1. Introduction

In the last few decades, terahertz (THz) spectroscopy has been shown to be a powerful tool for studying charge carriers in a wide variety of materials [1]. Most semiconductors have plasma frequencies in the terahertz regime, providing distinct spectral signatures in systems [2], which are relevant to modern electronics. With a photon energy much below the band gap of conventional semiconductors, THz radiation allows for a non-invasive measurement of only the carriers, which contribute to conductivity. THz time-domain spectroscopy (THz-TDS) and imaging are well-established techniques in which the electric field of the THz pulses are detected in the time domain, preserving all spectral information [3–5]. THz-TDS can provide a quantitative characterization of a semiconductor’s doping profile, but with a limited spatial resolution due to the diffraction limit [6]. Thus, conventional THz techniques are not able to spatially resolve nanostructures, although THz spectroscopy can reveal the average changes in the electrical properties of a collection of nanostructures, such as semiconductor nanowires [7]. In an effort to overcome the diffraction limit and extract information about individual nanostructures, THz imaging has been adapted to various types of near-field microscopy techniques [8–16]. In one promising approach, the incoming radiation encounters a sharp metal tip that confines the probing spot to an area of size on the order of the tip diameter known as scattering-type Scanning Near-field Optical Microscopy (s-SNOM) [17,18]. This method allows for a significant background suppression by performing lock-in detection referenced to harmonics of the oscillation frequency of the metallic tip. In the mid-IR and THz range, researchers have used s-SNOM to demonstrate a spatial resolution down to the nanoscale [19,20], and also for combining both a high spatial and temporal resolution [21]. With the application and development of s-SNOM within the field of THz and infrared imaging, several groundbreaking studies of physical phenomena on the nanoscale have been published [22–28]. Using an s-SNOM configuration, we recently demonstrated the nanoscale version of laser THz emission microscopy (LTEM) [29]. LTEM has previously been shown to be a powerful tool for examining photo-induced charge carrier dynamics in integrated circuits [30] and for spectroscopic studies of surfaces [31,32]. To facilitate the extraction of the dielectric function of a sample from an s-SNOM experiment, analytical models have been developed to describe the tip-sample near-field system; most notably, the point dipole model (PDM) which represents the tip as a conductive metal sphere [17], and the finite dipole model (FDM), which represent the tip as a conductive metal prolate spheroid [33,34]. The latter is widely used, for example in extracting local carrier concentration.

Here, we have implemented a THz-TDS system in the s-SNOM configuration along with the experimental setup for LTEN demonstrated in [29]. This allows us to perform THz-TDS with nanometer resolution, in the same experimental configuration as LTEN measurements; for the remainder of this discussion, we refer to this nanoscale THz-TDS imaging technique as THz nanoscopy. To show the complementarity of the two methods, we provide a direct experimental comparison of near-field tip-based THz nanoscopy and LTEN by performing back-to-back imaging of a structured InAs surface. For the THz nanoscopy and the LTEN imaging, we are able to record waveforms modulated up to the 6^th and the 10^th harmonic of the tip oscillation frequency, respectively, which gives an extremely tight confinement of the THz fields in both cases. To support our experimental results, we implement a computational approach for precisely determining the local field confinement in each case. Finally, we demonstrate the broad-spectrum applications of the FDM for THz nanoscopy, and the first use of the FDM for predicting THz pulses generated from LTEN.

2. Setup

The experimental setup is illustrated in Fig. 1, and is based on a Ti:Sapphire femtosecond (fs) laser providing near-infrared (NIR) laser pulses with a center wavelength of 820 nm, pulse duration of 100 fs, and a repetition rate of 80 MHz. The near-field microscope is a commercial Atomic Force Microscopy (AFM) based s-SNOM system (Neaspec) operated in tapping mode. The AFM probe tips (Rocky Mountain Nanotechnology) are coated with PtIr, with a conical shank approximately 80 µm long and which tapers to a rounded apex with radius of approximately 20 nm. For LTEN, the NIR beam is focused on the tip-sample junction, illuminating a region that is much larger than the tip, and generating a macroscopic THz dipole in the sample through the photo-Dember effect.. Further details about the LTEN technique can be found in [29].

