Quantification of Carrier Density Gradients along Axially Doped Silicon

Article Cite This: ACS Photonics 2019, 6, 1744−1754

pubs.acs.org/journal/apchd5

Quantification of Carrier Density Gradients along Axially Doped Silicon Nanowires Using Infrared Nanoscopy Lena Jung,†,‡ Julian Pries,†,‡ Tobias W.W. Maß,†,‡ Martin Lewin,†,‡ Dmitriy S. Boyuk,§ Amar T. Mohabir,§ Michael A. Filler,*,§ Matthias Wuttig,†,‡ and Thomas Taubner*,†,‡ †

I. Institute of Physics (IA), RWTH Aachen University, 52056 Aachen, Germany Jülich Aachen Research Alliance - Fundamentals of Future Information Technology (JARA-FIT), 52056 Aachen, Germany § School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, United States Downloaded via UNIV PARIS-SUD on August 5, 2019 at 05:57:21 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

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S Supporting Information *

ABSTRACT: Doped semiconductor nanostructures are interesting for the fabrication of nanoscale electronic and photonic devices. Here, we use scattering-type scanning near-field optical microscopy (s-SNOM) to characterize axial carrier density gradients in phosphorus-doped silicon nanowires. We quantitatively determine the carrier density and length of the doped segment as well as the functional form of the charge carrier gradient in the transition region between doped and nominally undoped segments. These measurements are enabled by understanding and accounting for the influence of the native oxide on the near-field optical contrasts in the transition region. Our results are supported by correlative energy dispersive X-ray spectroscopy (EDS) measurements. This work demonstrates the ability of s-SNOM to directly probe nanoscale charge carrier density transitions through thin surface layers, a capability that is important for a variety of doped semiconductor systems. KEYWORDS: near-field microscopy, s-SNOM, doped semiconductor, nanowire, charge carrier

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native oxide or other surface layers exist. In s-SNOM, highly confined near-fields are generated by laser-illumination of a metallized atomic force microscopy (AFM) tip. When the tip and sample are brought into close proximity, there is a change in the backscattered light due to tip−sample near-field interactions. s-SNOM measures the optical amplitude and phase of the backscattered light using interferometric detection.24 The near-field interactions are sensitive to the local dielectric function of the sample and, in the mid-infrared range, depend on molecular vibrations, free charge carriers, and phonon excitations.25−33 The accessible charge carrier density range is only limited by the availability of suitable light sources in the range close to the material’s plasma edge, where sSNOM is most sensitive (Figure S1). For the quantum cascade laser system used within this work (ν = 900 cm−1 to ν = 1875 cm−1), charge carrier densities between N ≈ 0.6 × 1019 cm−3 and N ≈ 30.9 × 1019 cm−3 are accessible, depending on the material and type of doping. The ranges for some common semiconductor materials are summarized in Table S1. To reach lower charge carrier densities, THz laser sources are desired. Despite some challenges, promising progress has been obtained in the last years and several s-SNOM measurements using THz sources have been reported.34−38 Independent of

sound understanding of semiconductor nanostructure process−structure−property relationships is essential to engineer future electronic and photonic devices.1−7 In this regard, there is a need for characterization techniques to determine the spatial arrangement of charge carriers with nanoscale resolution. A variety of high-resolution techniques e.g. atom probe tomography (APT)8,9 or energy dispersive Xray spectroscopy (EDS)10−12can locally determine dopant atom concentration. However, there are few techniques capable of mapping the density of activated charge carriers contributing to electrical conductivity.13−17 Most of these methods are limited in their applicability by several factors. Electron holography,15 for example, requires vacuum conditions. Scanning spreading resistance microscopy13 relies on a conductive substrate. Kelvin probe force microscopy14,16 can only yield quantitative results at the surface, which is, under ambient conditions, likely to be covered by a native oxide layer. Scattering-type scanning near-field optical microscopy (sSNOM)18,19 is a promising optical method that combines the high resolution of scanning probe techniques with the sensitivity of infrared spectroscopy to free charge carriers. It is a nondestructive technique capable of measuring nanostructures at ambient conditions. Carrier properties can be obtained without the need for electrically contacting a sample. Furthermore, subsurface imaging has been demonstrated20−23 and makes this technique applicable even in situations where © 2019 American Chemical Society

Received: March 26, 2019 Published: June 20, 2019 1744

DOI: 10.1021/acsphotonics.9b00466 ACS Photonics 2019, 6, 1744−1754

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density on amplitude and phase signals within the finite dipole model can be found in the Supporting Information (Figure S1). The Si nanowires studied here are grown via the bottom-up Au-seeded vapor−liquid−solid (VLS) mechanism using disilane (Si2H6) and phosphorus trichloride (PCl3) as the Siand P-containing precursors, respectively. Additional information can be found in the Methods section and elsewhere.52 The resulting i-n-i Si nanowire morphology is depicted schematically in Figure 1a (top). Nanowires grown with the same

the spectral range used, the tip radius (∼ 25 nm), not the laser wavelength,27,32 limits the lateral resolution of s-SNOM. Several reports reveal an increased interest in s-SNOM investigations of doped semiconductors as sample material and resonant probes39 and underscore the potential of s-SNOM to map local charge carrier concentration.17,32,33,40−46 Timeresolved investigations of photoexcited carriers47,48 as well as studies discussing the role of growth conditions and postgrowth annealing of doped semiconductor nanostructures on near-field signal exist.49,50 However, many important aspects of doping in semiconductor nanostructures, such as the carrier density profile in the transition region between regions of different doping, remain poorly understood. Prior work estimated the transition width from the change in near-field amplitude from a single wavelength image.49 However, as we will illustrate later, the near-field signal does not necessarily increase with increasing charge carrier density.33,42 Instead, quantifying the transition region requires measurements of the complex-valued near-field signals at multiple frequencies. Furthermore, doped semiconductors are usually covered with a thin native oxide layer. When extracting the charge carrier density in the spectral vicinity of oxide absorption, the question arises how this influences the near-field signals. This influence has not yet been studied. The goal of this study is to quantitatively extract the charge carrier density profile, including any axial gradients, in silicon nanowires containing phosphorus-doped and nominally undoped segments. We combine spectroscopic measurements and theoretical modeling to answer the following important questions: (a) How does a native oxide layer influence the measurement of local charge carrier density? (b) How can we quantify a charge carrier density gradient, especially when a native oxide layer is present? Our results are supported by correlative electron energy loss spectroscopy (EELS) and energy dispersive X-ray spectroscopy (EDS) measurements with a scanning transmission electron microscope (STEM).

