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Electron viscosity and device-dependent variability in four-probe electrical transport in ultra-clean graphene field-effect transistors

Richa P. Madhogaria, Aniket Majumdar, Nishant Dahma, Pritam Pal, Rishabh Hangal, Kenji Watanabe, Takashi Taniguchi, Arindam Ghosh

TL;DR

This work demonstrates that even in simple rectangular four-terminal graphene devices, strong device-to-device variability governs four-probe transport, complicating signatures of viscous electron flow. The authors combine transport measurements with a phenomenological network model to extract the viscous conductance and contact-coupling terms, revealing that electronic viscosity η generally scales approximately as $1/T$ but exhibits nontrivial, device-dependent density dependencies that challenge conventional Fermi-liquid predictions. They also report WF-law violations and Poiseuille-like width scaling, reinforcing the presence of viscous hydrodynamics while emphasizing the crucial role of boundary conditions and nonlocal contact effects. The study provides a practical methodology to quantify viscous contributions in ultra-clean graphene FETs, offering a bridge between diverse hydrodynamic observations and enabling systematic comparisons across device architectures.

Abstract

Hydrodynamic electrons in high-mobility graphene devices have demonstrated great potential in establishing an electronic analogue of relativistic quantum fluid in solid-state systems. One of the key requirements for observing viscous electron flow in an electronic channel is a large momentum-relaxation path, a process primarily limited by electron-impurity/phonon scattering in graphene. Over the past decade, multiple complex device geometries have been successfully employed to suppress momentum-relaxing scattering mechanisms; however, experimental observations have been found to be sensitive to the device fabrication process and architecture, raising questions about the signature of electron hydrodynamics itself. Here, we present a study on multiple ultra-clean graphene field-effect transistors (FETs) in a simple, rectangular four-terminal device architecture. Using electrical transport measurements, we have characterised the pristine quality of the graphene FETs and examined the variation of electrical resistance in the doped regime as a function of carrier density and temperature. Our results reveal strong device-dependent variability even in the most simple architecture that we attribute to competing momentum-conserving and momentum-relaxing scattering mechanisms, as well as coupling to contacts. Further, we have proposed a phenomenological method for analysing the results, which yields transport parameters in accordance with recent experiments. This simple experimental strategy and analysis can serve as an efficient tool for extracting the viscous electronic contribution in state-of-the-art high-mobility graphene FETs.

Electron viscosity and device-dependent variability in four-probe electrical transport in ultra-clean graphene field-effect transistors

TL;DR

This work demonstrates that even in simple rectangular four-terminal graphene devices, strong device-to-device variability governs four-probe transport, complicating signatures of viscous electron flow. The authors combine transport measurements with a phenomenological network model to extract the viscous conductance and contact-coupling terms, revealing that electronic viscosity η generally scales approximately as but exhibits nontrivial, device-dependent density dependencies that challenge conventional Fermi-liquid predictions. They also report WF-law violations and Poiseuille-like width scaling, reinforcing the presence of viscous hydrodynamics while emphasizing the crucial role of boundary conditions and nonlocal contact effects. The study provides a practical methodology to quantify viscous contributions in ultra-clean graphene FETs, offering a bridge between diverse hydrodynamic observations and enabling systematic comparisons across device architectures.

Abstract

Hydrodynamic electrons in high-mobility graphene devices have demonstrated great potential in establishing an electronic analogue of relativistic quantum fluid in solid-state systems. One of the key requirements for observing viscous electron flow in an electronic channel is a large momentum-relaxation path, a process primarily limited by electron-impurity/phonon scattering in graphene. Over the past decade, multiple complex device geometries have been successfully employed to suppress momentum-relaxing scattering mechanisms; however, experimental observations have been found to be sensitive to the device fabrication process and architecture, raising questions about the signature of electron hydrodynamics itself. Here, we present a study on multiple ultra-clean graphene field-effect transistors (FETs) in a simple, rectangular four-terminal device architecture. Using electrical transport measurements, we have characterised the pristine quality of the graphene FETs and examined the variation of electrical resistance in the doped regime as a function of carrier density and temperature. Our results reveal strong device-dependent variability even in the most simple architecture that we attribute to competing momentum-conserving and momentum-relaxing scattering mechanisms, as well as coupling to contacts. Further, we have proposed a phenomenological method for analysing the results, which yields transport parameters in accordance with recent experiments. This simple experimental strategy and analysis can serve as an efficient tool for extracting the viscous electronic contribution in state-of-the-art high-mobility graphene FETs.
Paper Structure (9 sections, 2 equations, 4 figures, 1 table)

This paper contains 9 sections, 2 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Device configuration
  3. Experimental Results
  4. Discussion
  5. Viscous electron flow in ultra-clean graphene
  6. Phenomenological Analysis
  7. Temperature and density dependence of electronic viscosity in doped graphene
  8. Summary
  9. Acknowledgement