 

Fig. 1. (a): Schematic of the near-field experiments; NIR pulse is shown in red while the THz beam is shown in blue. A delay stage and a ZnTe EO crystal provide the coherent E-field amplitude detection. (b): Time-domain waveforms recorded with THz nanoscopy (blue) and LTEN (red) by lock-in detection to the 2^nd harmonic of the tapping frequency. (c): Amplitude spectra corresponding to the recorded waveforms in (b).

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Here, we perform THz nanoscopy and LTEN back-to-back in immediate succession to facilitate a direct comparison of the two techniques under the same experimental conditions. For the THz nanoscopy experiments, NIR pulses are incident on a large area photoconductive emitter (Tera-SED, Laser Quantum), which generates vertically polarized, sub-picosecond single-cycle THz pulses. These pulses are collimated with an off-axis parabolic mirror (reflected focal length: 15 mm), and then sent into the microscope with an ITO-coated beam splitter to reflect the vertically polarized THz pulses and transmit the horizontally polarized NIR pulses. The incident beams are focused onto the sample surface, directly under the tip apex using another off-axis parabolic mirror, which also collects and collimates the outgoing (including the scattered) THz pulses. For both techniques, this outgoing radiation is detected via electro-optic (EO) sampling [35] and lock-in detection referenced to the tip tapping frequency (approximately 45 kHz) or its harmonics, while the beam for the other technique is blocked. The tapping amplitude for these experiments is approximately 100 nm.

The off-axis parabolic mirror located inside the near-field microscope focuses the incident THz or NIR beam to a spot whose size is approximately diffraction-limited and therefore significantly larger than the ∼20 nm tip apex. As a result, a large portion of the light does not interact at all with the tip-sample system, and a significant background contribution with no near-field information is introduced. As first demonstrated by Knoll et al. [17], this background can be distinguished from the near-field signal by a Fourier decomposition of the signal strength vs tip-sample distance, since the two contributions vary differently with tip-sample separation. Experimentally, this corresponds to demodulating our lock-in detection to higher harmonics of the tapping frequency. Higher-order harmonics increasingly isolate the near-field signal, suppressing the background more effectively at the expense of signal-to-noise. As shown below, we are able to detect near-field signals even up to the 6^th harmonic of the tapping frequency for THz nanoscopy and the 10^th harmonic for LTEN. However, we have found that, at least for the experiments shown here, the 2^nd and 3^rd harmonics are generally sufficient for discriminating the near-field information from the background, and thereby obtain a sufficiently high spatial resolution comparable to the tip diameter.

The EO sampling technique allows for coherent, time-resolved detection of the THz electric field; we may keep the sample at a fixed location while recording the waveforms as shown in Fig. 1(b), here modulated to the 2^nd harmonic of the tip oscillation. In this case, we retain all spectroscopic information; Fig. 1(c) shows an example of the obtained amplitude spectra. For recording images as shown below, we instead keep the delay stage at a fixed position to measure the THz peak electric field and raster-scan the sample, generating a 2D map of the THz nanoscopy or LTEM signal. In either case, the sample topography is recorded simultaneously, which also gives the possibility to ensure that the same spatial region is imaged for each experiment. All images and time-traces shown here are recorded under ambient conditions with a lock-in time-constant of 100 ms.

