Terahertz s-SNOM reveals nonlocal nanoscale conductivity of graphene

Abstract

As photonic and electronic technologies approach nanometre length scales and terahertz operating speeds, electrical conductivity can no longer be treated as a purely local material parameter. In this regime, charge transport becomes intrinsically nonlocal, with conductivity depending on both frequency and momentum, $σ(ω,q)$, fundamentally limiting field confinement, dispersion, and loss in nanoscale devices. Here, we directly measure the nonlocal nanoscale conductivity of graphene using terahertz scattering-type near-field optical microscopy. By combining broadband THz near-field spectroscopy with quantitative electrodynamic modelling, we extract the complex conductivity of single- and few-layer graphene with $\sim$50 nm spatial resolution. We find that nonlocal response dominates the terahertz conductivity of monolayer graphene even at length scales comparable to practical device dimensions. These results establish nonlocal conductivity as a measurable and design-relevant material property in the terahertz regime, providing a quantitative foundation for predicting performance limits in ultracompact photonic and electronic systems.

Figures (10)

• Figure 1: $|$THz-frequency scattering-type Scanning Near-Field Optical Microscopy (THz-SNOM) instrumentation and imaging of exfoliated graphene flakes.a. Schematic of the tip-sample interaction, featuring a tapping-mode AFM adapted for in- and out-coupling of single-cycle THz pulses to a metallic tip. b. Time-domain waveforms of the scattered THz pulse at the tip apex (orange) versus substrate reference (blue) (b). c. Corresponding spectral amplitudes spanning 0.5-1.5 THz. d. Tip–sample approach curves for demodulation orders m = 2–5, showing enhanced near-field confinement at higher harmonics; m = 2 (grey) is used for all subsequent measurements for optimal signal-to-noise ratio. e,f. Representative SEM images of dented AFM tips, revealing radii of $\sim$385 nm and non-spherical apex shapes with $\sim$226 nm radius, produced by gentle denting to boost the scattering signal. g,h,i. Optical micrographs of mechanically exfoliated graphene on SiO$_2$/Si for samples S1-S3: monolayer (1L) in S1; predominantly monolayer with a narrow bilayer (2L) region in S2; and a central monolayer region flanked by bi- and trilayer (3L) regions in S3. Polymer residues (bright blue) and bulk graphite (bright region, top left of g) are also visible. j,k,l. White-light mode THz-SNOM contrast images at $\omega_c/2\pi=1.2\text{~THz}$ for S1, S2 and S3, recorded with the temporal delay parked at the substrate peak; distinct layer-dependent contrast and intralayer variations are clearly resolved.
• Figure 2: $|$THz-SNOM spectroscopy of monolayer graphene conductivity.a. Time-domain THz waveforms recorded on the bare substrate at reference position R (blue) and on graphene sample S1 at position P2 (red), overlaid with the finite-dipole model (FDM) simulation using the nonlocal Lovat conductivity (green) and the absolute difference $|S_{\rm meas}-S_{\rm sim}|$ (grey). b. Same experimental data fitted with a local Drude-like conductivity, showing markedly poorer agreement. c,d. Spectrally resolved amplitude (c) and phase (d) contrasts, respectively, averaged over four graphene positions (P1-P4) relative to references for samples S1 (blue) and S2 (red); shaded regions denote $\pm$1 standard deviation between measurements. e. Real and imaginary parts of the complex conductivity recovered via inversion of the FDM for both samples. Solid curves show Lovat-model fits to the frequency-domain spectra, while dashed curves are the model conductivities obtained from the time-domain minimisation in a. f. Inset: Lovat-model conductivity spectra extended to 0.1-5.5 THz; grey shading indicates the spectral coverage of the instrument, and envelopes represent the fit standard deviation between points P1-P4.
• Figure 3: $|$Field distribution and nonlocal response in THz‐SNOM.a. Finite-element calculation of the near-field enhancement at 1 THz in the apex region of an AFM tip (ellipsoidal cross-section in black) with radius $R = 400\text{~nm}$, positioned 20 nm above the sample surface. The white dashed curve shows the circular approximation near the apex. False-colour shading plots the field enhancement $|E(z = 10~nm)|/E_0$, revealing a confinement with full-width at half-maximum (FWHM) $\approx 90\text{~nm}$ (grey line), much smaller than the tip radius. The real-space nonlocal response of the Lovat-BGK model at $\omega/2\pi = 1\text{~THz}$ ($\tau = 100\text{ fs}, E_\text{F} = 200\text{ meV}$), $|\sigma_\text{BGK}(x)|$ is overlaid in orange, showing that the nonlocal "reach" is comparable to, and even exceeds the extent of the THz field. b. Spatial profiles of $|\sigma_\text{BGK}(x,\omega)|$ obtained by inverse Fourier transform of $\sigma_\text{BGK}(q,\omega)$ over the frequency range $\omega = 10^{11}–10^{15} s^{-1}$. The dashed white curves trace the FWHM of the nonlocal response at each $\omega$, illustrating a reach of a few hundred nanometres in the THz (green band: 0.5–1.5 THz) that collapses to only a few nanometres in the mid-IR. c. Imaginary part $\text{Im}(\sigma_\text{BGK}(q,\omega)$ plotted as a function of momentum $q$ and frequency $\omega$. The white dashed line marks $\omega = v_\text{F}q$, below which $\text{Im}(\sigma_\text{BGK})$ becomes negative, a clear signature of nonlocality, while the dotted line indicates the onset of interband transitions at $\hbar\omega = 2E_\text{F}$. The THz-SNOM band is shown in green, and the weighted in-plane momentum distributions (FDM weight) for tip radii $R = 200\text{~nm}$ (blue) and $R = 50\text{~nm}$ (orange) illustrate the $q$-values probed in the experiment and used in the Lovat model evaluation, respectively.
• Figure 4: $|$Conductivity and material parameter maps of three graphene samples.a,b,c. Absolute value of the complex conductivity extracted from white-light (WL) amplitude images combined with phase-resolved spectra at points P1–P4, averaged over 0.5–1.5 THz. Sample S1 (a) shows landscape-like variations with a pronounced low-conductivity region along the lower left edge. Sample S2 (b) exhibits generally lower, more uniform monolayer conductivity and enhanced bilayer conductivity. Sample S3 (c) clearly differentiates mono-, bi- and trilayer regions. d,e. Maps of Fermi energy $E_\mathrm{F}$ and scattering time $\tau$ for sample S1, obtained by inverting the Lovat-BGK analytical conductivity expression using both amplitude and average phase shift $\Delta\phi_\mathrm{av}$. f,g. Histograms of $E_\mathrm{F}$ and $\tau$ over the area in (d,e). Data are fitted with a double Gaussian distribution, where the main peak and shoulder correspond to the central flake region and the left-edge region, respectively.
• Figure 5: Maps of nonlocal conductivity in the hydrodynamic Drude model (a,b: real and imaginary part, respectively) and in the Lovat formulation of the BGK model (c,d: real and imaginary part, respectively), plotted as a function of $q$ and $\omega$. Horizontal lines indicate frequencies 0.5 and 1.5 THz, see Fig. \ref{['am:lovat_vs_hdm_1d']}. The diagonal dot-dashed line indicates $\omega=v_F q$ (see text).