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RESULTS AND DISCUSSION For the characterization of doped semiconductors using infrared spectroscopy, the dielectric function ε(ν) connects the optical properties with charge carrier properties. We describe the dielectric function of Si with the Drude model (Methods section). Using few assumptions (cf. Methods section), plasma frequency and dielectric function only depend on the charge carrier density. To analyze the carrier properties from infrared near-field spectra on the nanometer scale, we apply an s-SNOM system (see Methods section). For the prediction of near-field contrasts, theoretical modeling of a dipole interaction of tip and sample can be used. The finite dipole model used in this study can quantitatively predict the optical near-field amplitude s2 and phase φ2 signals from the dielectric function ε(ν) of the sample.51 Further details about the model can be found in the Methods section. s-SNOM is most sensitive to the wavenumber range slightly below the plasma frequency of a doped semiconductor. In this range, the near-field signals reveal characteristic resonance curves with a Lorentzian peak in phase and a derivative line shape in amplitude (cf. Figure S1). Since the plasma frequency shifts to larger values for higher charge carrier densities, the characteristic near-field resonance curves also shift to higher frequencies. This shift can be used for a determination of charge carrier densities by performing spectroscopic s-SNOM measurements. Further information about the impact of the charge carrier

Figure 1. (a) Schematic of the Si nanowire i-n-i structure (top) and AFM topography image (bottom) of a nanowire with diameter d ∼100 nm lying on a Ge substrate. (b) Optical amplitude and (c) phase images of the nanowire shown in (a). The optical signals are recorded at ν = 1400 cm−1 and normalized to the signal of the Ge Ge substrate (sGe 2 and φ2 ). Black arrows in the amplitude image mark the position of the line profiles extracted and shown in Figure 4.

dimensions, but with two different doping levels in the nregion, are investigated in this work. Nanowires are postsynthetically transferred to a Ge substrate via drop-casting. The frequency-independent optical response of Ge enables normalization of the s-SNOM signals. Figure 1 shows the topography image of a representative Si nanowire lying on a Ge substrate (Figure 1a), as well as the corresponding optical amplitude s2 (Figure 1b) and phase φ2 (Figure 1c) images at a single frequency (ν = 1400 cm−1). The optical signals are normalized to the signal of the Ge substrate (s2Ge and φ2Ge). The topography image reveals a largely homogeneous nanowire with the Au seed particle located on the right-hand end. A topography line profile is shown in Figure S2. The diameter of the nanowire slightly decreases from 110 nm at its bottom (left end) to 100 nm at its top (right end). The amplitude and phase images clearly show the n-doping in the center segment. The doped segment appears 1745


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Figure 2. (a) Optical amplitude (black) and phase (red) signals extracted from the middle of the doped segment (marked by the blue dot in the inset) as a function of wavenumber. (b) Calculated spectroscopic near-field response employing a finite dipole model for doped silicon with a charge carrier density of N = 1.5 × 1020 cm−3 (solid line) and employing a multilayer finite dipole model with a 2 nm silicon dioxide layer (dashed curve). Gray dash-dotted lines are guides to the eye that show the correspondence between spectral features observed in experiment with those stemming from silicon dioxide in the calculation. Corresponding experimental data (c) and calculations (d) for a lower doped nanowire and a charge carrier density of N = 0.7 × 1020 cm−3. While the plasmon resonance of the free charge carriers is red-shifted for the lower doped nanowire, the position of the features arising from the oxide is not changed.

curves in Figure 2b,d represent the calculated amplitude (black) and phase (red) for a charge carrier concentration of N = 1.5 × 1020 cm−3 and N = 0.7 × 1020 cm−3, respectively. The Drude calculation confirms a broad near-field resonance for the lower doped nanowire within the investigated frequency range (Figures 2c,d), while for the higher doped nanowire the resonance is shifted to slightly above the investigated frequency range (Figures 2a,b). The resonance in the latter case is only indicated by an increase in phase toward higher frequencies. Both amplitude and phase are nicely reproduced by the Drude calculations for high frequencies for both nanowires, even if the experimental resonances are still broader than the calculations suggest. A source for this deviation might be an inhomogeneous radial distribution of charge carriers. This could be due to a redistribution of charges at the native oxide/silicon interface or inhomogeneous doping. Averaging over this inhomogeneous distribution, as it is done in the experiment, could result in a further broadening of the experimental spectra compared to the calculations. In contrast to the high frequency range, the experimental results do not match the Drude calculation for frequencies below ν ≈ 1300 cm−1; here, an additional contribution leading to the sharper spectral features observed at low frequencies needs to be considered. A native oxide layer with a thickness of a few nanometers is expected since these nanowires have been exposed to ambient conditions.55,56 We confirm the presence of the oxide by obtaining an EELS oxygen map of a nanowire cross-section (Figure S3). We account for the phonon near-field resonances of the native oxide (dashed lines in Figure 2b)57,58 using our

bright in the amplitude signal, while only its boundaries appear bright in the phase signal. Among all investigated nanowires on the sample, we observe similar amplitude and phase contrasts. Spectroscopic measurements are needed to quantitatively determine charge carrier densities and gradients as well as identify the surface oxides that potentially alter the near-field signals. Figure 2 shows s-SNOM measurements performed for a series of excitation frequencies in the range between 900 and 1875 cm−1 for two representative nanowires with different doping levels in the n-region. The near-field amplitude (black) and phase (red) spectra are extracted from the middle of the doped segment (blue dot in the inset of Figure 2a) for a nanowire with higher (Figure 2a) and lower (Figure 2c) doping in the phosphorus-doped region. Both nanowires exhibit a series of sharper features between 1100 and 1250 cm−1 in amplitude and phase, as denoted by gray dash-dotted lines. These features are at the same spectral position for both nanowires; however, the nanowires differ, especially their phase response, in the high frequency range. While the phase slightly increases toward high frequencies for the higher doped nanowire (Figure 2a), a broad peak with a maximum around 1400 cm−1 is observed for the lower doped nanowire (Figure 2c). These results suggest a broad resonance stemming from the Drude response of the free charge carriers (cf. Figure S1) overlaid in both cases by some additional dopant-independent contribution, which we will show stems from the native silicon oxide layer. A broad and thus quite weak Drude response is expected due to the low mobility of electrons in Si53 compared to other semiconductors as, e.g., InP or InAs.54 The solid 1746


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Figure 3. Calculated carrier density dependence of the near-field amplitude (a,c) and phase (b,d) signal evaluated at two different wavenumbers inside (a,c) and outside (b,d) of the silicon oxide phonon resonance. The solid lines are calculated with the bulk finite dipole model, the dashed lines with the multilayer finite dipole model including a 2 nm thin silicon dioxide cover layer. While the oxide influences the signal height, the positions of the peaks do not change between the calculations with and without the oxide (vertical gray dashed lines). Blue shaded areas mark the regions outside the sensitivity range for the given wavenumber, where the near-field signals barely change anymore within the experimental error.

recently developed multilayer finite dipole model59 (see Methods section) assuming a conformal silicon dioxide layer with a thickness of 2 nm surrounding the nanowire. Importantly, the positions of the calculated phase peaks (see dash-dotted gray lines) are consistent with the experimentally observed spectral features. Slight differences between the calculated and experimental amplitudes could be due to the fact that we used a SiO2 dielectric function for room temperature-sputtered silicon dioxide thin films from literature,57 which might be different from the native SiOx present on our nanowires. The higher frequency range is less affected by the phonon resonances of the oxide, despite a further broadening of the calculated amplitude Drude peak. The agreement between our calculations and experiment for higher frequencies permits us to extract carrier density values of N = 1.5 × 1020 cm−3 and N = 0.7 × 1020 cm−3 for the higher and lower doped segments, respectively. These values are consistent with earlier results obtained by Chou et al.60 in a far-field study. To get an even better fitting of theory and experiment, a vertical inhomogeneous charge carrier distribution could also be modeled in the future. Very recently, the multilayer approach using up to five layers59 has been employed for the detection of 5 nm thin surface states on the topological insulator (Bi0.5Sb0.5)2Te3 with s-SNOM.61 However, without additional insights, e.g., from other techniques or imaging of cross sections, a large parameter space for such a multilayer modeling opens up. Thus, there is a high risk of overinterpretation of the data. This is why we have not gone so far within the scope of our study. Nonetheless, our results underscore the potential of s-SNOM to probe the signature of an even thinner surface layer and a weak carrier response of the underlying nanowire simultaneously.