3. Results

3.1 Imaging with THz nanoscopy and nanoscale LTEM

To rigorously compare the THz nanoscopy and nanoscale LTEM methods, we image a 3.5 µm x 3.5 µm area of a structured, lightly p-doped (n ≈ 10¹⁶, plasma frequency ν_p ≈ 2 THz, damping constant γ ≈ 4.3 × 10³ s⁻¹) InAs sample. In Fig. 2, we show the images recorded when referencing to the 2^nd harmonic of the tapping frequency. The AFM topography images recorded for both experiments appear identical, so we only show one of them here. As seen in Fig. 2(a), (a) groove in the surface of the InAs sample is imaged with a total height variation of 95 nm. A clear contrast is observed in both the THz nanoscopy and LTEN images shown in (b) and (c), which are plotted on the same false-color scale for the measured THz peak electric field. Considering these, a slightly higher contrast is seen for the LTEN image. As described in [29], the photo-Dember effect, where the THz signal strongly depends on the mobility of the carriers [36–38], is the THz generation mechanism which is expected to be responsible for the LTEN signal obtained from a lightly p-doped InAs sample.

 

Fig. 2. Images recorded of a groove in an bulk InAs surface. (a): AFM image recorded over an area of 3.5 × 3.5 µm² with associated scale bar on right. (b): Peak THz Nanoscopy signal. (c): Peak LTEN signal. Associated color scale for (b), (c), is shown on the right of (c).

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To study the contrast deviations of the THz nanoscopy and the LTEN techniques further, Fig. 3(a) and (b) show the two images shown in Fig. 2(b) and (c) superimposed onto their corresponding topography. Although the material is uniform bulk InAs, the strength of the signal will be highly dependent on the tip-sample distance, which may change slightly as the tip scans over a rapidly, changing height profile - and with the presence of extra reflections. It can be seen in Figs. 2(b), 2(c) and 3(a), 3(b) that both images exhibit a bright band inside the groove. This narrow crevice may generate extra reflections and/or additional field enhancement in the vicinity of the tip. However, once the tip scans past the groove and returns to a flat region, we observe remarkably different signal behavior. This can be seen clearly in Fig. 3(c), which shows line cuts perpendicular to the groove, averaged over a uniform stripe with a width of 1.6 µm. The THz nanoscopy signal shows a local maximum just before and after the groove, with a decrease in magnitude in the flatter regions. Meanwhile, the LTEN signal is strongest far from the groove and exhibits local minimum at the rightmost groove edge.

 

Fig. 3. (a): THz nanoscopy and (b): LTEN image superimposed on the AFM topography. (c): Projections onto a plane perpendicular to the groove of the data in a center region of the image (1.6 × 5 µm²).

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Although a rigorous theoretical approach to tip-based LTEM is yet to be developed, the differences in signal behavior between THz nanoscopy and LTEN relative to local topography emphasize that the two techniques may yield complementary information regarding the local charge carriers and quality of sample material. More specifically, the nanoscale LTEN technique is anticipated to be highly sensitive to the mobility of the carriers in the subsurface of the sample responsible for the THz generation [29,37], while the THz nanoscopy is expected to be predominantly sensitive to the local scattering properties of the surface due to its dielectric function, topography or even surface contamination. We expect that the latter is responsible for the drop in signal for the THz nanoscopy images at the rightmost side of the groove. Therefore, this also suggests that LTEN is a more suitable technique for characterizing charge carriers in a material with a photo-Dember response in the presence of topographical defects or surface impurities.

3.2 Higher harmonics

In early s-SNOM results using mid-IR sources, Knoll and Keilmann demonstrated the ability to discriminate the near-field signal from the background to a greater extent when referencing to a higher harmonic of the tip oscillation frequency [17]. Here, we evaluate the near-field confinement of the THz fields obtained from THz nanoscopy and nanoscale LTEM for different harmonic orders.

Figure 4 depicts the same groove as in Figs. 2 and 3, with the THz peak LTEN signal recorded with the lock-in referenced to the 1^st, 2^nd, 3^rd, and 4^th harmonics of the tip tapping frequency. We observe a sharpening of the imaged feature when the harmonic is increased; this suggests an increased spatial resolution, but at the expense of a decrease in the signal-to-noise with increasing harmonic order. The contrast at the 4^th harmonic is very small although it should be noted that the THz peak field signal here is still above the noise floor. The THz nanoscopy images recorded at higher harmonics (not shown here) show a similar trend. Evidently, for samples with features in this size range (∼100 nm or larger), the 2^nd or 3^rd harmonic are acceptable choices for sufficient background suppression.

