Investigating Charge Transport in Semiconducting Single‐Walled Carbon Nanotube Networks by Charge Modulation Microscopy

Solution‐processed networks of semiconducting single‐walled carbon nanotubes (SWCNTs) hold promise as active layers for large‐scale digital circuits, thermoelectric devices, and healthcare applications. Yet, the combined effects of local network properties and (n,m) species composition need to be addressed to improve the performance and reproducibility of state‐of‐the‐art devices. Charge modulation microscopy (CMM) is used to investigate charge transport in field‐effect transistors (FETs) based on monochiral (6,5) SWCNT networks and multichiral networks containing five semiconducting SWCNT species with different band gaps. By mapping the charge‐modulated signal within the FET channel with sub‐micrometer resolution, the spatial distribution of free carriers and its evolution during the switching of the FET are revealed. The CMM maps provide direct evidence that holes and electrons are transported preferentially through the same percolation paths. A moderate positive correlation between the SWCNT density in the monochiral (6,5) network and the charge density in subthreshold regime is demonstrated. In multichiral networks, the charge transport paths are on average less fragmented when involving low band gap species. CMM emerges as a valuable technique to estimate the size of preferential percolation domains associated with the average distance traversed by charge carriers on SWCNTs of the same chirality before tunneling onto other species.


Introduction
Networks of semiconducting single-walled carbon nanotubes (SWCNTs) have great potential for applications in electronic circuits due to their outstanding electrical properties, such as in suitable solvents for devices fabrication. One example is polymer-sorted (6,5) SWCNTs, which can be produced and processed into networks from toluene dispersion. [11] Thus, even though charge transport is more effective in networks with only a single nanotube species (i.e., monochiral networks), polydisperse samples of semiconducting nanotubes with different species (i.e., multichiral networks) are still commonly employed.
The effects of the network composition and density on the charge transport have been investigated in several studies by temperature-dependent measurements of conductivity and field-effect mobility, [20][21][22] along with thermoelectric parameters. [23] Moreover, spectroscopic techniques such as photoluminescence (PL), [24] electroluminescence (EL), [25,26] and Raman [27] are valuable methods to study the charge transport in multichiral networks by exploiting characteristic optical transitions and monitoring the spectral evolution in the presence of charge carriers. The significant energetic disorder due to the presence of multiple species represents the main obstacle for achieving efficient charge transport in multichiral networks. [22,28] Semiconducting SWCNTs with large diameter, hence with small electronic band gap, provide higher intrinsic charge carrier mobility, lower junction resistance, [29] and smaller injection barriers when compared to small-diameter nanotubes. [30] The SWCNT density, [16,31] the degree of alignment, [32] and the presence of bundles [33] are additional network parameters that must be carefully optimized to improve charge transport. Recently, Zorn et al. employed charge modulation spectroscopy (CMS) and charge-modulated PL to study charge transport in FETs based on random networks of monochiral (6,5) SWCNTs and of a mixture of five chiralities. [34] CMS is a lock-in-based technique that has been widely employed to study charge transport in organic semiconductors. [35][36][37][38][39] In the CMS experiment, the charge density at the interface between the semiconducting and the dielectric layer is modulated by applying a gate potential composed of a constant offset potential, V os , and of a sinusoidal waveform with peak-to-peak intensity V pp and modulation frequency ω. By correlating the mobile carriers with the ground-state bleaching and charge-induced absorption of the different SWCNT chiralities, it has been shown that small band gap SWCNTs dominate the charge transport especially at low carrier concentrations, while the contribution of large band gap species increases at higher gate voltages. [34] In this work, we deepen the investigation of charge transport in monochiral and multichiral random networks of SWCNTs by charge modulation microscopy (CMM), which allows us to image CMS features with a sub-micrometer spatial resolution using a confocal microscopy setup. A schematic layout of the CMM system is presented in Figure 1a, while a detailed description of the setup is provided in Figure S1, Supporting Information. CMM mapping of the FET channel area has already enabled direct visualizations of the charge density distribution in films of organic semiconductors, allowing to analyze the impact of local film properties (e.g., morphology, density, and degree of alignment) on charge transport. [40][41][42][43][44] By tracking the evolution of the charge carrier distribution in a monochiral (6,5) SWCNT network as a function of the applied gate voltage in the subthreshold regime, that is, when the conductive channel is being formed, we are able to detect a progressive activation of percolation paths and highlight the impact of the network density. In networks composed of five different semiconducting SWCNT species, our CMM analysis shows that all of them clearly take an active role in charge transport. Being able to directly visualize charge transport paths, we show that charges distribute more homogeneously on the species with larger diameters (i.e., smaller band gap). Furthermore, we provide a method to estimate in a comparative way relevant length scales related to the size of percolation domains in the different species, thus providing insights for the design and dimensionality of electronic devices based on random SWCNT networks.