To better understand the influence of the oxide on the nearfield signals in the doped/undoped transition and enable the quantification of charge carrier density gradients in this region, we examine the near-field signals not only as a function of frequency (Figures 2 and S1) but also as a function of carrier density. Figure 3 shows the calculated near-field amplitude (black) and phase (red) with respect to the charge carrier density, for a wavenumber value inside (ν = 1225 cm−1, Figures 3a,c) and outside (ν = 1700 cm−1, Figures 3b,d) of the spectral regime influenced by the oxide phonons. The solid lines represent a calculation of bulk silicon using the finite dipole model, while a calculation that includes the oxide using the multilayer finite dipole model is shown with dashed lines. In both cases, the amplitude signal reveals a derivative lineshape with a dip for a certain charge carrier density, followed by a peak (gray dashed lines in Figures 3a,c). The phase peaks at a certain charge carrier density in between the dip and peak in amplitude (gray dashed lines in Figures 3b,d). For very low and very high charge carrier densities, the near-field signals barely change compared to the experimental error (cf. Methods section). Blue shaded areas mark these regions that are thus outside the sensitivity range for the given wavenumber. The amplitude signal is reduced by the oxide for all charge carrier densities (Figure 3a,c), most notably near ν = 1225 cm−1 (Figure 3a). The phase signal is strongly increased at the frequency within the spectral region influenced by the oxide phonons (Figure 3b) and is largely unaffected elsewhere (Figure 3d). Inclusion of the oxide does not change the charge carrier densities at which the highest amplitude and phase signals occur (gray dashed lines), meaning that the peaks in amplitude and phase (as a function of charge carrier density) are robust against any influence of the oxide. 1747


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knowledge of the exact dielectric function of the native oxide surface layer. The red data points in Figure 6a are the extracted carrier densities reproduced from Figure 5d. The blue dashed curves are guides to the eye, assuming a carrier density of N = 1.5 × 1020 cm−3 in the doped part as extracted from the spectroscopy presented in Figure 2 and using linear regression through the data points for the slope of the transitions. Notably, the data points are extracted without any assumption on the shape of the transition. From this, we can derive an almost linear relationship. Our investigated frequency range enables, using the phase peaks, the extraction of carrier densities between N ≈ 0.4 × 1020 cm−3 and N ≈ 1.4 × 1020 cm−3 (cf. Figure 5b), thus allowing only for a determination of the slope in the transition region within these carrier density values. Lower frequencies would be necessary to measure even lower charge carrier densities. However, we can still measure the width d50% (the width within which the charge carrier density has dropped to one-half of its maximal value) of each transition. We find that d50% ≈ 70 nm and d50% ≈ 190 nm for the bottom and top transitions, respectively. We attribute this difference in transition width to the kinetics of phosphorus atom delivery to and removal from the AuSi droplet. Upon initiating flow of the phosphorus-containing precursor (i.e., PCl3), there is a transient during which the AuSi droplet reaches its steady-state phosphorus atom concentration. Upon terminating the flow of the precursor, the phosphorus atoms remaining in the AuSi droplet are depleted either via incorporation into the nanowire, diffusion to the sidewall, or back reaction into the gas phase. This process of depletion is often referred to as the “reservoir effect”.10,62,63 Our measurements make clear, for the precursors and growth conditions used here, that the dopant transient is longest upon precursor termination. A follow-up study on the comparison of different growth conditions is of high interest for the nanowire growth community. The changes in growth rate and temperature, as well as the corresponding i-n-i morphology, are illustrated in Figure 6c. We determine the axial dopant atom profile via EDS in an effort to corroborate our s-SNOM measurements of the carrier density profile. Figure 6b shows the line profile of phosphorus atoms along the axis of a nanowire and clearly reveals the doped segment. The length of the doped segment (∼160 nm, indicated by vertical gray dashed lines) and differing steepness of the two transitions are in very good agreement with that determined from s-SNOM. A high background level, largely stemming from the overlap of the phosphorus peak with the tails of the nearby silicon and gold peaks (cf. Figure S5b), prohibits an accurate determination of the transition width with EDS. Nonetheless, the width in which the counts drop by 50% suggests that the gradient in dopant atom concentration is on the same order of magnitude as the carrier density gradient obtained from s-SNOM. It is important to note that s-SNOM probes free charge carriers while EDS probes dopant atoms. Thus, the transition details measured in each case could very well be distinct. It is well-known in semiconductor science that trap states exist at silicon/silicon oxide interfaces, resulting in a depletion region near the silicon surface.64,65 However, any such charge redistribution along the same axis as the doping gradient has remained largely unexplored.15 This interplay between dopant atoms and free charge carriers is an exciting opportunity for future investigations, especially for electrically contacted

For a gradual transition, such as between the differently doped segments of Figure 1, the carrier density axis in Figure 3 can simply be seen as equivalent to an axis marking the position along the nanowire. In a single wavelength image, a peak in phase can thus be assigned to a corresponding resonant charge carrier density N(x) at a certain position x along the nanowire. The comparison of calculations with and without including the oxide proves that this assignment can be done despite any uncertainty about the dielectric function of the native oxide. Figure 4 displays line profiles extracted from the optical amplitude and phase for the nanowire with higher doping at

Figure 4. Extracted amplitude and phase line profiles taken along the nanowire axis as indicated by black arrows in Figure 1b for several frequencies. The line profiles are normalized to the signal of the Ge substrate, marked by the gray base lines corresponding to s2/sGe 2 = 1 and φ2 − φGe 2 = 0, and vertically shifted by 1 (relative contrast units) to better visualize changes as a function of wavenumber. Gray arrows visualize the increasing distance of the peaks in phase and amplitude.

different frequencies, which can be used to quantitatively characterize the transition region between the doped/undoped segments. The line profiles are averaged ±15 nm from the nanowire centerline to exclude any distortion of the signals by the topography at the edges of the nanowire. The amplitude line profiles exhibit a single, broad peak at high frequencies corresponding to the doped segment. This peak splits into two peaks, whose separation increases, at lower frequencies (gray arrows). For the phase line profiles, all frequencies exhibit two peaks with increasing separation toward low frequencies (gray arrows); however, at ν = 1875 cm−1, these two peaks have largely merged. We now quantify the carrier density profile in the transition region between the doped and undoped segments by a multistep process outlined in Figure 5 for an exemplary frequency of ν = 1275 cm−1: The carrier densities yielding the highest phase signal are first determined using data analogous to that in Figure 3 at each frequency (Figures 5a,b). These carrier densities N are then cross-correlated with the peak positions x extracted from the phase line profiles shown in Figure 4 at each frequency ν (Figures 5c,d). The distinct nature of the peak maxima in the phase line profiles (Figure 4) allow their positions x to be extracted with high accuracy (±10 nm). As mentioned above, this extraction does not rely on the 1748


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Figure 5. Determination of the charge carrier density profile. (a) Calculated near-field phase signal in dependence of the charge carrier density as shown in Figure 3, here exemplarily shown for ν = 1275 cm−1. The carrier density giving the highest phase signal has been extracted (gray dashed line) for each investigated wavenumber and is plotted in (b). (c) For each phase line profile, as shown in Figure 4, the spatial positions of the phase peaks have been extracted. This is shown here exemplarily for the phase line profile for ν = 1275 cm−1 with dashed gray lines marking the spatial positions of the phase maxima. (d) Results obtained in (b) and (c) are combined (cf. continuations of gray dashed lines) and allow for an assignment of the charge carrier density for each of the extracted positions, yielding the plotted carrier density profile.

materials68−70 with carrier densities that result in plasma frequencies in the mid-infrared wavelength range.

devices where gating can allow for additional charge redistribution.

