Ballistic–Hydrodynamic–Ohmic Crossover in Graphene

• Ballistic–hydrodynamic–Ohmic crossover is defined by the interplay of momentum-conserving (ℓₑₑ) and momentum-relaxing (ℓₘᵣ) scattering lengths relative to the device width (W), delineating distinct transport regimes.
• Experimental studies in graphene constrictions reveal a temperature-driven transition from ballistic wall scattering at 4.5 K to viscous, super-ballistic conductance at 77 K, with clear kinetic and hydrodynamic signatures.
• Kinetic formulations and hydrodynamic equations, combined with scanning tunneling potentiometry, provide actionable insights into non-Ohmic voltage patterns and validate the additive conductance model across the crossover.

Searching arXiv for recent and foundational papers on ballistic–hydrodynamic–Ohmic crossover in electronic transport, especially graphene constrictions and related kinetic/hydrodynamic diagnostics. arXiv search query: "ballistic hydrodynamic ohmic crossover graphene constriction Gurzhi conductance scanning tunneling potentiometry" The ballistic–hydrodynamic–Ohmic crossover denotes the evolution of charge transport as the hierarchy among the momentum-conserving mean free path $\ell_{ee}$, the momentum-relaxing mean free path $\ell_{mr}$, and a geometric scale such as the channel width $W$ is varied. In the standard hydrodynamic-transport language used for clean electronic systems, the ballistic side corresponds to the Knudsen regime $\ell_{ee}\gg W$ with $\ell_{mr}\gg W$; the hydrodynamic side corresponds to the Gurzhi regime $\ell_{ee}\ll W\ll \ell_{mr}$; and the Ohmic limit is recovered when viscous terms become negligible, equivalently $l_G\ll W$ with $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$ (Krebs et al., 2021). In graphene, this crossover has been directly imaged around electrostatically defined constrictions, where a temperature increase from $4.5\ \text{K}$ to $77\ \text{K}$ drives a Knudsen-to-Gurzhi crossover, while the fully Ohmic regime remains the limiting case of the theory rather than the main experimental focus (Krebs et al., 2021).

1. Regime hierarchy and control parameters

The crossover is controlled by three basic ingredients: a momentum-conserving scattering length $\ell_{mr}$0, a momentum-relaxing scattering length $\ell_{mr}$1, and a geometric scale $\ell_{mr}$2, typically the narrowest cross-section of a constriction. The most compact organizing parameter is the Knudsen number $\ell_{mr}$3, together with the Gurzhi length $\ell_{mr}$4. In this language, the hydrodynamic equation

$\ell_{mr}$5

makes the crossover explicit: when $\ell_{mr}$6, the viscous term can be neglected and one recovers $\ell_{mr}$7, whereas $\ell_{mr}$8 implies viscous dominance (Krebs et al., 2021).

Regime	Length-scale hierarchy	Leading transport character
Ballistic / Knudsen	$\ell_{mr}$9, $W$0	Wall scattering and geometric collimation dominate
Hydrodynamic / Gurzhi	$W$1, $W$2	Viscous flow and super-ballistic conductance
Ohmic / diffusive	$W$3	Local Drude behavior

This hierarchy is not merely terminological. In clean graphene constrictions, the ballistic side is realized at $W$4, where $W$5 at carrier density $W$6, much larger than channel widths of tens to hundreds of nm; at $W$7, $W$8 becomes short enough that $W$9, placing the constriction in the viscous regime (Krebs et al., 2021).

2. Kinetic and hydrodynamic formulations

A standard microscopic description separates momentum-relaxing and momentum-conserving scattering already at the level of the Boltzmann equation. In a high-mobility GaAs/AlGaAs two-dimensional electron system, this is written as

$\ell_{ee}\gg W$0

where $\ell_{ee}\gg W$1 is a local stationary Fermi distribution and $\ell_{ee}\gg W$2 a drifting Fermi distribution (Gupta et al., 2020). This formulation is useful precisely because it interpolates continuously between the limits $\ell_{ee}\gg W$3 and $\ell_{ee}\gg W$4.

The hydrodynamic limit replaces the full kinetic description by a Stokes or linearized Navier–Stokes equation. In the graphene constriction problem, the flow is modeled by

$\ell_{ee}\gg W$5

which leads directly to the current equation above (Krebs et al., 2021). The first term is viscous stress; the second is Ohmic momentum loss. The hydrodynamic regime is therefore not defined by low dissipation alone, but by the dominance of momentum-conserving redistribution over momentum-relaxing decay.