Monochiral (6,5) SWCNT Networks
Using selective polymer-wrapping [11] we prepared a nearly monochiral dispersion of (6,5) SWCNTs in toluene from CoMoCAT SWCNT raw material with the wrapping polymer poly[(9,9dioctylfluorenyl-2,7-diyl)-alt-(6,6′-(2,2′-bipyridine))] (PFO-BPy; details provided in the Experimental Section). We fabricated FETs in bottom-contact, top-gate configuration (scheme displayed in Figure 1a) by depositing the SWCNTs dispersion via spin-coating onto pre-patterned gold electrodes. As confirmed by atomic force microscopy (AFM) (Figure 1b), this leads to dense networks of randomly distributed (6,5) SWCNTs. Due to a thin silver top gate, the FETs are semi-transparent, thus enabling optical measurements in transmission mode. The characteristic transfer and output curves of a representative device are shown in Figure 1c and in Figure S2, Supporting Information, respectively. The transfer curve exhibits a nearly ideal ambipolar behavior, high on/off current ratio above 10 6 , and negligible hysteresis. The field-effect mobility of the monochiral network, as extracted from the slope of the linear fit of I ds (V g ), reaches about 6.6 cm 2 V −1 s −1 for holes and 9.8 cm 2 V −1 s −1 for electrons. The threshold voltages in hole and electron accumulation regimes amount to −5.3 and 4.1 V, respectively ( Figure S4, Supporting Information).
By using CMS, Zorn et al. [34] previously analyzed the differential changes in the optical absorption of the SWCNT network upon modulation of the gate potential in FETs that were identical to those fabricated for this study. The CMS signal originates from the differential transmittance, namely ΔT/T, induced by the mobile charges upon the sinusoidal modulation. A positive CMS signal corresponds to an increased transmittance (∆T > 0) due to the bleaching of the absorption of neutral SWCNTs. A negative CMS signal is observed when there is a decreased transmittance (∆T < 0) because of charge-induced absorption. The modulated spectra presented very high signal intensities and signal-to-noise ratios, enabling clear observation of charge-induced exciton bleaching of the E 11 transition and charge-induced absorption features (related to a polaron or trion) caused by mobile carrier modulation. In this study, we perform CMM to record local charge-modulation spectra with sub-micrometer resolution and acquire microscopy images of the CMS signal by fixing the incident wavelength and scanning a large area of the sample. Figure 1d shows the local charge-modulation spectrum (V os = −1 V, V pp = 0.2 V) acquired in the middle of the transistor www.advmatinterfaces.de channel by means of the confocal microscope setup with a focal spot of ≈0.8 µm ( Figure S1b, Supporting Information). Two positive peaks are present: a pronounced and sharp peak at around 1010 nm, which is assigned to the bleaching of the E 11 absorption peak ( Figure S3, Supporting Information), and a phonon side band (PSB) at around 860 nm. [34] The negative band in the NIR is due to the absorption of positively charged nanotubes (polarons or trions, X + ). Trions are three-body particles (i.e., charged excitons) that eventually form upon optical excitation of doped carbon nanotubes. [45,46] The local spectrum is in very good agreement with the macroscopic CMS spectrum measured from the entire device ( Figure S5a,b, Supporting Information) and with the CMS spectra previously reported for monochiral (6,5) networks. [34] Furthermore, local spectra acquired from different positions of the sample exhibit the same CMS features, with only slight variations in the peak intensities, which is a consequence of the differences in the local distribution of free charges throughout the active area ( Figure S5c, Supporting Information). Importantly, the local CMS spectra measured in the center of the FET channel and close to the electrodes are virtually identical ( Figure S5d, Supporting Information), thus excluding the presence of any spurious electro-absorption effect that would be more pronounced at the electrode edges. [39,42] The CMS signal intensity shows a characteristic trend as a function of the applied offset potential V os , which is found also for local CMS spectra (Figure 1e,f). The maximum of the E 11 bleaching peak is associated to a critical value of the charge carrier density, d*, which is reached at V os * ≈ −1.2 V for the FET based on the (6,5) network ( Figure 1f). For |V os | < |V os *|, Figure 1. a) Schematic diagram of the charge modulation microscopy (CMM) setup. The inset shows a scheme of a SWCNT-based field-effect transistor architecture. b) AFM topography of a monochiral (6,5) SWCNT network. The scale bar is 500 nm. c) Transfer curve (V ds = −0.1 V) for the monochiral (6,5) SWCNT-based FET plotted on logarithmic (black) and on linear (red) scales. The device has channel length L = 20 µm and channel width W = 1 cm. d) Local charge modulation spectrum of monochiral (6,5) SWCNTs (V os = −1 V, V pp = 0.2 V). The focal spot is ≈0.8 µm wide. The x-axis is interrupted at 1100 nm because the spectrum before and after this wavelength is acquired with different optical fibers and acousto-optic tunable filter (AOTF). e) Trend of the local E 11 bleaching signal at different V os and fixed V pp = 0.2 V. f) Evolution of the CMS signal at 1005 nm (E 11 bleaching peak) as a function of the applied offset potential V os . www.advmatinterfaces.de the E 11 