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METHODS Scattering-Type Scanning Near-Field Optical Microscope. A commercial s-SNOM system (neaspec GmbH) with a pseudoheterodyne detection scheme24 is used. Line-tunable cw quantum cascade lasers (Daylight Solutions) provide access to wavenumbers between ν = 1/λ = 900 and 1875 cm−1. All amplitude sn and phase φn signals are evaluated at the second demodulation order (n = 2) of the tapping frequency to extract the near-field signal without any background contributions. To get an estimate of the experimental error in s2 and φ2, measurements at exemplary wavenumbers have been repeated several times, as can be seen in Figure 2a, e.g., for 1000 cm−1. This estimation results in an error of ∼5% for the amplitude and ∼15% for the phase signals. While the error on the amplitude is already quite small, an improvement of the readout technology might be beneficial for a decrease of the error on the phase.71 PtIr-coated Si tips were used and operated with a tapping amplitude of ∼35 nm. The scanned area is 2 μm × 0.5 μm. Possible sample drift between measurements performed at different wavenumbers has been monitored and corrected using the edges of the nanowire in the topography images as reference points. Silicon Nanowire Growth. The silicon nanowires investigated were grown via the Au catalyzed vapor−liquid− solid (VLS) technique in a two-step process. First, the temperature of the Au covered substrate was ramped to and held at 620 °C for 5 min while maintaining a Si2H6 (Voltaix, 99.998%) pressure of 4 × 10−5 Torr to achieve Au droplet nucleation and ripening. The sample temperature was

CONCLUSION We have examined two important questions regarding the capabilities of infrared s-SNOM to characterize the charge carrier density distribution in doped semiconductor nanostructures: (a) How does a native oxide layer influence the measurement of local charge carrier density? (b) How can we quantify a charge carrier density gradient, especially when a native oxide layer is present? We first identify the spectroscopic signature of the phonon resonances resulting from few nm native oxide layer around phosphorus-doped silicon nanowires. By modeling the amplitude and phase of the complex-valued near-field signals, we prove that characteristic peak positions in phase line profiles are robust against any influence from the presence of the oxide. We then determine that the carrier density profile in the region between doped and undoped segments is approximately linear for carrier densities between N ≈ 0.4 × 1020 cm−3 and N ≈ 1.4 × 1020 cm−3. We can quantify different widths for the first and second transition by the drop of the carrier density to 50% of its value in the fully doped region to d50% ≈ 70 nm and d50% ≈ 190 nm. Mapping of the full transition width is only limited by the spectral range of the available laser sources. Determination of even lower charge carriers might be possible in the future due to the progress employing THz radiation sources. Our results are supported by correlative EDS measurements. Our study underscores the potential of s-SNOM to yield detailed information about the charge carrier distributions in nanostructures, including those with thin surface layers. The methods developed herein are applicable to any nanostructures44,66,67 and semiconductor 1749


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(N ≈ 1014 cm−3) that functions as a reference for the s-SNOM measurements. In order to drop-cast the nanowires onto the germanium, the substrate with the upstanding nanowires was placed face down into a glass vial filled with methanol, followed by a sonication for 5 min. The nanowires are dispersed into the methanol as they break from the substrate due to the sonication. Finally, the liquid is pipetted onto the Ge substrate. After evaporation of the methanol the nanowires are randomly distributed on the substrate. EDS and EELS. In transmission electron microscopy (TEM), an electron beam is used for imaging. For TEM investigations in this study, a FEI Tecnai F20 featuring an electron energy of 200 keV is used. Because of this high energy, a beam electron may scatter inelastically with inner shell electrons of the sample material atoms removing one of the inner shell electrons. The energies required for the removal of one electron from one of the inner shells are characteristic for every element. Upon inelastic scattering, the beam electron loses this characteristic amount of energy. After passing the sample, the beam’s energy spectrum may be investigated utilizing an energy filter (Gatan GIF 2000). In the resulting electron energy loss spectrum (EELS) the characteristic loss or absorption energy is obtained from the sudden increase (often edges-like) in intensity. Energy filtering now permits to visualize electrons only from within a certain energy loss interval corresponding to a specific element creating chemical contrast. Thereby, the elemental image is obtained. When an electron from an outer shell of an atom transfers to the inner shell after the beam was scattered inelastically, an Xray photon of characteristic energy is released. In energydispersive X-ray spectroscopy (EDS), the elements a specimen consists of can be identified from the peaks in intensity at the characteristic X-ray energy (cf. Figure S5a). In this study, EDS spectra are taken by the FEI Tecnai F20 in scanning mode (STEM). The focused electron beam illuminates a certain position on the sample, while the spectrum corresponding to this position is taken by the EDS detector. Afterward, the acquisition of an EDS-spectrum is repeated at the next sample position and so on (cf. Figure S5b). When the intensity at a characteristic X-ray energy is plotted as a function of sample position, the EDS line profile is obtained, exposing relative changes in local elemental concentration. For the results presented in Figure 5b, the resulting X-ray counts that lie in the energy range of the phosphorus Kα1 and Kα2 peaks72 (1960 to 2060 eV) are extracted for each position and normalized to the P counts in the region of the doped segment. For the EDS measurements in a transmission electron microscope (TEM), the nanowires have been transferred onto an electron transparent carbon film of ∼8 nm thickness supported by a copper grid. For EELS elemental mapping, a cross section of a nanowire was cut using focused ion beam (FIB) milling after embedding the nanowire in a 1.5 μm thick platinum film. Dielectric Function of Doped Silicon. We describe the dielectric function ε(ν) of Si with the Drude model:

Figure 6. (a) Charge carrier density profile along the nanowire axis extracted from the positions of the maxima in the phase line profiles presented in Figure 4. The blue dashed lines are guides to the eye marking the carrier density level in the doped part extracted from Figure 2 and linear regression to the data points for the transitions. (b) Normalized phosphorus (P) counts obtained from EDS spectra taken along the nanowire axis. The blue line is the original data; the black line is a moving average along the blue line serving for better visibility of the profile. Vertical gray dashed lines in a and b mark the length of the fully doped segment. (c) Illustration of nanowire growth rate and temperature, as well as the resulting i-n-i morphology of the nanowire, corresponding to the doped/undoped and transition regions, as obtained from the results in (a) and (b).

