Response regimes in on-chip THz spectroscopy

Abstract

On-chip THz spectroscopy enables quantitative measurements of the optical conductivity of sub-wavelength 2D materials by tightly confining THz fields in metallic transmission line structures interfaced to the material. However, because the probed structures are smaller than the THz wavelength, finite-size and environmental effects can strongly influence the measured response. Here, we identify the conditions under which a metallic sample exhibits a genuine Drude response and when finite-size and environmental effects must be considered. We further introduce and characterize an additional regime, the Phantom-Drude response, which mimics Drude behavior but instead originates from the superposition of multiple finite-momentum plasmonic resonances. If unrecognized, this regime can lead to misinterpretation of intrinsic material properties. We systematically show how the Phantom-Drude response can emerge and demonstrate its sensitivity to sample dimensions, transmission line geometry, material shape, and gate properties, providing practical guidelines to avoid this regime in future on-chip THz measurements.

Figures (4)

• Figure 1: Probing the near-field optical conductivity of 2D materials with on-chip THz spectroscopy: a Schematic of the on-chip THz circuit architecture: Two identical THz pulses (dark blue) propagate along symmetric transmission lines; one interacts with a 2D material (blue gradient) and is detected on the left, while the other serves as a reference on the right. Comparing both signals yields the material’s near-field optical conductivity. b 2D materials embedded in on-chip THz spectroscopy circuitry form plasmonic self-cavities, featuring one or more resonances in the near-field optical conductivity spectrum kipp2024cavity. These resonances arise from finite-momentum standing waves of current density spanning the sample, as illustrated for the three lowest frequency modes of $\sigma_{\text{near-field}}$ (blue). When the resonance linewidth $\gamma$ and the frequency resolution are broader than the mode spacing, overlapping resonances merge into a broad spectrum that mimics a Drude response (red), despite being fundamentally different. A Drude fit to the red spectrum (gold dashed curve) yields misleading parameters: the extracted scattering rate $\gamma$ and Drude weight $D$ (gold) differ significantly from the intrinsic material values (blue). We discuss the origin of this effect and mitigation strategies in this work. Sample parameters for calculations: $W_0=$ 0µm (2D material width exceeding the metal strips), $W_1=$ 10µm (width of each metal trace) and $W_2=$ 10µm (width of the gap between strips), $d_{\text{gold}}=$ 285nm (gold thickness), $d_{\text{hBN}}=$ 20nm (hBN thickness), $d_{\text{sapphire}}=$ 2mm (sapphire substrate thickness), $D=9.74mS\times THz \times rad$ (Drude weight of the 2D material) and $\gamma=0.2THz$ (scattering rate of the 2D material).
• Figure 2: Impact of transmission line design and material conductivity on the measured response: Three transmission line designs are investigated. a In the first design, a 2D material is electrically insulated from the transmission line by a hBN layer of thickness $d_{\text{hBN}}=$ 20nm. The structure is embedded within a transmission line of total width $W=2W_1+W_2$ with $W_1=W_2$. No material is extending beyond the transmission line ($W_0=0$, see SI). b In the second design, the transmission line has a small gap of width $W_2=$ 100nm that is kept constant while tuning the overall width $W$. c In the third design, the hBN spacer is removed and the 2D material is placed in perfect ohmic contact to the transmission line of total width $W=2W_1+W_2$ with $W_1=W_2$. d displays the real parts of the near-field optical conductivity as function of frequency corresponding to the markers in panels (e). e The nature of the response (whether plasmonic cavity, Phantom-Drude, or Drude) is calculated as a function of $W$ and the intrinsic Drude weight $D$ of the material, assuming a fixed scattering rate of 0.1THz, for each of the three designs illustrated in (a)–(c). For the design shown in (a), any of the three responses (plasmonic cavity, phantom-Drude, or Drude) can be observed depending on $W$ and $D$. In contrast, the design in (b) permits only cavity responses or broad spectral features that do not resemble a Drude response. The design in (c), featuring an ohmic contact, consistently produces a Drude response regardless of $W$ or $D$. The black curves in the left (data adapted from Fig. 2d of Ref. kipp2024cavity) and center panel mark parameter sets ($D$, $W$) for which the real part of the near-field optical conductivity, measured via on-chip THz spectroscopy, deviates less than 10% from the intrinsic 2D conductivity at 10GHz. Notably, the phantom-Drude regime (red-shaded area in (e)) expands with increasing linewidth, while the black curve remains fixed. Note that for design (c), full 3D electromagnetic simulations were used in place of the analytical theory (see SI).
• Figure 3: Inhomogeneous linewidth broadening due to irregularly shaped 2D materials: a An irregularly shaped 2D material embedded in on-chip THz spectroscopy can be decomposed into slices of fixed width (here, 2.5µm). Since the frequencies of the self-cavity modes strongly depend on the local width $W_0$ of the part of the 2D material extending the transmission line, the overall near-field optical conductivity spectrum of the irregular flake can be approximated as the sum of the responses from each individual slice. b Full 3D electromagnetic simulations are performed for both the entire irregular flake and its constituent slices. c Full 3D electromagnetic simulations of the current density distribution in the irregularly shaped sample at 0.686THz and a phase of 103^∘. Distinct plasmonic modes of different wavelengths, and thus different resonance frequencies, form in different slices of the sample, whose combined response gives rise to the inhomogeneously broadened high-frequency peak shown in (b). Sample parameters: $W_1=W_2=$ 3µm, $D=$ 24.5mS THz rad, $d_{\text{hBN}}=$ 25nm and $\gamma =1.5/(2\pi)$ THz.
• Figure 4: The challenge of low-conductive gates: a In on-chip THz spectroscopy of gate-tunable vdW heterostructures, plasmonic modes in the vdW layer and nearby gate strongly couple due to their nanometer separation, forming hybrid symmetric (in-phase) and antisymmetric (out-of-phase) excitations. b Calculated resonance frequencies of hybrid and uncoupled modes versus gate-to-vdW material Drude weight ratio. Highly conductive gates yield weak hybridization, while low-conductive gates cause strong mode mixing and large frequency shifts. c Near-field conductivity spectra for gate-tunable vdW heterostructures in a transmission line of design (a) (see Fig. \ref{['fig:fig2']}a). Two cases are compared: a perfectly screened vdW layer with a thick metallic gate (dashed red) and a vdW layer with a lower conductive gate where $D_{\text{gate}}/D_{\text{vdW material}} = 1$ (red–blue). In the latter, antisymmetric (vdW material-like, red) and symmetric (gate-like, blue) hybrid plasmon modes appear. The antisymmetric modes shift to lower frequencies due to coupling with the symmetric modes, a result of resonant interaction at the dielectric boundaries provided by the metal trace edges, as described in Ref. kipp2024cavity. d When the scattering rate is increased to $\gamma_{\text{gate}} = \gamma_{\text{vdW material}}$, distinct spectral features from multiple resonances remain visible (red-blue curve). However, if the gate's scattering rate becomes much larger than that of the vdW material, both the symmetric (blue) and antisymmetric (red) modes broaden significantly, merging into a smooth, featureless Phantom-Drude response (red curve). Sample parameters for panel (b): momentum $q=\pi/(39µm)$, $D_{\text{vdW material}}=$ 43.35mS THz rad and $d_{\text{hBN,1}}=$ 20nm. Sample parameters for panels (c) and (d): $W_0=W_1=W_2=$ 13µm, $d_{\text{hBN,1}}=$ 20nm, $d_{\text{hBN,2}}=$ 100nm, $D_{\text{vdW material}}=D_{\text{gate}}=$ 34.05mS THz rad. Scattering rates in (c): $\gamma_{\text{gate}}=\gamma_{\text{vdW material}}=0.02THz$. Scattering rates in (d): $\gamma_{\text{gate}}=\gamma_{\text{vdW material}}=0.12THz$ (red-blue) and $\gamma_{\text{gate}}=5\times\gamma_{\text{vdW material}}=0.60THz$ (red).