bleaching peak grows with the offset potential because the quantum capacitance of the carbon nanotubes increases, thus resulting in higher modulated charge density. Instead, for |V os | > |V os *|, the CMS signal drops due to the reduction of the oscillator strength of the E 11 transition when the charge carrier density is greater than d*. [47,48] Keeping the wavelength of the incident laser beam fixed at 1005 nm, we scan a 10 × 10 µm 2 area within the FET channel to obtain CMM maps of the E 11 bleaching peak as a function of V os , which provide information on the local density distribution of free carriers when varying the offset potential ( Figure S6, Supporting Information). Importantly, by applying a positive/ negative offset voltage we can visualize the CMM maps that correspond selectively to hole/electron accumulation regimes, as shown in Figure 2a,b. The difference of the average intensity of the CMS signal in the two CMM maps is consistent with the general trend of the signal for |V os | > |V os *|. Indeed, the average signal in the map recorded for V os = −2.0 V (hole accumulation regime) is higher than in the one measured for V os = 2.2 V (electron accumulation regime). Remarkably, the two CMM maps present the same local features. Hence, both holes and electrons accumulate preferentially in the same regions (those with higher CMS signal), meaning that there are favorable percolation paths in the network regardless of the polarity of the charge carriers. This is consistent with the nearly ideal ambipolar characteristics of the device and with previous observations of electron-hole recombination paths by electroluminescence experiments. [25,49] A convenient way to quantify the similarity in spatial distribution of the signal in two different CMM maps consists in computing the zero-mean normalized cross-correlation (ZNCC) coefficient. With this approach, the two maps are first standardized by subtracting the mean and dividing by the standard deviation of the signal, and then they are cross-correlated (details in the Experimental Section). This method is frequently used in image-processing and template-matching applications to recognize common patterns in images where the brightness and contrast vary because of different lighting conditions. The ZNCC coefficient ranges from −1 to 1 and represents the 2D version of the Pearson correlation coefficient, which is a measure of the linear correlation between two signals or datasets. Therefore, when the ZNCC coefficient is close to 1, the two images are positively correlated, when it approaches −1 they are inversely correlated, while when it is close to 0 the two images are uncorrelated. Intermediate values (about ±0.5) indicate a moderate correlation. In the case of the CMM maps in hole and electron accumulation regimes ( Figure 2a,b), the ZNCC coefficient is 0.93, thus confirming the high level of positive correlation (i.e., similarity of the charge density distribution).
As a next step toward understanding the impact of local network properties on the charge transport, we assess whether the non-uniform carrier distribution in the channel can be correlated with the presence of local orientational order at the micrometer scale within the random network, or with the local density of the SWCNT network. Hence, we performed polarized CMM measurements, and quantified the local degree of orientational order of the SWCNT active layer as previously shown by Martino et al. [41] This analysis is detailed in the Figure S7 (and discussion therein), Supporting Information, and clearly indicates that there is no sufficient local alignment of the SWCNTs network to support a positive correlation with the signal intensity. In addition, we carefully measured the local optical density (OD) of the film to reveal any possible correlation with the network density. In fact, previous studies have demonstrated that the density of SWCNTs has a strong impact on the charge carrier percolation. [24,50] Since the examined SWCNT networks are only few nanometers thin, the film is highly transparent. Hence, we averaged 20 OD maps acquired by scanning the same area to obtain reliable values (details in Figure S8, Supporting Information). It is important to highlight that the contrast of signal intensity in the CMM maps is more difficult to be rationalized when |V os | > |V os *|, since the signal starts to drop because of high electrostatic doping ( Figure S6 and discussion therein, Supporting Information). However, it is fairly straightforward for |V os | < |V os *|, where the E 11 bleaching peak increases monotonously with the offset potential, that is, with the carrier density. For this reason, we analyze in Figure 3a-c how the CMM map in hole accumulation regime evolves by subsequently tuning V os from −0.6 to −0.8 and to −1.0 V. Here, we report the standardized OD on the z-axis, which is identical for all the panels since the mapped area is the same. Thus, the structuration on the z-axis allows

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to visually compare pixel-by-pixel the SWCNT network density with the local CMS signal (colormap). It is possible to detect a good correlation between those regions of the maps where the CMS signal is high and peaks in OD (e.g., at the center of the scan and at the top corner), and between regions with low CMS signal and valleys of SWCNT density (e.g., the bottom corner).