subsequently lowered to 540 °C at a rate of approximately 3 °C/s in a constant Si2H6 pressure environment. Second, the nanowire growth was maintained at 540 °C and constant Si2H6 pressure of 4 × 10−5 Torr until reaching a desired length as a function of time. A single phosphorus-doped nanowire segment, approximately 200 nm in length, was grown during the nanowire elongation step by introducing PCl3 (Strem Chemicals, 99.999%) for 40 min at 470 °C. Nanowires with two different doping levels within the doped segment were grown. The PCl3 pressure was kept at 3 × 10−6 Torr for the higher doped nanowires investigated and discussed in the main text and 4 × 10−7 Torr for a lower doping of the nanowires investigated and discussed in Figures 2c,d. The resulting i-n-i Si nanowire morphology is shown schematically in Figure 1a (top) of the main text. For the near-field investigations the nanowires have been transferred to an undoped Ge substrate

ij ν 2 yz ε(ν) = ε∞jjj1 − 2 P zzzz; j ν + iγν z{ k

νp(N ) =

1 2πc

Ne 2 ε0ε∞m*

where ε∞ is the high-frequency permittivity,73 νp the plasma frequency (in wavenumbers), N the charge carrier density, 1 e γ(μ) = 2πc μm* the mobility-dependent damping (in wave1750


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numbers), and m* = 0.28m0 the effective mass.74 For the mobility, we use a charge carrier density-dependent formula empirically obtained from Hall measurements of bulk samples.53 Since mobilities measured with infrared light (i.e., high frequency measurements) are known to be smaller relative to the case of Hall measurements, we scale our mobilities by a factor of 0.83.75 Using these assumptions, plasma frequency and dielectric function and thus near-field amplitude and phase signals only depend on the charge carrier density. Finite Dipole Model. If an electromagnetic field is incident on a metalized s-SNOM tip, it is polarized mainly along the shaft of the tip. This is accounted for in the finite dipole model51 by treating the tip as an elongated spheroidal dipole, leading to a large spatial separation of the opposite dipole charges. Due to the large separation only the monopole charge at the apex of the tip is assumed to play a role for the tip−sample near-field interaction. A self-consistent treatment of the interaction of this charge producing a mirror charge in the sample leads to the following equation for the effective polarizability of the tip−sample system: αeff ≈

component of the wavevector in the order of the inverse of the tip radius (q = 2.5 × 105 cm−1), which dominates the nearfield coupling. The assumption of one wavevector component dominating the near-field interaction has also been reported by Fei et al.77 and using it for the multilayer modeling has proven to nicely reproduce experimental data even for resonant sample systems.78 The multilayer model has been used in this study for including a thin native oxide layer on top of doped silicon. Notably, any influence of the Ge substrate on the near-field signals can be neglected for nanowires of diameter greater than ∼100 nm. This assumption is supported by calculations comparing nanowires treated as bulk Si with those including the Ge substrate (Figure S4) using the multilayer finite dipole model. Thus, in all calculations presented in the main text, the substrate has been neglected.

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.9b00466. Additional information about the carrier density dependence of the s-SNOM signals, a summary of important material parameters for a selection of common semiconductor materials and the resulting carrier density range accessible with the laser system used in this work, a topographic line profile taken along the nanowire axis, EELS oxygen map obtained on a nanowire cross section, calculations proving the negligible substrate influence, and STEM image of the nanowire investigated with EDS and corresponding EDS energy spectrum (PDF)

βf0 1 +1 2 (1 − βf1 )

where f 0 and f1 describe the geometric properties of the system:

f0,1

ij ρ + 2H(t ) + W0,1 yz ln zz = jjjg − zz j 2L k {

(

4L ρ + 4H(t ) + 2W0,1

)

( )

ln

ASSOCIATED CONTENT

4L ρ

■

ε+1

In these formulas, β = ε − 1 denotes the quasi-static reflection coefficient in dependence of the samples dielectric function and H(t) = A(1 + cos(Ωt)) the time-dependent height of the tip above the sample, which is sinusoidally driven at a tapping amplitude A and frequency Ω. W0 = 1.31ρ and W1 = 0.5ρ describe the positions of induced charges within the tip in dependence of the tip radius ρ. Additionally, two empirical parameters are needed: the length of the spheroid L, and g, describing the portion of the near-field induced charge in the tip that contributes to the near-field interaction. For our calculations, we used the following values for the mentioned parameters: ρ = 25 nm, L = 300 nm, A = 35 nm, Ω = 250 kHz, g = 0.7e0.06i. Thus, the values for f 0 and f1 oscillate, dependent on the time, between 0.11 < |f 0| < 0.40 and 0.13 < |f1| < 0.52. The optical near-field amplitude and phase of the n-th demodulation order are proportional to the absolute value and argument of the n-th Fourier component of the effective polarizability, respectively. The finite dipole model has been chosen as it has been shown to quantitatively predict near-field contrasts without the need for a complex electrodynamic treatment that includes the actual tip geometry.76 Multilayer Finite Dipole Model. In the multilayerextension of the finite dipole model,59 the tip is treated in the same way as in the regular finite dipole model. The only change is the calculation of β, describing the samples response. Instead of taking the quasi-static limit of the reflection coefficient, the field at the position of the tip stemming from the induced charge distribution within the sample is described by a quasi-electrodynamic treatment of the reflection coefficient. This is calculated for an effective in-plane

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: mfi[email protected]. ORCID

Lena Jung: 0000-0002-3377-1309 Martin Lewin: 0000-0003-4036-2252 Michael A. Filler: 0000-0003-4239-8558 Matthias Wuttig: 0000-0003-1498-1025 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Camilla La Torre for helpful discussions about the interplay of dopant atoms and charge carrier distribution. Additionally, the authors thank the GFE (Gemeinschaftslabor fü r Elektronenmikroskopie) of the RWTH Aachen for the chance to perform EELS and EDX measurements at their facilities. L.J., M.L., J.P., M.W., and T.T. gratefully acknowledge funding from the DFG (German Science Foundation) within the SFB 917 “Nanoswitches” and the Excellence Initiative. M.A.F., D.S.B., and A.T.M. acknowledge support from the U.S. National Science Foundation (#1510934).

■

REFERENCES

(1) Garnett, E.; Yang, P. Light Trapping in Silicon Nanowire Solar Cells. Nano Lett. 2010, 10, 1082−1087.