A complementary constriction theory shows that the ballistic-to-viscous crossover can be formulated as an additive conductance relation

$\ell_{ee}\gg W$6

with the viscous term dominating in the hydrodynamic limit and permitting conductance to exceed the ballistic Landauer limit in a viscous point contact (Guo et al., 2016). This additive structure is one of the cleanest theoretical statements of the crossover: ballistic and viscous channels are not combined by a Matthiessen rule for resistivities, but by a conductance enhancement.

3. Electrostatic dams and scanning-tunneling potentiometry in graphene

A particularly direct realization of the crossover uses single-layer graphene on hBN, with electrostatic “dams” written by the STM tip through a $\ell_{ee}\gg W$7–$\ell_{ee}\gg W$8 minute, $\ell_{ee}\gg W$9 pulse that ionizes defects in the underlying hBN and creates smooth electrostatic wells. For suitable back-gate voltages, each well is surrounded by a circular $\ell_{mr}\gg W$0–$\ell_{mr}\gg W$1 ring, and two such rings define a gate-tunable constriction whose width $\ell_{mr}\gg W$2 varies continuously from pinch-off to roughly $\ell_{mr}\gg W$3 (Krebs et al., 2021).

The key probe is scanning tunneling potentiometry (STP). A source–drain bias drives lateral current through a graphene flake of width $\ell_{mr}\gg W$4 and total current path length $\ell_{mr}\gg W$5. At each pixel, the feedback is turned off, local $\ell_{mr}\gg W$6–$\ell_{mr}\gg W$7 curves are measured, and the zero-current tip bias yields the local electrochemical potential,

$\ell_{mr}\gg W$8

up to sign conventions. The method achieves a potential resolution of roughly $\ell_{mr}\gg W$9 and spatial resolution set by STM, effectively nm-scale for STP maps (Krebs et al., 2021).

On bare graphene/hBN, STP maps under $\ell_{ee}\ll W\ll \ell_{mr}$0 show an almost linear drop with slope $\ell_{ee}\ll W\ll \ell_{mr}$1, from which the Drude analysis gives $\ell_{ee}\ll W\ll \ell_{mr}$2. Once electrostatic dams are written, the same local probe directly resolves how the potential landscape reorganizes as the constriction opens and as temperature drives the system from ballistic to viscous transport (Krebs et al., 2021).

4. Local flow signatures across the crossover

The local electrochemical-potential maps reveal non-Ohmic structure on both sides of the crossover. At both $\ell_{ee}\ll W\ll \ell_{mr}$3 and $\ell_{ee}\ll W\ll \ell_{mr}$4, the upstream side of an electrostatic barrier shows an increase in $\ell_{ee}\ll W\ll \ell_{mr}$5, while the downstream side shows a decrease, forming in-plane charge dipoles identified as Landauer residual resistivity dipoles. In the ballistic regime these dipoles coexist with ray-like “streams” of high potential emerging downstream of the constriction, interpreted as collimated classical trajectories. At $\ell_{ee}\ll W\ll \ell_{mr}$6, those rays are much less evident, while the potential landscape becomes smoother and more fluid-like (Krebs et al., 2021).

These structures are non-Ohmic in several distinct senses. The potential does not follow a uniform gradient, local maxima and minima develop near the barriers, and the upstream accumulation/downstream depletion pattern is characteristic of scattering centers in ballistic and hydrodynamic flows rather than a locally uniform conductivity model. A purely viscous continuum description reproduces the qualitative dipole structure near the constriction, while the ballistic rays require a kinetic interpretation (Krebs et al., 2021).

The same diagnostic issue appears in nonlocal transport. In a high-mobility GaAs/AlGaAs 2D electron system, negative nonlocal resistance and current vortices occur in both the predominantly ballistic regime at $\ell_{ee}\ll W\ll \ell_{mr}$7, where $\ell_{ee}\ll W\ll \ell_{mr}$8 and fitted $\ell_{ee}\ll W\ll \ell_{mr}$9–$l_G\ll W$0, and in the hydrodynamic regime at $l_G\ll W$1, where $l_G\ll W$2–$l_G\ll W$3 (Gupta et al., 2020). This establishes an important interpretive point: negative nonlocal resistances and current vortices are not exclusive to only the hydrodynamic regime.

Magnetotransport adds a sharper local discriminator. In a narrow channel under weak non-quantizing magnetic field, the curvature of the transverse electric field $l_G\ll W$4 has opposite signs in the ballistic and hydrodynamic regimes: at weak field the ballistic profile has positive curvature near the center, while the hydrodynamic profile has negative curvature. At the special field $l_G\ll W$5, ballistic and near-ballistic transport also generate distinctive peaks or cusps in $l_G\ll W$6, whereas the true hydrodynamic profile is smooth and parabolic-like (Holder et al., 2019). This indicates that current density alone can be insufficient; higher angular harmonics encoded in the Hall field remain sensitive to ballistic nonlocality even when the current profile appears Poiseuille-like.