This trend is further corroborated by plotting the distribution of ΔT/T versus the standardized OD for all the pixels ( Figure S9, Supporting Information). Such a correlation suggests that regions with higher network density present more connections among the carbon nanotubes, favoring the formation of percolative pathways and therefore preferential transport of charges. Still, there are some locations such as the top-right corner where ΔT/T is high, but OD is average. To quantify the degree of correlation between the charge density distribution and the SWCNT density, we computed the ZNCC coefficients of the CMM maps at V os = −0.6, −0.8, −1.0 V with the OD map. In agreement with previous visual observations, the obtained ZNCC coefficients are 0.39, 0.41, and 0.37, respectively, which indicate a moderate positive correlation of the charge carrier density with the SWCNT network density. Thus, SWCNT density alone cannot explain the distribution of free carriers in the channel, which must be influenced by other local properties such as bundles, or residues of wrapping polymer, or screening from the gate field.
Moreover, it is interesting to investigate how the charge distribution evolves by progressively increasing the offset potential. This information is valuable to understand the early stages of device switching and possibly relate it to parameters like subthreshold swing and threshold voltage. Although the CMM maps in Figure 3a-c generally present a similar signal distribution, the increase of the CMS signal with |V os | is not uniform. Hence, we analyze the variation of the CMS signal as a function of V os in Figure 3d-f, showing that the charge density varies locally in different ways.
From −0.6 to −1.0 V (Figure 3f), there is an increase of charge density nearly everywhere in the mapped area. Remarkably, the increase of the CMS signal is higher in the regions with the lowest SWCNT density (see for instance the map at x,y < 5 µm). In contrast, a negative variation of the CMS signal when increasing |V os | (blue regions in Figure 3d-f) can indicate two opposite phenomena: 1) the charge density locally decreases even if |V os | is raised because of the activation of different percolation paths at higher |V os |, or 2) the charge density increases with |V os | in agreement with the average trend for the entire active area, and locally reaches values greater than d*.
The former mechanism is at the origin of the few blue spots in Figure 3f, where the CMS signal variation is tracked from −0.6 to −1.0 V. The reduction of the local CMS signal in these areas is likely due to a drop of the local charge density from −0.6 to −0.8 V (Figure 3d), followed by a partial recovery from −0.8 to −1.0 V (Figure 3e). The decrease of the local charge density from −0.6 to −0.8 V can be rationalized considering that the percolation paths at −0.6 V become less effective once other pathways are being activated at −0.8 V. This effect might be attributed to a non-linear activation of tunneling processes among the SWCNTs that depends on the specific bias point (V os ) and on the local morphological properties of the network.

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The latter mechanism can explain the blue spots in Figure 3e (negative signal variation from −0.8 to −1.0 V). In these regions, the local charge density likely reaches values greater than d* already at −1.0 V. Indeed, the CMS signal grows in these regions from −0.6 to −0.8 V (Figure 3d), locally reaching an absolute intensity higher than average at −0.8 V (red spots in Figure 3b). Hence, it is plausible that the charge density keeps on increasing in these zones as |V os | is progressively raised from 0.6 to 0.8 and then to 1.0 V.
As a matter of fact, these CMM maps capture an evolution of the quasi-static charge density distribution as the transistor is gradually switching to its on-state. By increasing |V os |, the charge density does not rise homogeneously, but a redistribution of the most favorable percolation paths occurs likely because of a nonlinear activation of tunneling processes among the SWCNTs. Overall, the areas with the lowest OD (i.e., the lowest density of SWCNTs) are those experiencing the highest increase in CMS signal as |V os | raises, with up to 60 % positive variation (see also Figure S9, Supporting Information). These variations indicate that even in a monochiral nanotube network charge transport paths are complex and vary with the applied gate voltage.