 

Fig. 4. Recorded LTEN images from 1^st to 4^th harmonic.

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Although no image contrast for either technique was observed at harmonics higher than the 4^th, it was still possible to record THz electric fields for higher harmonics. To study the limitations of the measurable harmonics, a fixed sample position in the upper left corner of Fig. 2 was chosen since the topography here is flat and both the LTEN and the THz nanoscopy signals are relatively high. Figure 5(a) shows the waveforms of the THz pulses recorded at the lowest (solid curves) and the highest (dotted curves) harmonic orders; these were the 6^th harmonic for the THz nanoscopy measurement (blue curves) and the 10^th harmonic for the LTEN measurement (red curves). We note a phase shift of approximately π/2 when demodulating to the highest possible harmonics in both techniques; subsequent experiments with different tips did not reproduce this phenomenon, so we attribute it to variations in tip shape and size which affect the resonance response and tip-field coupling. We plot the peak-to-peak values of the THz pulses recorded at all measured harmonic orders in Fig. 5(b). An approximately exponential decay is observed for both LTEN and THz nanoscopy, with a slightly faster decay for the latter. The black curves in Fig. 5(b) represent our theoretical prediction, as discussed below.

 

Fig. 5. (a): Waveforms of the lowest (solid lines) and highest (dotted lines) detectable harmonics for LTEN (red) and THz nanoscopy (blue). (b): THz peak-peak signal of all detectable harmonics.

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Confinement of the THz electric field below the AFM tip is a key element for obtaining an image resolution that is comparable to the diameter of the tip. To estimate the THz field confinement, we measured approach curves on the same fixed sample position, in which the LTEN or THz nanoscopy peak electric fields are measured as the tip-sample distance z is increased. A pure near-field signal should decay exponentially in the region where 0 ≤z ≤ r_tip, and eventually taper off to a noise-limited background when the tip is far enough away from the sample surface. This behavior is shown as an example for LTEN at the 1^st (red) and 5^th (gray) harmonic in Fig. 6(a). The field confinement is defined as the 1/e width of the approach curves; double-exponential fits are exploited for an experimental estimation of this parameter (the solid lines). The double exponential fits represent both a quickly decaying term and a slowly decaying term, a reasonable method for lower harmonics (1^st, 2^nd) where a slowly-decaying background is observed. However, this background is effectively suppressed at higher harmonics, leaving only the quickly decaying near-field signal.

 

Fig. 6. (a): Double exponential fits (black solid lines) to the approach curve recorded for the 1^st (red dots) and 5^th (gray dots) harmonics of the LTEN peak field signal together with the associated 95% confidence interval (shaded areas). The dashed line black line shows the 1/e-level of where there field confinement is estimated. (b): Field confinements estimated as shown in (a) from all harmonic approach curves for LTEN and THz nanoscopy. Error bars are estimated from the 1/e-level of the confidence intervals.

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Using these fits, we determine the 1/e width and the associated error bars from the 95% confidence intervals, shown as the shaded areas in transparent colors in Fig. 6(a). These are shown for both the THz nanoscopy (blue bars) and LTEN (red bars) in Fig. 6(b). For THz nanoscopy, the background noise approaches the 1/e level for the 4^th harmonic, yielding a significant error bar for the 1/e width. For this reason, it was not possible to measure the field confinement for THz nanoscopy at higher harmonics. In both cases, the confinement tightens for higher-harmonic demodulation, but we observe the 1/e width to be significantly smaller at all harmonics for LTEN than for THz nanoscopy. We interpret this as an illustration of one of the key advantages of the nanoscale LTEN technique: having the THz source within the sample, and hence, in a much closer proximity to the tip, results in a much stronger coupling of the THz field to the tip.

