 This behavior is observable in ultrapure material samples, and has been demonstrated through transport measurements in both PdCoO\_2\cite{moll2016evidence} and GaAs,\cite{de1995hydrodynamic} where other viscous, fluid-like behaviors have also been observed.\cite{braem2018scanning}
    
Recently, it has been predicted that such phenomena can also occur in graphene, where strong el-el interactions and low Umklapp scattering rates allow the quasiparticles to form viscous Fermi or Dirac fluids. \cite{guo2017higher, narozhny2017,lucas2018hydrodynamics, levitov2016electron, neto2009electronic,ho2018theoretical,principi2016bulk,torre2015nonlocal}   This fluid-like behavior could lead to several interesting phenomena to appear in graphene, including vortex formation, vortex shedding, and perhaps even electronic turbulence. \cite{mendoza2011preturbulent, levitov2016electron}  In order to observe these effects, a number of experimental methods have been implemented.  Transport measurements through a series of lithographed constrictions and strategically contacted samples have recently observed signatures of superballistic conductance as well as negative backflow - a possible indicator of vortex formation.\cite{berdyugin2019measuring,kumar2017superballistic,bandurin2018fluidity,bandurin2016negative}  Meanwhile, scanned single electron transistor and nitrogen vacancy measurements of etched, encapsulated graphene devices have imaged flow profiles with spatial resolutions as small as 50 nm, and observed signatures of Poiseuille flow, one potential sign of viscous flow behavior.\cite{sulpizio2019visualizing,ella2019simultaneous,jenkins2020imaging,ku2020imaging}  

In this work we use scanning tunneling potentiometry (STP) to image---with nm-scale spatial resolution---the electrochemical potential profile associated with quasiparticle flow in graphene around electrostatic barriers that are `drawn' using voltage pulses from the tip of the scanning tunneling microscope (STM).\cite{velasco_nanoscale_2016} This methodology allows for the creation of smooth barriers defined by in-plane p-n junctions which confine the particle flow without introducing diffusive scattering, or other momentum-relaxing processes which would occur in lithographed samples.\cite{kiselev2019boundary} Moreover, we are able to vary the width of the conduction channels from $\mu$m-scale to pinch-off (where the barriers form `electrostatic dams' that suppress flow). We also probe graphene/hBN samples that are non-encapsulated, which reduces charge screening, enhancing el-el interactions and allowing viscous flow behavior to be observable at shorter lengthscales.

Our results reveal how quasiparticle flow through constrictions changes as the carrier density, channel width, and temperature are varied.    We observe multiple signatures of non-Ohmic behavior, including a small drop in potential across the graphene, mean free paths that exceed 3 $\mu$m, and the formation of Landauer residual resistivity dipoles characterized by charge build-up and depletion on the upstream and downstream side of barriers, respectively.\cite{landauer1957spatial}  At 4.5 K, the flow behaves ballistically and we are able to observe ray-like streams of charge passing through the opened dam, with the channel conductance matching the Sharvin formalism.  Meanwhile, at 77 K, we observe a profile that more closely resembles viscous flow behavior, and we measure a channel conductance that is super-ballistic.  We find that our observations can be qualitatively-described by numerical simulations of a Stokesian fluid. We are able to estimate key parameters of the fluid, including $l_{ee}$, which we measure to be $\sim$100 nm at 77 K and a kinematic viscosity $\nu$ of $\approx 2.5 \times 10^3$ cm$^2$/s. These measurements represent a fundamentally new way of probing viscous electronic fluids that allows for controllable geometric effects on the flow to become observable.