Multichiral SWCNT Networks
Starting from the knowledge acquired from monochiral SWCNT networks, in the following we will analyze CMM measurements on FETs based on a multichiral network composed of a mix of five different SWCNT species (Table S1, Supporting Information). The aim is to unveil the role of the network composition on the charge transport. The dispersion was obtained by selective polymer-wrapping from HiPco SWCNT raw material (details in the Experimental Section), which produces a mix of only five semiconducting species, namely (7,5), (7,6), (8,6), (8,7), and (9,7) SWCNTs, which are identified from their characteristic E 11 transition at 1045, 1135, 1200, 1285, and 1350 nm, respectively ( Figure S3, Supporting Information). [10] We fabricated FETs based on the multichiral network using the same architecture as for the devices with monochiral (6,5) nanotubes. A transfer curve is shown in Figure 4a, and a representative AFM image of the active layer is displayed in Figure 4b. The mix of different SWCNTs species introduces some non-idealities, such as a marked hysteresis and a threshold voltage shift. Also, the field-effect mobility is lower than in the previous case (about 0.5 cm 2 V −1 s −1 for both electrons and holes, with a strong dependence on the gate potential as already seen for the monochiral network, see Figure S4, Supporting Information).
A typical local CMS spectrum of the multichiral network ( Figure 4c) presents five positive peaks, which correspond to the bleaching of the E 11 transitions of each SWCNT species (absorption spectrum provided in Figure S3, Supporting Information). The trend of each E 11 bleaching peak as a function of V os is analogous to that analyzed for the monochiral network ( Figure S10, Supporting Information). The negative band above 1350 nm is assigned to charge-induced absorption (see Table  S1, Supporting Information). The (9,7) bleaching peak overlaps with the (8,6) trion peak, complicating the correct quantification of its intensity. In agreement with reports by Zorn et al., [34] the relative intensity ratio between the CMS bleaching peaks of the five SWCNT species varies as a function of the applied V os ( Figure S10a, Supporting Information). When normalizing the spectra at the bleaching peak of the (8,7) species (Figure S10b, Supporting Information), the (7,6) and (8,6) bleaching peaks grow with respect to the (8,7) peak when |V os | is increased, while the (9,7) bleaching peak decreases.
These observations confirm at the micrometer scale that SWCNT species with a low band gap dominate the charge transport at low gate voltage, while the contribution of larger band gap species becomes more relevant as the potential increases and the applied bias favors tunneling between nanotubes of different chirality. Yet, at the micrometer scale the intensity ratios among the peaks associated to the E 11 bleaching of the various chiralities depend on the probing position. The latter can be observed by comparing local CMS spectra measured in different locations of the same device in Figure 4d. CMS spectra measured on the entire device area result from a statistical average of the local spectra, highlighting the significant role of the local composition of the networks on charge transport ( Figure S11 and discussion therein, Supporting Information). Note that within one measurement spot of ≈0.8 µm width, ≈25-35 nanotubes are located and contribute to the signal.
Figures 4e and 4f depict the CMM maps corresponding to E 11 bleaching of the (7,6) species (1135 nm) in hole and electron accumulation regimes, respectively. The two maps are very well correlated (ZNCC coefficient is equal to 0.80), which means that holes and electrons share the same percolation paths as in the case of the monochiral network. The CMS spectra reported in Figure S12, Supporting Information, acquired from the entire device also support this conclusion, as the ratios among the characteristic E 11 peaks of the five chiralities are the same in hole and electron accumulation regimes.
We map the E 11 bleaching peak in hole accumulation regime for all the five species in a 10 × 10 µm 2 area (Figure 5a-e). All the maps present characteristic patterns that indicate the presence of different percolative pathways involving all the chiralities. Importantly, we perform the CMM measurements at V os = −0.5 V (|V os *| > 0.6 V for all the species), so to correlate the intensity of the CMS signal with the density of charge carriers accumulated on the specific SWCNT chirality in the FET subthreshold regime. In addition, at such low potential the intensity of the polaron/trion peak could be considered much lower than the (9,7) bleaching peak, allowing us to include all the chiralities in the analysis. We further investigate the complex charge carrier distribution originating from the presence of energy barriers by comparing the CMM maps and the OD maps corresponding to the E 11 peak of the five SWCNT species (Figure 5f-j).