The confinement estimates shown in Fig. 6(b) are listed in Table 1. For the 2^nd harmonic measurement for the LTEN signal, the field confinement is already comparable to the expected tip diameter (∼ 20 nm); this is the case for the 3^rd harmonic measurement of the THz nanoscopy signal. The tightest field confinement of 11 nm is observed for the 5^th harmonic of the LTEN signal; this is well below the expected tip diameter, even considering the error bars from the confidence interval (8 nm / 17 nm), which here are heavily influenced by the low signal-to-noise.

Table 1. Field confinements estimated from the 1/e-widths.

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3.3 Theoretical approach

With the approach curves obtained above for THz nanoscopy and LTEN, we develop a theoretical framework to explore the impact of the different nature of the two methods. An electrostatic approach can be used to analytically describe the near-field interaction between the probe, sample, and incident radiation. The FDM [33] approximates the AFM tip as a conducting spheroid illuminated by a uniform electric field as illustrated in Fig. 7. The analysis below is based on the FDM as reported in [33].

In the far-field, the scattered electric fields are proportional to the incident field i.e. ${E_{sca}} \propto {\alpha _{eff}}{E_{0}}$, where ${\alpha _{eff}}$ is the effective polarizability

 

Fig. 7. Conductive spheroid approximation used within the FDM (solid line) and conductive sphere approximation used in the PDM [2] (dashed line). Selected parameters of the FDM include the spheroid semi-major axis L, the apex radius of curvature R, the tip-sample separation z, and the magnitude of the uniform external field E_inc.

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The expression for ${\alpha _{eff}}$ contains the sample material’s frequency-dependent dielectric function within the parameter $\beta = \frac{{\varepsilon (\omega ) - 1}}{{\varepsilon (\omega ) + 1}}$ and is therefore, in general, complex. Previous applications of these analytical models have been limited to a single frequency, with a continuous-wave light source in mind. We demonstrate the FDM’s utility for predicting a broadband response as appropriate for pulsed THz sources in a THz-TDS system. The dielectric function of our sample (lightly p-doped InAs, $n \approx 3 \times {10^{16}}c{m^{ - 3}}$) can be described with the Drude model,

Tunable parameters are introduced when describing the dimensions of the ellipsoid which represents the AFM probe, as well as the ellipsoid’s charge distribution, induced by the external THz field. These parameters are not known a priori; indeed, it has been shown that the dimensions of the hypothetical ellipsoid do not correspond directly with the tip shank geometry, in general [33]. For a THz nanoscopy experiment with the known tapping amplitude of 103 nm as used above, we determine through manual tuning that the FDM very accurately recreates the THz nanoscopy approach curves using values of L = 485 nm, R = 20 nm, and g = 0.9e^0.06i. A small complex contribution is included in g to describe the slight phase shift induced by radiation resistance and finite probe conductivity, as in prior studies [33]. This agreement is shown in Fig. 8(a). Here, the fits based on our model are shown as the solid lines together with the corresponding measured data points.

 

Fig. 8. Theoretical approach: Experimental (dots) and FDM predicted (solid lines) 1^st, 2^nd, 3^rd, and 4^th harmonic-demodulated approach curves are shown for THz Nanoscopy (a) and LTEN (b). These curves, representing the peak amplitude of the THz waveform in the time domain, are all normalized to peak in-contact signal at 1st harmonic demodulation. For superior fits which recreate the approach curves’ decay rates and the higher harmonic signals’ relative strengths, values of L = 485 nm and R = 20 nm are used for the THz Nanoscopy curves, while L = 305 nm and R = 10 nm are used for the LTEN curves.