\begin{figure}[t!]
    \centering
    \includegraphics[width
    =6.5in,keepaspectratio]{Figures/OverviewFig.png}
    \caption{(A) Schematic of the STP experimental setup. $V_{sd}$ drives current in the sample while $V_s$ determines the difference between the sample and tip electrochemical potentials. The carrier density (and $E_F$) is globally modified through the use of an electrostatic gate electrode $V_g$. (B) A energy diagram depicting how the electrostatic potential and sample electrochemical potential vary along the flow direction across a potential well. (C) Simultaneously acquired topography [upper left] and spatial map of the electronic LDOS [lower left] ($V_s$ = -10 mV) of the electrostatic dam. All scale bars are 100 nm. Example STP map [upper right] ($V_g$ = -2 V) align with the current flow. Example tunneling I-V curves [lower right] acquired under transport. I = 0 (black dashed line) corresponds to $\mu_{ec} = -V_s$ (orange vertical line); by fitting many such curves (blue solid line) the sample electrochemical potential is mapped spatially. (D) Electrostatic dam shown for different gating conditions, from low to high electron doping (top to bottom). $E_{CNP}$ tracks the electrostatic potential in the graphene sheet (here shown moving across the channel).}
    \label{fig:overview}
\end{figure}

A schematic of our STP measurement geometry is shown in Fig. \ref{fig:overview}A.   In STP, a source-drain bias is used to drive current laterally through a thin sample, and the subsequent spatially varying electrochemical potential, $\mu_{ec}$, is measured locally using an STM tip.\cite{kirtley1988direct,briner1996local,chu1989scanning,muralt1987scanning,muralt1986scanning, druga2010versatile}  To measure $\mu_{ec}$, the feedback of the STM tip is turned off and the tip bias needed to zero the tunneling current is determined by performing a linear fit of the tip-sample I-V curve measured near the zero crossing (Fig. \ref{fig:overview}C).  This allows $\mu_{ec}$ to be measured with $\sim$ 10 $\mu$V potential resolution, and with the \AA-scale spatial resolution of standard STM. Fig. \ref{fig:overview}B illustrates how $\mu_{ec}$ varies across the sample under transport conditions.  When $V_{sd} = 0$, all local accumulations of charge which affect the chemical potential, $\mu$, are offset by changes in electrostatic potential $\phi$, such that $\mu_{ec}$ is constant across the surface. For $V_{sd} \neq 0$, meanwhile, $\mu_{ec}$ will change continuously across the sample, with a spatially varying slope that depends on the local conductance; meanwhile, any changes in local charge accumulation that are due to active carrier scattering or ballistic transport will affect $\mu_{ec}$ and be visible in STP measurement.\cite{bevan2014first,morr2017scanning} 

Previous STP measurements of graphene devices on SiC substrates have revealed sharp drops in potential associated with monolayer-bilayer boundaries, as well as sub-surface crystal steps.  In some cases, Landauer residual-resistivity dipoles could be observed near defect features, which could be used to model the electron-barrier scattering mechanisms.\cite{ji2012atomic,giannazzo2012electronic,wang2013local,willke2015spatial,clark2013spatially, sinterhauf2020substrate}  STP measurements have also been performed on graphene nanoribbons on SiC, where signatures of ballistic transport were observed.\cite{de2020non}  To the best of our knowledge, all previous STP measurements have been performed on SiC subtrates with large dielectric constants, which strongly screen el-el interactions.  Those measurements also utilized topographic features to act as scattering barriers, which are known to introduce artifacts in STP measurements due to tip convolution.\cite{pelz1990tip}

In this work, we probe ultraflat graphene/hBN samples with electrostatic barriers that are introduced by `drawing' them with the STM tip using a methodology developed by Velasco et al. \cite{velasco_nanoscale_2016, lee_imaging_2016, velasco2018visualization}. Each individual barrier was created by introducing sub-surface charges in the underlying hBN by applying a 1--2 minute, 5 V pulse with the STM tip which serves to ionize defects in the underlying hBN substrate.  Those defects create an electrostatic potential well in the plane of the graphene sheet that scatters incident holes and electrons. Within a suitable range of negative gate voltages (when the graphene is hole doped), a circular p-n (outside-inside) junction forms on the periphery of the potential well, which acts as a reflective boundary. By placing two of these p-n junctions in close proximity, we build a small channel that current can flow through when a source-drain bias is applied. Moreover, as shown in Fig. \ref{fig:overview}D, the width of this current-carrying channel can be tuned by using an electrostatic backgate to adjust the Fermi energy, $E_F$,  which alters the radius of the p-n junction barrier; the p-n junctions considered here decrease in radius with higher hole concentrations, which leads to an increased channel width.    

\begin{figure}[t!]
    \centering
    \includ