The OD maps display similarities in the distribution of SWCNT between species with close values of the band gap, while they seem uncorrelated for those with large band gap difference (e.g., (7,5) and (9,7) SWCNTs). Such a distribution can be rationalized based on previous literature suggesting that aggregation is favorable among species having similar band gap. [51] The cross-correlation analysis among the five OD maps corroborates this observation ( Figure S13a, Supporting Information). We note that for some spots, the OD of a certain chirality is close to zero. While the absorption spectrum ( Figure S3,

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Supporting Information) already shows a low abundance of (7,5) SWCNTs and some local inhomogeneities are expected, it seems unlikely that species with a higher abundance (such as (8,7) and (9,7) SWCNTs) are not present in extended regions of the dense networks. Some fluctuations might arise during the transmission measurements of the highly transparent films or due to minor shifts of the E 11 absorption peaks for different dielectric environments. By selecting the probing wavelength corresponding to the E 11 peak of each of the species, we could map the local CMS signal arising from the modulation of the charge density of each chirality independently (Figure 5a-e). Although the local CMS features in these maps appear generally more fragmented than for the (6,5) monochiral network, it is remarkable that charge carriers distribute on all the species and throughout the 10 × 10 µm 2 mapped area. Thus, all five SWCNT species clearly take an active role in charge transport. Effective percolation pathways across the FET channel likely involve hopping of charge carriers among regions where the transport proceeds preferentially on different chiral species. In contrast to the monochiral (6,5) SWCNT network, there is no clear correlation between the SWCNT density (proportional to the OD maps of Figure 5f-j) and the accumulated free carrier density (proportional to the CMS signal in Figure 5a-e). Consistently, the ZNCC coefficients of each CMM map with the corresponding OD map range from 0.09 to −0.20 for all the species ( Figure S13b, Supporting Information), thus indicating extremely low linear correlation as opposed to the ZNCC coefficient of ≈0.40 obtained for the monochiral network. Such evidence suggests that the bottleneck to the charge transport in the multichiral network is not represented by the number of www.advmatinterfaces.de tunneling junctions among SWCNTs, but by the energy barriers arising between nanotubes with different diameters and band gaps. [52]

Average Size of Percolation Pathways
To corroborate this hypothesis, we compare the average dimension of percolation pathways in the monochiral and multichiral networks. We aim to identify a characteristic length scale of regions of the CMM maps displaying similar signal intensity, as this corresponds to areas with uniform charge carrier density. The rationale is that a homogeneous charge carrier distribution arises in presence of extended effective percolation pathways, while a heterogenous charge density is associated to inefficient fragmented percolation pathways. To this purpose, we compute the 2D autocorrelation function (ACF) of the CMM maps and derive the autocorrelation length. The autocorrelation length of an image, ξ, represents the average distance at which a pattern is repeated, and so it refers to the typical size of domains on an image. The procedure is illustrated in Figure 6a for the CMM map of the (6,5) monochiral network measured at V os = −0.6 V (Figure 3a). First, we calculated the 2D ACF of the CMM map, which is equivalent to the ZNCC of the CMM map with itself shifted along the x-and y-axis of d x and d y , respectively. Thus, the colormap represents the ZNCC as a function of the spatial shift (d x , d y ), that is, the relative distance between the two CMM maps that are being cross-correlated (computational details reported in the Experimental Section). Then, we calculate the ACFs related to the magnitude of the spatial shift ( = + ), by averaging in all directions the points of the 2D autocorrelation map as a function of R. The ACF is identically equal to 1 at zero shift (R = 0), being the correlation of the map with itself, and then it drops exponentially with R. Hence, we interpolate the ACF with an exponential function, f(R) = k·exp(−R/ξ), where k ≈ 1, and ξ denotes the correlation length. The value of the correlation length for the CMM map of the (6,5) monochiral network is ξ = 0.89 µm. As defined, ξ corresponds to the shift R at which the ACF decays in value to 1/e. Importantly, ξ is not to be considered as a characteristic transport length (e.g., it is not a mean free path between two tunneling events). It represents the average distance between two pixels of the CMM map at which the local CMS signal preserves a high correlation. Hence, ξ is correlated to the size of areas of the semiconducting film displaying very similar conductivity and it can be used to compare the extension of percolation paths in networks with different composition. Therefore, in Figure 6b we adopt the same procedure to compute the 2D ACFs of the CMM maps for the multichiral network (Figure 5a-e) and we derive the correlation lengths associated to percolation domains of all the five (n,m) species composing the network. The values of the correlation lengths are reported in Figure 6b and range from 0.52 to 0.83 µm. Except for the (7,6) species, ξ grows from (7,5) to (9,7) SWCNTs: small band gap species are those contributing the most to charge transport, hence the extent of the percolation domains is larger than for large band gap species, whose percolation paths are more fragmented as evident for the CMM map of the (7,5) species (Figure 5a). Consistently, the largest ξ is obtained for the monochiral (6,5) SWCNT network (Figure 6a), for which no energy barriers due to a mixed composition are present. The large value of ξ for the (7,6) species is difficult to be rationalized and it might originate from the complex dependence of the charge transport on the specific local composition of the network and on the correlation with the applied gate potential.