5. Conductance, super-ballistic transport, and extracted parameters

The conductance side of the crossover is obtained from STP by measuring the electrochemical-potential drop $l_G\ll W$7 across the constriction and estimating the current through the channel from

$l_G\ll W$8

so that

$l_G\ll W$9

At $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$0, the data follow a Sharvin-type ballistic law

$l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$1

with an empirically determined geometric factor $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$2 (Krebs et al., 2021).

At $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$3, the measured conductance exceeds this ballistic benchmark. The excess is interpreted as a Gurzhi contribution,

$l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$4

so that ballistic scaling $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$5 crosses over to a stronger-than-linear increase with $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$6, characteristic of viscous transport (Krebs et al., 2021). This is the experimentally accessed Knudsen-to-Gurzhi crossover.

Solving for the electron–electron scattering length gives

$l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$7

and the extracted values imply $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$8 at $l_G=\sqrt{\nu\tau_{mr}}=\sqrt{\ell_{ee}\ell_{mr}}$9, up to the unknown geometric factor $4.5\ \text{K}$0. The same analysis yields a kinematic viscosity

$4.5\ \text{K}$1

and, with $4.5\ \text{K}$2, a Gurzhi length

$4.5\ \text{K}$3

comparable to or larger than the channel widths used, which is consistent with $4.5\ \text{K}$4 and therefore with viscous dominance (Krebs et al., 2021).

The broader significance of this super-ballistic regime was formulated theoretically for a viscous point contact, where $4.5\ \text{K}$5 dominates over $4.5\ \text{K}$6 in the hydrodynamic limit and allows conductance to exceed the fundamental Landauer ballistic limit $4.5\ \text{K}$7 (Guo et al., 2016). In this sense, the crossover is not a monotonic degradation of transport with increasing scattering; momentum-conserving scattering can enhance conduction before momentum-relaxing processes restore the Ohmic decline.

6. Interpretive issues, extensions, and related settings

A recurrent misconception is to identify hydrodynamics from any single nonlocal feature. The literature represented here argues against that shortcut in several independent ways. Negative nonlocal resistance and current vortices are compatible with both predominantly ballistic and hydrodynamic transport in mesoscopic GaAs/AlGaAs, and purely hydrodynamic approaches become insufficient in the Gurzhi regime once a magnetic field is introduced, because nonlocal kinetic correlations still control observables such as $4.5\ \text{K}$8 (Gupta et al., 2020, Holder et al., 2019). A plausible implication is that regime identification requires simultaneous access to geometry, boundary conditions, and at least one observable that is sensitive to higher angular harmonics.

Alternative diagnostics have therefore been proposed. In anisotropic conductors at zero magnetic field, the transverse channel voltage can change sign as the system crosses from ballistic to hydrodynamic transport, because the boundary-stress contribution and the momentum-relaxing contribution enter with opposite signs and different $4.5\ \text{K}$9 dependence (Wang et al., 2024). This suggests that zero-field transverse voltage may complement nonlocal resistance and local Hall-field imaging in systems where ballistic and viscous current maps are deceptively similar.

The same crossover language extends beyond electronic viscosity in graphene constrictions. Suspended graphene phonons display a crossover among ballistic, hydrodynamic, and diffusive regimes, quantified by separating momentum destruction due to direct diffuse boundary scattering, boundary shear generated by normal scattering, and umklapp scattering; the phonon Knudsen minimum emerges when the hydrodynamic window is robust (Li et al., 2018). In one-dimensional momentum-conserving fluids, the analogous sequence is ballistic $77\ \text{K}$0 kinetic/Ohmic $77\ \text{K}$1 anomalous hydrodynamic, with a crossover length $77\ \text{K}$2 separating normal and anomalous transport (Lepri et al., 2021). These cases do not identify the hydrodynamic sector with the same constitutive law, but they preserve the central principle: transport regimes are distinguished by the competition between collision lengths and device scale, not by a single phenomenological signature.

The ballistic–hydrodynamic–Ohmic crossover is therefore best understood as a hierarchy problem. Ballistic transport is boundary-limited and trajectory-resolved; hydrodynamic transport is collective and viscosity-dominated when momentum-conserving collisions are fastest; Ohmic transport is recovered when momentum-relaxing processes dominate and $77\ \text{K}$3. What changes across the crossover is not only the magnitude of conductance, but the structure of the local electrochemical potential, the morphology of current flow, the relevance of higher angular harmonics, and the proper level of description itself.