Near-field terahertz probes with room-temperature nanodetectors for subwavelength resolution imaging

Near-field imaging with terahertz (THz) waves is emerging as a powerful technique for fundamental research in photonics and across physical and life sciences. Spatial resolution beyond the diffraction limit can be achieved by collecting THz waves from an object through a small aperture placed in the near-field. However, light transmission through a sub-wavelength size aperture is fundamentally limited by the wave nature of light. Here, we conceive a novel architecture that exploits inherently strong evanescent THz field arising within the aperture to mitigate the problem of vanishing transmission. The sub-wavelength aperture is originally coupled to asymmetric electrodes, which activate the thermo-electric THz detection mechanism in a transistor channel made of flakes of black-phosphorus or InAs nanowires. The proposed novel THz near-field probes enable room-temperature sub-wavelength resolution coherent imaging with a 3.4 THz quantum cascade laser, paving the way to compact and versatile THz imaging systems and promising to bridge the gap in spatial resolution from the nanoscale to the diffraction limit.


Determination of the Seebeck coefficient for sample A
The Seebeck coefficient can be retrieved using the Mott formula: where k is the Boltzmann constant, T is the temperature, σ is the channel conductivity, ε is the energy and E F is the Fermi energy. The electrical behavior of sample A shows that the device operates in the linear transport regime in the range of gate voltages: V G = -0.35 V -0.25 V, as shown in the transconductance curve ( Figure S1a). In this regime, the carrier mobility is constant and Eq. S1) can be re-written as [36]: Where C gw is the capacitance between the nanowire and the gate electrode, normalized to the gated volume, and n the carrier density. The derivative dn/dE F can be estimated from the electron density in the InAs nanowire: is the 3-dimensional density of states (this approximation is valid for nanowires with radius larger than 20 nm) and f(E) is the Fermi-Dirac Distribution: where E = 0 corresponds to the bottom of the conduction band, and m* = 0.023 m 0 is the electron effective mass in InAs, Equation (S3) allows us to retrieve the trends of Fig. S1,b and Fig. S1,c. Under the assumption of constant mobility (μ = 500 cm 2 /Vs across the linear region, as estimated from the measurement of transcoductance), the channel conductivity σ and the electron density n are related via the following equations:

Determination of the Seebeck coefficient for sample B
While analyzing the transport behavior of sample B, two different transport regimes can be identified ( Figure   S2a): region I (blue shaded linear region in the logarithmic plot of Fig. S2b) corresponding to the subthreshold regime (V G = 0.5 V-1 V); region II corresponding to the linear transport regime (V G = -1.5 V -0.5 V). To extract the Seebeck coefficient S (blue curve in Fig. S2d) at V G > 0.5 V we investigated the subthreshold swing , which is ≈ β [38]: Here β can be extracted from the linear fit of the logarithm of the current vs V G in the subthreshold region.
Conversely, in region II (gray region in Figure S2b) we can make the same assumptions used for the calculation of the Seebeck coefficient for sample A, after substituting in Eq. S2, C gw with the gate to channel capacitance per unit area C gc , and by retrieving the density of carriers per unit area via the equation [39]: The energies E ! ! here indicate the subbands arising in a BP flake of finite thickness as an effect of the confinement in the out-of-plane z direction. The n(E F ) plot calculated via Eq. S8 is shown in Fig. S2c.
Combining Eqs. S6 and S8 we derived dn/dE f and then S b (red curve in Figure S2c) via Eq. S2.
The overall Seebeck coefficient of sample B, plotted in Fig. 5b of the main text, can be then achieved combined the two curves discussed above and correspond to the red curve (Fig. S2d) below V G = 0 V and to the blue curve (Fig. S2d) above V G = 0 V. The two curves well combine at V G = 0 V, if we set the mobility of the BP flake at 900 cm 2 /Vs across the linear region, a value which is expected for BP flakes, having thickness comparable with the present experimental case.

Phase--sensitive interferometric setup
The interferometric setup exploited for phase sensitive experiments is depicted in Figure 4 in the main text.
A transparent chip carrier has been used, so that the detector can be simultaneously excited by two optical beams (Figure 4 right, upper panel), one impinging on the aperture (I 1 , signal) and the other on the back-side (I REF , reference) through the Si substrate. The QCL beam is first collimated by a Picarin lens having focal length of 3 cm and then hits a double-side polished intrinsic silicon wafer which acts as beam-splitter (BS).
The estimated BP reflectance is 65%, whereas its transmittance is 35% (assuming the material lossless). This

Evaluation of the spatial resolution
The spatial resolution of a 20 μm aperture near-field probe has been assessed by mapping the THz intensity distribution formed at the tips of two metallic needles separated by ~ 10 μm, brought in close proximity (~ 2 μm) to the aperture (see inset of Figure S3) and placed in the focus of the beam [40]. The spatial size of the confined THz field is determined by the distance between the two tips and is narrower than the aperture size.
Thus, the signal profile obtained by moving the detector in front of the needles shown in Figure S3 is the near-field detector response to a point-like source. The full width at half maximum (FWHM) of the intensity profile then represents the spatial resolution of the ~ 17 μm aperture detector.