Conclusions
We studied the charge transport in FETs based on nearly monochiral (6,5) SWCNT networks and on a multichiral network with five different chiralities on a microscopic scale using CMM. By measuring local charge modulation spectra and mapping the signal distribution within the FET channel, we could visualize how free carriers are spatially distributed in the device under operating conditions. In particular, we provide direct evidence that both holes and electrons are preferentially moving along the same percolation paths. For the case of the monochiral network, we highlight non-linear local changes of Figure 5. a-e) CMM maps (V os = −0.5 V) and f-j) optical density maps for a multichiral SWCNT network (scale bars are 2 µm). From left to right the maps show the E 11 peak for (7,5), (7,6), (8,6), (8,7), and (9,7) species, respectively. www.advmatinterfaces.de charge density distribution in the subthreshold regime when the gating potential is progressively increased. While in monochiral networks we observed a moderate dependence of the charge distribution on the network density, in multichiral networks the charge transport is influenced more by the energy barriers between SWCNTs with different band gaps. We analyzed the 2D ACFs of the CMM maps for the (6,5) monochiral network and for all the five nanotube chiralities present in the mixed network. This allowed us to estimate a correlation length, ξ, associated with the average extent of charge percolation paths. The value of ξ for the CMM map of the monochiral network is larger than for the maps of the five species in the multichiral network, which is consistent with the presence of more extended percolation pathways beneficial to charge transport. The correlation length derived from the CMM maps of each species present in the multichiral network can be related to the average distance traversed by a charge carrier on nanotubes of the same chirality before it eventually tunnels onto another species. Such a distance is on average longer for SWCNTs with large diameters (hence, narrow band gap), while the most fragmented percolation paths are observed for species with the smallest diameter (larger band gap). Overall, CMM emerges as a valuable technique to investigate the local charge transport properties of random SWCNT networks and to identify critical length scales in solution-processed thin film devices.

Experimental Section
Monochiral (6,5) SWCNT Dispersion: Essentially monochiral dispersions of (6,5) SWCNTs in toluene were obtained from CoMoCAT SWCNT raw material (CHASM Advanced Materials Inc., SG65i-L58) via selective polymer-wrapping with PFO-BPy (American Dye Source, M w = 34 kg mol −1 ) under shear force mixing as described previously. [11] Shear force mixing (Silverson L2/Air mixer) was applied at 10 230 rpm for 72 h at a constant temperature of 20 °C. Then, the dispersion was centrifuged twice for 45 min at 60 000 g (Beckman Coulter Avanti J26XP centrifuge), and the supernatant was filtered with a polytetrafluoroethylene (PTFE) syringe filter (pore size 5 µm) to remove the undispersed material and aggregates. SWCNTs were collected by filtration through a PTFE filter (Millipore JVWP, pore size 0.1 µm) and the filter cake was washed with hot toluene to remove excess unwrapped polymer. Ultimately, the filter cake was immersed in 1 mL of toluene and bath sonicated for 30 min to obtain the ink, which appears of purple color with OD of 10 cm −1 at the E 11 transition.
Multichiral SWCNT Dispersion: Multichiral SWCNT dispersions in toluene were obtained by polymer-wrapping with poly(9,9-dioctylfluorene) (PFO, Sigma Aldrich, M w > 20 kg mol −1 ) from HiPco SWCNT raw material (Unidym Inc., batch no. 2172) using bath sonication (45 min) as exfoliation method. The resulting suspension was centrifuged for 45 min at 60 000 × g (Beckman Coulter Avanti J26XP centrifuge), and the supernatant was collected. The pellet was recycled twice, and the supernatants were combined to increase the overall yield. The PFO-wrapped SWCNTs were sedimented by ultracentrifugation for 20 h at 284 600 × g (Beckman Coulter Optima XPN-80 centrifuge), and the obtained SWCNT pellet was washed with tetrahydrofuran (THF) and re-dispersed in 1 mL of toluene via bath sonication for 30 min to obtain a greenish dispersion.  Figure 3a). First, the 2D autocorrelation function is obtained by computing the zero-mean normalized cross-correlation of the CMM map with itself as a function of the spatial shift (d x , d y ). Hence, the autocorrelation as a function of the distance from the origin, 2 2 R d d x y = + , is computed by averaging the points of the 2D autocorrelation map at the same R. The data are fitted with an exponential function, f = k·exp(−R/ξ), where k ≈ 1 and ξ is the correlation length. b) The same procedure is applied to the CMM maps of the five species composing the multichiral network (Figure 5a-e), deriving the corresponding 2D autocorrelation functions and correlation lengths.

