the expected current profile, obtained using a parameter-free, purely Ohmic model, and shows good agreement with the data. The dashed orange line plots the expected constriction current profile in the small ${\ell }_{ee}$ limit with a Gurzhi length ${\ell }_g=208\mathrm{nm}$ given by the fit to the measured current density profile in a long, $2.7\mu \mathrm{m}$-wide channel (see the Supplemental Material [18]). This model takes into account broadening of the features in $|j|$ due to the finite distance between the NV and the graphene. (f) The simulated Ohmic current in the device without the NV filter function applied. (g) A linecut taken along the dashed white line in (f) displays a pronounced double-peak feature indicative of Ohmic transport.

Figure 2

Boltzmann transport model. (a) Four examples of current flow profiles for a slit of finite thickness and a width $w=3\mu \mathrm{m}$ and thickness $t=0.4\mu \mathrm{m}$. The free parameters in the model are ${\ell }_{ee}$ and ${\ell }_{\mathrm{mr}}$. The current density is normalized by the average current density through the slit. Line profiles of the current across the center of the slits are shown underneath. Region I corresponds to the Ohmic regime where ${\ell }_{\mathrm{mr}}={\ell }_{ee}\approx 0$. Region II is in the deep ballistic regime with ${\ell }_{\mathrm{mr}}={\ell }_{ee}=10\mu \mathrm{m}$. The hydrodynamic regime is represented by Region III (${\ell }_{ee}=1\mu \mathrm{m}$, ${\ell }_{\mathrm{mr}}=10\mu \mathrm{m}$) where the current shape assumes a more elliptical profile. A final region, IV (${\ell }_{ee}\approx 0$, ${\ell }_{\mathrm{mr}}=10\mu \mathrm{m}$), corresponds to when electron-electron scattering becomes so dominant that it begins to reduce the effective current relaxation length and results in viscous bunching near the slit edges. (b) The curvature of the current, $d^2|j|/d{x}^2$, at the center of the slit, normalized by the average current density through the slit. The changes in profiles in the regimes of Regions II and III are subtle to resolve when uncertainty in the slit width is taken into account.

Figure 3

Local breakdown of Ohmic transport. (a) Flow profiles across the center of the slit at room temperature (top row) and at $T=128\mathrm{K}$ (bottom row). The red dashed line at CNP and $T=298\mathrm{K}$ is the current profile expected from a purely Ohmic model as in Fig. 1. The dashed red lines in the plots at $5\times {10}^{11}{\mathrm{cm}}^{-2}$ and $-1\times {10}^{12}{\mathrm{cm}}^{-2}$ are the best-fit flow profiles, with ${\ell }_{\mathrm{mr}}$ and ${\ell }_{ee}$ given by the corresponding red circles in (b). At room temperature, Ohmic double-peaked current profiles are always visible, indicating the importance of momentum-relaxing scattering processes. ${\ell }_{\mathrm{mr}}$ can be tightly constrained at room temperature, and the current flow is insensitive to ${\ell }_{ee}$ in the regime ${\ell }_{ee}>500\mathrm{nm}$. For lower temperatures, at finite density we observe the breakdown of the double-peaked profile and observe a rounder, single-peaked profile—the qualitative signature of the absence of strong momentum-relaxing scattering. (b) Maps of the RMS error per pixel between QBE simulations and the reconstructed current densities, normalized by the average pixel uncertainty in the current reconstruction (see the Supplemental Material [18]). At room temperature, the fits are consistent with diffusive transport and inconsistent with a hydrodynamic or ballistic flow. For lower temperatures, our data are inconsistent with both the deeply hydrodynamic regime and inconsistent with the diffusive regime—an indication that the device sits near the boundary between the weakly hydrodynamic and ballistic regimes. The white dashed lines correspond to fits where the RMS error per pixel is equal to the uncertainty in the current reconstruction. We take the areas bounded from above by these contours as consistent with the experimental data. The red dot corresponds to the absolute best fit in these data.