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In addition to expanding the utility of this model to apply for pulsed THz sources rather than single-frequency sources, we also examine the case of nanoscale LTEN, where the THz pulses emitted by our sample via the photo-Dember effect [29]. We again find the FDM provides accurate predictions of the emitted THz signal, both as a function of harmonic demodulation order (Fig. 5(b)) and as the tip-sample distance is increased (Fig. 8(b)). However, the ellipsoid fit parameters must be adjusted from the THz Nanoscopy case in order to properly recreate the behavior of the LTEN approach curves. We find that the values L = 305 nm and R = 10 nm are best for optimizing the fits in Fig. 8 (using the same value for g as noted above). We note that both of these values are smaller than those required to describe the THz nanoscopy measurement in Fig. 8(a). This apparent tip sharpening and shortening effect is likely due to the fact that the LTEN experiment involves a nonlinear optical interaction at the tip apex. The THz nanoscopy interaction involves purely elastic scattering and is linear with the electric field; however, LTEM is a nonlinear process involving frequency conversion from the input 800 nm pulse to the outgoing THz pulse. It has been shown that the THz field strength from the photo-Dember effect in InAs is linear with the incident NIR power (i.e., a quadratic nonlinearity) when observed outside a s-SNOM configuration [41]. In a tip-based approach, the complicated form of ${\alpha _{eff}}$ introduces an additional nonlinear relationship between the input power (linear with carrier density) and the strength of the coupling between the subsurface THz dipole to the AFM probe. However, we have determined to the extent of our measurement precision that the LTEN electric field strength and the NIR input power are approximately linearly related below a saturation threshold (about 50mW) [29]. We perform our experiments at power levels well below this limit, and therefore suggest that our nanoscale LTEM results can in fact be understood in the context of the inherently linear FDM by considering it a second-order nonlinear process. Therefore the best-fit spheroid geometry in the THz nanoscopy case should also be accurate in representing the LTEN case if we instead examine the behavior of the squared E field; that is, we find the ideal ellipsoid parameters by performing the THz nanoscopy fits, and the best LTEN fit parameters will be those where the E-field confinement most closely approximates the THz nanoscopy E-field­-squared confinement. We substantiate this notion by performing frequency-domain finite-element (FD-FEM) simulations at an example frequency of 1 THz (chosen because 1 THz is a dominant frequency in both our THz nanoscopy and LTEN signal pulses); results are shown in Fig. 9. Here, we show the vertical component of the electric field |E_z|for a tip of R = 10 nm and L = 305 nm (Fig. 9(b)), and the vertical squared electric field |E_z|² for a tip of R = 20nm and L = 485 nm (Fig. 9(c)), extracted from simulations. We extract the line profiles as shown in Fig. 9(a) and observe that the confinement profiles for these two cases are nearly identical in the highest-field region very close (within 10 nm) to the tip apex, where most of the near-field behavior originates. This similarity supports the idea that the differing fit parameters originate from the dominantly 2^nd-order nonlinear THz generation mechanism in the case of nanoscale LTEM.

 

Fig. 9. (a): Line profiles of 1 THz vertical E-field enhancement from FEM simulation under conductive tip; R = 10 nm and L = 305 nm ellipsoid E_z indicated by the red solid line, R = 20 nm and L = 485 nm E_z² indicated by blue solid line. Profiles are normalized to same height to depict confinement width. (b) and (c) show R = 10 nm, L = 305 nm and R = 20 nm, L = 485 nm simulations, respectively. Line profiles are extracted along white dashed lines. X-Z axes are shown in white.

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4. Conclusions

We have experimentally demonstrated a back-to-back comparison of THz nanoscopy and LTEN, using p-InAs as a prototypical sample for obtaining both THz time-domain waveforms and images. We have obtained a slightly better contrast for the LTEN images and detected measurable signals at harmonic orders as high as the 10^th harmonic of the tapping frequency. From the recorded approach curves, we have also extracted the field confinement of the measured THz peak field signals and measured a field confinement down to 11 nm (8 nm / 17 nm) for the LTEN signal modulated by the 5^th harmonic. We present a mathematical approach, based on the FDM, for estimating the spatial electric field confinement in the region of the tip apex. For this, we modify the FDM to account for our broadband THz signals. We also adapt this model to the case of a LTEN measurement. From the parameters obtained by fitting the model’s predictions to our measurements for the case of LTEN, we observe a “tip sharpening” effect, which we attribute to the different experimental geometry and the nonlinear THz generation process for LTEN versus the more commonly employed case of THz nanoscopy.