1751


ACS Photonics

Article

Subsurface Near-Field Optical Microscopy. Opt. Express 2012, 20, 593−600. (22) Jung, L.; Hauer, B.; Li, P.; Bornhöfft, M.; Mayer, J.; Taubner, T. Exploring the Detection Limits of Infrared Near-Field Microscopy Regarding Small Buried Structures and Pushing Them by Exploiting Superlens-Related Effects. Opt. Express 2016, 24, 4431−4441. (23) Engelhardt, A. P.; Hauer, B.; Taubner, T. Visibility of Weak Contrasts in Subsurface Scattering Near-Field Microscopy. Ultramicroscopy 2013, 126, 40−43. (24) Ocelic, N.; Huber, A.; Hillenbrand, R. Pseudoheterodyne Detection for Background-Free Near-Field Spectroscopy. Appl. Phys. Lett. 2006, 89, 101124. (25) Kuzmany, H. Solid-State Spectroscopy: an Introduction; Springer: Berlin, Heidelberg, 2009. (26) Taubner, T.; Hillenbrand, R.; Keilmann, F. Nanoscale Polymer Recognition by Spectral Signature in Scattering Infrared Near-Field Microscopy. Appl. Phys. Lett. 2004, 85, 5064. (27) Taubner, T.; Hillenbrand, R.; Keilmann, F. Performance of Visible and Mid-Infrared Scattering-Type Near-Field Optical Microscopes. J. Microsc. 2003, 210, 311−314. (28) Govyadinov, A. A.; Amenabar, I.; Huth, F.; Carney, P. S.; Hillenbrand, R. Quantitative Measurement of Local Infrared Absorption and Dielectric Function with Tip-Enhanced Near-Field Microscopy. J. Phys. Chem. Lett. 2013, 4, 1526−1531. (29) Mastel, S.; Govyadinov, A. A.; de Oliveira, T. V. A. G.; Amenabar, I.; Hillenbrand, R. Nanoscale-Resolved Chemical Identification of Thin Organic Films Using Infrared Near-Field Spectroscopy and Standard Fourier Transform Infrared References. Appl. Phys. Lett. 2015, 106, 023113. (30) Bensmann, S.; Gaussmann, F.; Lewin, M.; Wueppen, J.; Nyga, S.; Janzen, C.; Jungbluth, B.; Taubner, T. Near-Field Imaging and Spectroscopy of Locally Strained GaN Using an IR Broadband Laser. Opt. Express 2014, 22, 22369−22381. (31) Muller, E. A.; Pollard, B.; Bechtel, H. A.; van Blerkom, P.; Raschke, M. B. Infrared Vibrational Nano-Crystallography and NanoImaging. Science Advances 2016, 2, No. e1601006. (32) Huber, A. J.; Keilmann, F.; Wittborn, J.; Aizpurua, J.; Hillenbrand, R. Terahertz Near-Field Nanoscopy of Mobile Carriers in Single Semiconductor Nanodevices. Nano Lett. 2008, 8, 3766− 3770. (33) Stiegler, J. M.; Huber, A. J.; Diedenhofen, S. L.; Gómez Rivas, J.; Algra, R. E.; Bakkers, E. P. A. M.; Hillenbrand, R. Nanoscale FreeCarrier Profiling of Individual Semiconductor Nanowires by Infrared Near-Field Nanoscopy. Nano Lett. 2010, 10, 1387−1392. (34) Chen, X.; Hu, D.; Mescall, R.; You, G.; Basov, D. N.; Dai, Q.; Liu, M. Modern Scattering-Type Scanning Near-Field Optical Microscopy for Advanced Material Research. Adv. Mater. 2019, 70, 1804774. (35) Maissen, C.; Chen, S.; Nikulina, E.; Govyadinov, A.; Hillenbrand, R. Probes for Ultrasensitive THz Nanoscopy. ACS Photonics 2019, 6, 1279−1288. (36) Mastel, S.; Lundeberg, M. B.; Alonso-González, P.; Gao, Y.; Watanabe, K.; Taniguchi, T.; Hone, J.; Koppens, F. H. L.; Nikitin, A. Y.; Hillenbrand, R. Terahertz Nanofocusing with Cantilevered Terahertz-Resonant Antenna Tips. Nano Lett. 2017, 17, 6526−6533. (37) Siday, T.; Natrella, M.; Wu, J.; Liu, H.; Mitrofanov, O. Resonant Terahertz Probes for Near-Field Scattering Microscopy. Opt. Express 2017, 25, 27874−27885. (38) Liewald, C.; Mastel, S.; Hesler, J.; Huber, A. J.; Hillenbrand, R.; Keilmann, F. All-Electronic Terahertz Nanoscopy. Optica 2018, 5, 159−163. (39) Sakat, E.; Giliberti, V.; Bollani, M.; Notargiacomo, A.; Pea, M.; Finazzi, M.; Pellegrini, G.; Hugonin, J.-P.; Weber-Bargioni, A.; Melli, M.; et al. Near-Field Imaging of Free Carriers in ZnO Nanowires with a Scanning Probe Tip Made of Heavily Doped Germanium. Phys. Rev. Appl. 2017, 8, 054042. (40) Knoll, B.; Keilmann, F. Infrared Conductivity Mapping for Nanoelectronics. Appl. Phys. Lett. 2000, 77, 3980−3982.

