We use a combination of a scattering-type near-field infrared microscope with a free-electron laser as an intense, tunable radiation source to spatially and spectrally resolve buried doped layers in silicon. To this end, boron implanted stripes in silicon are raster scanned at different wavelengths in the range from 10 to 14 µm. An analysis based on a simple Drude model for the dielectric function of the sample yields quantitatively correct values for the concentration of the activated carriers. In a control experiment at the fixed wavelength of 10.6 µm, interferometric near-field signals are recorded. The phase information gained in this experiment is fully consistent with the carrier concentration obtained in the spectrally resolved experiments.
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The quantitative and non-destructive analysis of buried doping profiles in semiconductors on the nanometer length-scale is an important challenge in nanometrology. A standard method used in semiconductor technology is secondary ion mass spectroscopy (SIMS), which, however, provides only depth resolution and is highly destructive. Electrical techniques providing lateral resolution as well are usually based on scanning probe techniques, such as scanning spreading resistance microscopy (SSRM) or scanning capacitance microscopy (SCM) [1,2]. In contrast to SIMS, these methods probe the carrier concentration, rather than the concentration of implanted ions. Since the activation of dopants during annealing processes is often not known accurately, SSRM and SCM techniques are extremely important characterization techniques. Optical near-field methods, also probing the carrier concentration, are even less invasive and do not require complex sample preparation. Good contrast due to different carrier concentrations can be especially expected at infrared wavelengths, in the vicinity of the free-carrier plasma frequency.
In 1997 Lahrech et al. showed that near-field optical microscopy (SNOM) is capable to probe changes in the charge carrier concentration in silicon . They explained this by a change of the refractive index between implanted and unimplanted Si, but were not able to extract the local carrier density from that optical contrast. Later on, Huber et al.  succeeded in determining the carrier concentration in a pnp-transistor by utilizing the contrast in optical phase and amplitude. However, both experiments were restricted to the fixed CO2 laser wavelength at 10.6 µm. Especially carrier densities below 3 × 1018 cm-3 cannot be probed using this wavelength due to a lack of contrast in the amplitude and phase (see Fig. 1 ).
Similar experiments with highly doped silicon were performed by Samson et al. utilizing an optical parametric oscillator which emits in the near infrared . Recently, the capabilities of near-field optical microscopy to probe the carrier density in InP nanowires have been demonstrated by Stiegler et al. . Furthermore, near-field microscopy has been extended to the terahertz frequency range, where the carrier concentration of uniform semiconductor samples has been measured [7, 8] and also doping profiles of transistors have been imaged .
In this paper we deduce the charge carrier concentration of sub-surface boron doped silicon by combining a scattering-type near field optical microscope (s-SNOM)  with the tunable IR-laser light source at the Forschungszentrum Dresden-Rossendorf . Near-field images are measured while the wavelength of the free-electron laser is tuned in the relevant range around the plasma resonance, i.e. from 10 to 14 µm for our sample. An analysis based on the Drude model allows for an accurate determination of the carrier concentration in the doped regions. Our results are corroborated by data obtained from interferometric near-field measurements at a fixed wavelength of 10.6 µm.
In order to understand the origin of phase and amplitude contrast in s-SNOM measurements one has to refer to the theory that describes the amplitude of the light scattered off the tip. The commonly accepted dipole model describes the interaction of a polarization in the tip, which is modeled as a conducting sphere of radius r, with an image charge induced in the sample . Here we use a recently proposed extension of this model, in which the indirect illumination of the tip by radiation reflected from the surface of the sample is taken into account . According to this model, the near-field scattering field E is given by
The dielectric function εs of a doped semiconductor is sufficiently described by the Drude model for free carriers , i.e.Eqs. (2) and (3)), yielding different intensities of the scattered signal. Thus, an optical contrast arises whenever areas of different dielectric functions are probed. In particular, the highest signals can be expected for the real part of the dielectric function approaching the value of −1. This is valid for z→∞ while the resonance will be found at slightly more negative values for finite tip-sample distances .
Sample preparation and experimental methods
The sample resonance was targeted to lie around 10.6 µm in order to allow an easy verification of the FEL-measurements using a fixed-wavelength CO2 laser operating at this wavelength. Therefore an n-type Si(100) wafer (resistivity of 10 Ωcm) was implanted with boron ions at a dose of 6.8 × 1014 cm-2 which corresponds to a carrier concentration of 6 × 1019 cm-3 at the peak of the implantation profile. The implantation and annealing behavior of Si:B is well established [16, 17]. The peak of the depth profile can be shifted by altering the implantation energy. We used an implantation energy of 30 keV to obtain the maximum concentration at 100 nm below the sample surface. The implantation profile simulated using Crystal-TRIM  is shown in Fig. 2 . The peak carrier concentration is about one order of magnitude larger as compared to both the concentration at the surface and the concentration in a depth of 200 nm. An implantation mask was used to create 4 µm wide stripes of doped silicon separated by 12 µm spacing of undoped silicon in (see Fig. 3 ). Subsequently the sample was tempered using rapid thermal annealing at 1000 °C for 60 s in a dry nitrogen atmosphere. The preparation gave rise to a tiny height difference between implanted and unimplanted silicon of approximately 2 nm, measured by AFM, where the implanted region is lower due to sputtering during implantation.