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Device Fabrication: SWCNT FETs were fabricated in a bottomcontact top-gate configuration. Source and drain interdigitated contacts (L = 20 µm, W = 10 mm) were patterned on glass substrates (Schott AG, AF32eco, 300 µm) using standard photolithography (LOR5B/ S1813 double-layer resist), electron-beam evaporation of chromium (2 nm) and gold (30 nm), and lift-off in N-methyl pyrrolidone. Before depositing the SWCNT films, samples were cleaned by ultrasonication in acetone and 2-propanol (10 min each), followed by UV ozone treatment (Ossila E511, 10 min). Dense, monochiral (6,5) and multichiral SWCNT networks were deposited from the dispersions via a spin-coating process (2000 rpm, 30 s) that was repeated for three times with brief annealing steps (100 °C, 2 min) in between. The SWCNT films were rinsed with THF and 2-propanol to remove excess polymer. The SWCNTs outside of the channel area were removed using photolithography as detailed above and oxygen plasma treatment (Nordson MARCH AP-600/30, 100 W, 2 min). Samples were annealed at 300 °C for 1 h in inert atmosphere before deposition of the double-layer gate dielectric. First, a thin PMMA layer (≈11 nm) was spin-coated (4000 rpm, 1 min) on the SWCNTs film from a solution in n-butyl acetate (6 g L −1 ). Then, atomic layer deposition (Ultratech Savanna S100) at 100 °C was used to deposit a hafnium oxide (HfO x ) layer (≈61 nm) which simultaneously served as encapsulant for the devices. The measured total capacitance of the dielectric layer was C = 120 nF cm −2 for the (6,5) SWCNT-based FET and C = 98 nF cm −2 for the multichiral SWCNT-based device. Finally, silver top gates (20 nm) were thermally evaporated through a shadow mask.
Characterization: AFM images were acquired on a Bruker Dimension Icon in ScanAsyst mode under ambient conditions. Baseline-corrected absorption spectra of SWCNT dispersions were measured on a Varian Cary 6000i UV-vis-NIR spectrometer. The transfer and output electrical characteristics of the FETs were measured with a semiconductor parameter analyzer (Agilent B1500A) in a nitrogen glove box on a Wentworth Laboratories probe station.
Charge Modulation Spectroscopy and Charge Modulation Microscopy: CMS measurements on the entire active area of the FETs were performed in vacuum (≈10 −6 mbar) by modulating the gate voltage with a waveform generator (3390, Keithley), while both the source and drain electrode were grounded. A tungsten lamp was used as light source in combination with a monochromator to select the wavelength during the measurement, and the light transmission was measured with an InGaAs photodiode. The electrical signal was amplified through a trans-impedance amplifier (Femto DHPCA-100) and read with a DSP Lock-in amplifier (Stanford Research Systems SR830), synchronized to the wave generator.
CMM measurements were performed with a home-made setup, whose details are provided in the Figure S1, Supporting Information.
Zero-Mean Normalized Cross-Correlation Calculation: The signal in every pixel i of the A and B maps was previously standardized according to: where µ corresponded to the mean signal of the map, and σ to its standard deviation.
The ZNCC map C of the CMM/OD map A with the map B was calculated as: where F denoted the Fourier transform. The 2D cross-correlation maps were plotted as a function of the spatial shift (often called "lag") along the x-and y-axis (d x and d y , respectively), so that the origin corresponds to the zero-lag crosscorrelation maps. The Matlab code that was implemented for this calculation is reported in the Supporting Information. Note that for A = B, the cross-correlation coincided with the 2D ACF of the map. From the 2D autocorrelation maps (2D-ACF), the radial average of the 2D-ACF was obtained by calculating the mean value of the ACF as a function of the distance R from the origin (

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.