(2) Thelander, C.; Agarwal, P.; Brongersma, S.; Eymery, J.; Feiner, L. F.; Forchel, A.; Scheffler, M.; Riess, W.; Ohlsson, B. J.; Gösele, U.; et al. Nanowire-Based One-Dimensional Electronics. Mater. Today 2006, 9, 28−35. (3) Li, Y.; Qian, F.; Xiang, J.; Lieber, C. M. Nanowire Electronic and Optoelectronic Devices. Mater. Today 2006, 9, 18−27. (4) Boyuk, D. S.; Chou, L.-W.; Filler, M. A. Strong Near-Field Coupling of Plasmonic Resonators Embedded in Si Nanowires. ACS Photonics 2016, 3, 184−189. (5) Chung, S.-W.; Yu, J.-Y.; Heath, J. R. Silicon Nanowire Devices. Appl. Phys. Lett. 2000, 76, 2068−2070. (6) Vallett, A. L.; Minassian, S.; Kaszuba, P.; Datta, S.; Redwing, J. M.; Mayer, T. S. Fabrication and Characterization of Axially Doped Silicon Nanowire Tunnel Field-Effect Transistors. Nano Lett. 2010, 10, 4813−4818. (7) Chou, L.-W.; Filler, M. A. Engineering Multimodal Localized Surface Plasmon Resonances in Silicon Nanowires. Angew. Chem., Int. Ed. 2013, 52, 8079−8083. (8) Perea, D. E.; Hemesath, E. R.; Schwalbach, E. J.; Lensch-Falk, J. L.; Voorhees, P. W.; Lauhon, L. J. Direct Measurement of Dopant Distribution in an Individual Vapour-Liquid-Solid Nanowire. Nat. Nanotechnol. 2009, 4, 315−319. (9) Chen, W.; Dubrovskii, V. G.; Liu, X.; Xu, T.; Lardé, R.; Nys, J.P.; Grandidier, B.; Stiévenard, D.; Patriarche, G.; Pareige, P. Boron Distribution in the Core of Si Nanowire Grown by Chemical Vapor Deposition. J. Appl. Phys. 2012, 111, 094909. (10) Christesen, J. D.; Pinion, C. W.; Zhang, X.; McBride, J. R.; Cahoon, J. F. Encoding Abrupt and Uniform Dopant Profiles in Vapor-Liquid-Solid Nanowires by Suppressing the Reservoir Effect of the Liquid Catalyst. ACS Nano 2014, 8, 11790−11798. (11) Kim, S.; Hill, D. J.; Pinion, C. W.; Christesen, J. D.; McBride, J. R.; Cahoon, J. F. Designing Morphology in Epitaxial Silicon Nanowires: the Role of Gold, Surface Chemistry, and Phosphorus Doping. ACS Nano 2017, 11, 4453−4462. (12) Cohen-Karni, T.; Casanova, D.; Cahoon, J. F.; Qing, Q.; Bell, D. C.; Lieber, C. M. Synthetically Encoded Ultrashort-Channel Nanowire Transistors for Fast, Pointlike Cellular Signal Detection. Nano Lett. 2012, 12, 2639−2644. (13) Ou, X.; Kanungo, P. D.; Kögler, R.; Werner, P.; Gösele, U.; Skorupa, W.; Wang, X. Carrier Profiling of Individual Si Nanowires by Scanning Spreading Resistance Microscopy. Nano Lett. 2010, 10, 171−175. (14) Vinaji, S.; Lochthofen, A.; Mertin, W.; Regolin, I.; Gutsche, C.; Prost, W.; Tegude, F. J.; Bacher, G. Material and Doping Transitions in Single GaAs-Based Nanowires Probed by Kelvin Probe Force Microscopy. Nanotechnology 2009, 20, 385702. (15) den Hertog, M. I.; Schmid, H.; Cooper, D.; Rouviere, J.-L.; Björk, M. T.; Riel, H.; Rivallin, P.; Karg, S.; Riess, W. Mapping Active Dopants in Single Silicon Nanowires Using Off-Axis Electron Holography. Nano Lett. 2009, 9, 3837−3843. (16) Minj, A.; Cros, A.; Auzelle, T.; Pernot, J.; Daudin, B. Direct Assessment of P-N Junctions in Single GaN Nanowires by Kelvin Probe Force Microscopy. Nanotechnology 2016, 27, 1−10. (17) Choi, W.; Seabron, E.; Mohseni, P. K.; Kim, J. D.; Gokus, T.; Cernescu, A.; Pochet, P.; Johnson, H. T.; Wilson, W. L.; Li, X. Direct Electrical Probing of Periodic Modulation of Zinc-Dopant Distributions in Planar Gallium Arsenide Nanowires. ACS Nano 2017, 11, 1530−1539. (18) Keilmann, F.; Hillenbrand, R. Near-Field Microscopy by Elastic Light Scattering From a Tip. Philos. Trans. R. Soc., A 2004, 362, 787− 806. (19) Atkin, J. M.; Berweger, S.; Jones, A. C.; Raschke, M. B. NanoOptical Imaging and Spectroscopy of Order, Phases, and Domains in Complex Solids. Adv. Phys. 2012, 61, 745−842. (20) Taubner, T.; Keilmann, F.; Hillenbrand, R. Nanoscale-Resolved Subsurface Imaging by Scattering-Type Near-Field Optical Microscopy. Opt. Express 2005, 13, 8893−8899. (21) Krutokhvostov, R.; Govyadinov, A. A.; Stiegler, J. M.; Huth, F.; Chuvilin, A.; Carney, P. S.; Hillenbrand, R. Enhanced Resolution in 1752


ACS Photonics

Article

(41) Stiegler, J. M.; Tena-Zaera, R.; Idigoras, O.; Chuvilin, A.; Hillenbrand, R. Correlative Infrared-Electron Nanoscopy Reveals the Local Structure-Conductivity Relationship in Zinc Oxide Nanowires. Nat. Commun. 2012, 3, 1131. (42) Arcangeli, A.; Rossella, F.; Tomadin, A.; Xu, J.; Ercolani, D.; Sorba, L.; Beltram, F.; Tredicucci, A.; Polini, M.; Roddaro, S. GateTunable Spatial Modulation of Localized Plasmon Resonances. Nano Lett. 2016, 16, 5688−5693. (43) Huber, A. J.; Kazantsev, D.; Keilmann, F.; Wittborn, J.; Hillenbrand, R. Simultaneous IR Material Recognition and Conductivity Mapping by Nanoscale Near-Field Microscopy. Adv. Mater. 2007, 19, 2209−2212. (44) Hauer, B.; Saltzmann, T.; Simon, U.; Taubner, T. Solvothermally Synthesized Sb2Te3 Platelets Show Unexpected Optical Contrasts in Mid-Infrared Near-Field Scanning Microscopy. Nano Lett. 2015, 15, 2787−2793. (45) Hill, D. J.; Teitsworth, T. S.; Ritchie, E. T.; Atkin, J. M.; Cahoon, J. F. Interplay of Surface Recombination and Diode Geometry for the Performance of Axial P-I-N Nanowire Solar Cells. ACS Nano 2018, 12, 10554−10563. (46) Lang, D.; Balaghi, L.; Winnerl, S.; Schneider, H.; Hübner, R.; Kehr, S. C.; Eng, L. M.; Helm, M.; Dimakis, E.; Pashkin, A. Nonlinear Plasmonic Response of Doped Nanowires Observed by Infrared Nanospectroscopy. Nanotechnology 2019, 30, 084003. (47) Eisele, M.; Cocker, T. L.; Huber, M. A.; Plankl, M.; Viti, L.; Ercolani, D.; Sorba, L.; Vitiello, M. S.; Huber, R. Ultrafast MultiTerahertz Nano-Spectroscopy with Sub-Cycle Temporal Resolution. Nat. Photonics 2014, 8, 841−845. (48) Wagner, M.; McLeod, A. S.; Maddox, S. J.; Fei, Z.; Liu, M.; Averitt, R. D.; Fogler, M. M.; Bank, S. R.; Keilmann, F.; Basov, D. N. Ultrafast Dynamics of Surface Plasmons in InAs by Time-Resolved Infrared Nanospectroscopy. Nano Lett. 2014, 14, 4529−4534. (49) Ritchie, E. T.; Hill, D. J.; Mastin, T. M.; Deguzman, P. C.; Cahoon, J. F.; Atkin, J. M. Mapping Free-Carriers in Multijunction Silicon Nanowires Using Infrared Near-Field Optical Microscopy. Nano Lett. 2017, 17, 6591−6597. (50) Lu, X.; Khatib, O.; Du, X.; Duan, J.; Wei, W.; Liu, X.; Bechtel, H. A.; D’Apuzzo, F.; Yan, M.; Buyanin, A.; et al. Nanoimaging of Electronic Heterogeneity in Bi2Se3 And Sb2Te3 Nanocrystals. Adv. Electron. Mater. 2018, 320, 1700377. (51) Cvitkovic, A.; Ocelic, N.; Hillenbrand, R. Analytical Model for Quantitative Prediction of Material Contrasts in Scattering-Type Near-Field Optical Microscopy. Opt. Express 2007, 15, 8550−8565. (52) Chou, L.-W.; Boyuk, D. S.; Filler, M. A. Optically Abrupt Localized Surface Plasmon Resonances in Si Nanowires by Mitigation of Carrier Density Gradients. ACS Nano 2015, 9, 1250−1256. (53) Masetti, G.; Severi, M.; Solmi, S. Modeling of Carrier Mobility Against Carrier Concentration in Arsenic-, Phosphorus-, and BoronDoped Silicon. IEEE Trans. Electron Devices 1983, 30, 764−769. (54) Sotoodeh, M.; Khalid, A. H.; Rezazadeh, A. A. Empirical LowField Mobility Model for III-V Compounds Applicable in Device Simulation Codes. J. Appl. Phys. 2000, 87, 2890. (55) Matteini, F.; Tutuncuoglu, G.; Rüffer, D.; Alarcon-Llado, E.; Morral, A. F. I. Untangling the Role of Oxide in Ga-Assisted Growth of GaAs Nanowires on Si Substrates. arXiv:1307.6113 [condmat.mtrl-sci], 2013. (56) Morita, M.; Ohmi, T.; Hasegawa, E.; Kawakami, M.; Ohwada, M. Growth of Native Oxide on a Silicon Surface. J. Appl. Phys. 1990, 68, 1272−1281. (57) Kischkat, J.; Peters, S.; Gruska, B.; Semtsiv, M.; Chashnikova, M.; Klinkmüller, M.; Fedosenko, O.; Machulik, S.; Aleksandrova, A.; Monastyrskyi, G.; et al. Mid-Infrared Optical Properties of Thin Films of Aluminum Oxide, Titanium Dioxide, Silicon Dioxide, Aluminum Nitride, and Silicon Nitride. Appl. Opt. 2012, 51, 6789−6798. (58) Zhang, L. M.; Andreev, G. O.; Fei, Z.; McLeod, A. S.; Dominguez, G.; Thiemens, M.; Castro-Neto, A. H.; Basov, D. N.; Fogler, M. M. Near-Field Spectroscopy of Silicon Dioxide Thin Films. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 075419.