Near-field measurements were performed using a home-built s-SNOM (see Fig. 3), operated in true non-contact mode. The cantilever oscillated at its resonance frequency of ω0 = 140 kHz with an oscillation amplitude of 60 nm. p-polarized infrared light at wavelengths tuned from 10 to 14 µm was focused onto a PtIr tip (PPP-NCLPt from NanoAndMore GmbH) used as the scatterer. The average power at the tip was approximately 10 mW. The angle between the k-vector of the beam and the tip axis is about 20°. Backscattered light was detected in reflection mode using a mercury cadmium telluride detector (Judson) and demodulated at the 3rd harmonic of the cantilever oscillation frequency to suppress the background components. For the measurements at 10.6 µm a pseudo-heterodyne setup similar to the one described by Ocelic et al.  was used in addition. For measurements the sample was scanned over an area containing at least one stripe of doped silicon, and the wavelength was changed stepwise every six lines. After normalizing the optical near-field amplitude signal to the incident power, the contrast was calculated as the amplitude signal difference between implanted and unimplanted silicon. In the following both these amplitude differences and phase information are displayed for the 3rd harmonic, since this reflects the strongest near-field signal free of any artifacts.
Results and discussion
Figure 4 illustrates the 3rd harmonic near-field contrast measured interferometrically over a 5 µm × 10 µm x-y-scan area with the CO2 laser fixed at 10.6 µm wavelength. We clearly identify the B-doped Si–area appearing as a dark stripe along the x-direction in this image. While the amplitude difference is weak (roughly 20 percent, not shown), the phase difference actually yields the contrast. In fact, the darker near-field optical contrast arises through a phase shift of ~50° ± 15° between the two different Si-regions, in excellent accordance with the calculated predictions (see Fig. 1). Moreover, this phase contrast allows quantifying the charge carrier concentration to (3.5 ± 0.5) × 1019 cm-3.
In our spectrally resolved measurements involving the FEL, only the optical amplitude of the near field signal is recorded non-interferometrically. The picture on the left in Fig. 5 is composed of lateral scans at 20 different wavelengths. The bright stripe represents the implanted silicon, whereas the unimplanted silicon appears dark. Furthermore the stepwise change of the wavelength from 11 µm to 13 µm (from left to right) is recognizable by the vertical lines of mismatching signals. While the contrast vanishes for short wavelengths, it reaches a maximum at around 13 µm. From these measurements the relative contrast is extracted and plotted versus wavelength (Fig. 6 ). Now, the theoretical cross-section difference was fitted using Eqs. (1), (2) and (3) to match the obtained data. We know from our experience that the tip radius is roughly 100 nm for a worn tip and used the experimental oscillation amplitude of 60 nm. The data was fitted by varying the carrier concentration N. Although the model assumes a homogenously doped material between the undoped regions, a very good agreement is observed as seen in Fig. 6. The main criterion for the determination of the carrier concentration was the wavelength where the contrast vanishes completely, since this point shifts with the concentration as depicted in the figure.
Furthermore, the wavelength dependence of the near-field signal is characteristic for a certain carrier concentration, especially near the plasmon resonance. This is shown in Fig. 6 for three different concentrations (dotted and solid lines). The combination of both aspects allows the estimation of the carrier concentration even without the phase information. Hence, we determined the carrier concentration to be (3.7 ± 0.3) × 1019 cm-3. This fits the expectations very well.
Finally we estimated the carrier concentration based on the annealing behavior of boron in silicon. Here we used the results from Solmi et al.  who applied secondary ion mass spectroscopy to analyze the implantation profile before and after annealing of such a boron implanted silicon sample. For a dose of 5 × 1014 cm-2 they reported a concentration of approx. 2 × 1019 cm-3 after annealing at 1000 °C for 10 sec. Furthermore they observed a slight broadening of the profile when the annealing time is increased. Thus the carrier concentration can be estimated to lie between 3 × 1019 cm-3 and approx. 5 × 1019 cm-3 for a dose of 6.8 × 1014 cm-2 which is consistent with our experimentally obtained data. We note that the good agreement is already achieved using the analysis with the simplified assumption of a 30 nm thick layer that is homogeneously doped in the direction perpendicular to the sample surface .
In conclusion, we have measured the wavelength dependent amplitude contrast between boron-implanted and unimplanted silicon using a widely tunable free-electron laser. With our IR scattering SNOM technique in combination with theory, we determined the carrier concentration of our sample in very good agreement with the value expected from ion implantation and subsequent annealing. We further confirmed this result by interferometric measurements of the optical phase contrast. In particular, these data imply that near-field optical microscopy allows to quantify the carrier concentration of an in-depth distribution at least 100 nm below the sample surface. Moreover, in combination with the free electron laser this technique is also applicable to probe much higher or lower dopant concentration simply by adapting the wavelength to match the resonances. This allows us to quantify the carrier concentration over a wide range of several orders of magnitude.
We gratefully acknowledge the funding by the German Science Foundation DFG (projects HE 3352/4-1 and EN 434/22-1). Furthermore, we thank the whole ELBE team for their dedicated support.
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