Figures (4)

  • Figure 1: Four-probe electrical resistance in an ultra-clean graphene channel: (a). Schematic of the hBN-encapsulated graphene (grey) device consisting of four electrical contacts (blue). (b) $R_\mathrm{4p}$ vs carrier density $n$ for devices D3S4, D1S5, D3S5 and D5S5 at $T = 100$ K. Left inset - Optical micrograph of a device. Scale-bar along both axes is $3$$\mu$m. Right inset - Normalized electrical conductivity ($\sigma/\sigma_\mathrm{min}$) vs $n$ for devices D1S5, D3S5 and D5S5 at $T = 100$ K. The horizontal dashed line indicates $\sigma = \sigma_\mathrm{min}$, and the solid line indicates $\sigma \propto \sqrt{n}$. The intersection of these two lines has been used to calculate the total charge inhomogeneity $n_\mathrm{min}$. (c) $R_\mathrm{4p}$ vs $n$ for devices D3S4 (class-I), D3S5 (class-II), and D1S4 (class-III) as a function of $T$ at intermediate and high densities. The horizontal dashed line indicates $R_\mathrm{4p} = 0$$\Omega$. (d) $R_\mathrm{4p}$ vs $n_\mathrm{min}(0)$ at $T = 100$ K and $n = 4\times10^{11}$ cm$^{-2}$. The horizontal dashed line indicates $R_\mathrm{4p} = 0$$\Omega$. (e) $R_\mathrm{4p}$ vs $T$ for devices D3S4, D1S5, D5S5 and D1S4 at $n = 5\times10^{11}$ cm$^{-2}$.
  • Figure 2: Evidence of viscous electron flow: (a) Violation of WF Law - Plot of $\mathcal{L}/\mathcal{L}_\mathrm{WF}$ as a function of $n$ for D2S1 at two different temperatures. (b) Current-induced decrease in differential resistance - [Left Panel] $dV/dI$ vs $J$ in D3S5 for different carrier densities near DP. [Right Panel] $dV/dI$ vs $J$ for $n = 10^{10}$ cm$^{-2}$ stacked across different temperatures. (c) Width dependence of electrical conductivity - Plot of $\sigma$ vs $W$ across $5$ different temperatures for $n = 10^{11}$ cm$^{-2}$. The solid line serves as a guide to the eye and is proportional to $W^\beta$, $\beta \simeq 1.8 \pm 0.1$.
  • Figure 3: The phenomenological model: (a) Schematic of different conductances used in the model. The grey area corresponds to the channel, connected to four blue-coloured electrical contacts ($1,~2,~3$ and $4$). Blue arrows represent $G_\mathrm{V}$ while white arrows represent $G_\mathrm{a},~G_\mathrm{\tilde{a}}$. (b) Experimentally obtained $R_\mathrm{4p}$ (black circles) as a function of $n$ for $\lvert n \rvert \ge 5 \times 10^{10}$ cm$^{-2}$ and $T = 105$ K for devices D1S5, D3S4 and D1S4, along with the best-fitted curves (red solid lines) as per our formalism. Left Panels - hole-doped, Right Panels - electron-doped. (c) $G_\mathrm{a}$ versus $T$ for different devices. (d) $G_\mathrm{a}$ versus $n_\mathrm{min}(0)$ for different $T$. The coloured bars in figures (c) and (d) indicate the range over which the fitted $G_\mathrm{a}$ varies. The yellow (beige) bar represents $G_\mathrm{a} < 0.1~(>1000)$ mS.
  • Figure 4: Estimation of the electronic shear viscosity: (a) Variation of $l_\mathrm{mfp}$ with $T$ for three devices, evaluated for $n = 10^{11}$ cm$^{-2}$. The solid lines represent $1/T^{0.7}$, $1/T$ and $1/T^{1.3}$ dependences. (b) $n$-dependence of $l_\mathrm{mfp}$ for four devices at $T = 105$ K. The purple and green dashed lines indicate $n$ and $\sqrt{n}$ dependence, respectively. (c) $\eta$ normalized with $\eta~(\mathrm{T=100~K})$ as a function of $T$ measured at $n = 10^{11}$ cm$^{-2}$. [Top to Bottom] Panel I: Devices D3S4 and D1S5 (Class-I), which display negative $R_\mathrm{4p}$ for $T \leq 260$ K. Panel II: Device D3S5 (Class-II), where $R_\mathrm{4p}$ makes a transition from negative to positive at $T \geq 160~\mathrm{K}$. Panel III: Devices D1S4 and D5S5 (Class-III) with positive $R_\mathrm{4p}$ throughout. (d) Plots of $\eta$ normalized with $\eta_0 = \eta~(n\ \mathrm{{=7\times10^{10}~cm^{-2}}})$ as a function of $n$ at $T = 150$ K. [Top to Bottom] The devices follow the same scheme as in (c).