(59) Hauer, B.; Engelhardt, A. P.; Taubner, T. Quasi-Analytical Model for Scattering Infrared Near-Field Microscopy on Layered Systems. Opt. Express 2012, 20, 13173−13188. (60) Chou, L.-W.; Near, R. D.; Boyuk, D. S.; Filler, M. A. Influence of Dielectric Anisotropy on the Absorption Properties of Localized Surface Plasmon Resonances Embedded in Si Nanowires. J. Phys. Chem. C 2014, 118, 5494−5500. (61) Mooshammer, F.; Sandner, F.; Huber, M. A.; Zizlsperger, M.; Weigand, H.; Plankl, M.; Weyrich, C.; Lanius, M.; Kampmeier, J.; Mussler, G.; et al. Nanoscale Near-Field Tomography of Surface States on (Bi0.5Sb0.5)2Te3. Nano Lett. 2018, 18, 7515−7523. (62) Clark, T. E.; Nimmatoori, P.; Lew, K.-K.; Pan, L.; Redwing, J. M.; Dickey, E. C. Diameter Dependent Growth Rate and Interfacial Abruptness in Vapor-Liquid-Solid Si/Si1-XGeX Heterostructure Nanowires. Nano Lett. 2008, 8, 1246−1252. (63) Li, N.; Tan, T. Y.; Gösele, U. Transition Region Width of Nanowire Hetero- and Pn-Junctions Grown Using Vapor-LiquidSolid Processes. Appl. Phys. A: Mater. Sci. Process. 2008, 90, 591−596. (64) Schmidt, V.; Senz, S.; Gösele, U. Influence of the Si/SiO2 Interface on the Charge Carrier Density of Si Nanowires. Appl. Phys. A: Mater. Sci. Process. 2006, 86, 187−191. (65) Sun, S. C.; Plummer, J. D. Electron Mobility in Inversion and Accumulation Layers on Thermally Oxidized Silicon Surfaces. IEEE J. Solid-State Circuits 1980, 15, 562−573. (66) Law, S.; Yu, L.; Rosenberg, A.; Wasserman, D. All-Semiconductor Plasmonic Nanoantennas for Infrared Sensing. Nano Lett. 2013, 13, 4569−4574. (67) Baldassarre, L.; Sakat, E.; Frigerio, J.; Samarelli, A.; Gallacher, K.; Calandrini, E.; Isella, G.; Paul, D. J.; Ortolani, M.; Biagioni, P. Midinfrared Plasmon-Enhanced Spectroscopy with Germanium Antennas on Silicon Substrates. Nano Lett. 2015, 15, 7225−7231. (68) Law, S.; Yu, L.; Wasserman, D. Epitaxial Growth of Engineered Metals for Mid-Infrared Plasmonics. J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 2013, 31, 03C121. (69) Fehrenbacher, M.; Winnerl, S.; Schneider, H.; Döring, J.; Kehr, S. C.; Eng, L. M.; Huo, Y.; Schmidt, O. G.; Yao, K.; Liu, Y.; et al. Plasmonic Superlensing in Doped GaAs. Nano Lett. 2015, 15, 1057− 1061. (70) Frigerio, J.; Ballabio, A.; Isella, G.; Sakat, E.; Pellegrini, G.; Biagioni, P.; Bollani, M.; Napolitani, E.; Manganelli, C.; Virgilio, M.; et al. Tunability of the Dielectric Function of Heavily Doped Germanium Thin Films for Mid-Infrared Plasmonics. Phys. Rev. B 2016, 94, 418. (71) Xu, X. G.; Gilburd, L.; Walker, G. C. Phase Stabilized Homodyne of Infrared Scattering Type Scanning Near-Field Optical Microscopy. Appl. Phys. Lett. 2014, 105, 263104. (72) Thompson, A. C.; Kirz, J.; Attwood, D. T.; Gullikson, E. M.; Howells, M. R.; Kortright, J. B.; Liu, Y.; Robinson, A. L.; Underwood, J. H.; Kim, K.-J. et al. X-Ray Data Booklet, 3rd ed.; Thompson, A. C., Ed.; Lawrence Berkeley National Laboratory, University of California, 2009. (73) Naik, G. V.; Shalaev, V. M.; Boltasseva, A. Alternative Plasmonic Materials: Beyond Gold and Silver. Adv. Mater. 2013, 25, 3264−3294. (74) Miyao, M.; Motooka, T.; Natsuaki, N.; Tokuyama, T. Change of the Electron Effective Mass in Extremely Heavily Doped N-Type Si Obtained by Ion Implantation and Laser Annealing. Solid State Commun. 1981, 37, 605−608. (75) Barta, E. Optical Constants of Various Heavily Doped P- and N-Type Silicon Crystals Obtained by Kramers-Kronig Analysis. Infrared Phys. 1977, 17, 319−329. (76) McLeod, A. S.; Kelly, P.; Goldflam, M. D.; Gainsforth, Z.; Westphal, A. J.; Dominguez, G.; Thiemens, M. H.; Fogler, M. M.; Basov, D. N. Model for Quantitative Tip-Enhanced Spectroscopy and the Extraction of Nanoscale-Resolved Optical Constants. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 085136. (77) Fei, Z.; Andreev, G. O.; Bao, W.; Zhang, L. M.; S McLeod, A.; Wang, C.; Stewart, M. K.; Zhao, Z.; Dominguez, G.; Thiemens, M.; 1753


ACS Photonics

Article

et al. Infrared Nanoscopy of Dirac Plasmons at the Graphene-SiO2 Interface. Nano Lett. 2011, 11, 4701−4705. (78) Hauer, B. Nano-Optical Mapping of Permittivity Contrasts and Electronic Properties at the Surface and Beneath; Dissertation, RWTH Aachen University: Aachen, 2015; pp 1−146.

1754


Quantification of Carrier Density Gradients along Axially Doped Silicon

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