Charge transport in semiconducting carbon nanotube networks

Efficient and controlled charge transport in networks of semiconducting single-walled carbon nanotubes is the basis for their application in electronic devices, especially in field-effect transistors and thermoelectrics. The recent advances in selective growth, purification, and sorting of semiconducting and even monochiral carbon nanotubes have enabled field-effect transistors with high carrier mobilities and on/off current ratios that were impossible a few years ago. They have also allowed researchers to examine the microscopic interplay of parameters such as nanotube length, density, diameter distribution, carrier density, intentional and unintentional defects, dielectric environment, etc., and their impact on the macroscopic charge transport properties in a rational and reproducible manner. This review discusses various models that are considered for charge transport in nanotube networks and the experimental methods to characterize and investigate transport beyond simple conductivity or transistor measurements. Static and dynamic absorption, photoluminescence and electroluminescence spectroscopy, as well as scanning probe techniques (e.g., conductive atomic force microscopy, Kelvin probe force microscopy), and their unique insights in the distribution of charge carriers in a given nanotube network and the resulting current pathways will be introduced. Finally, recommendations for further optimization of nanotube network devices and a list of remaining challenges are provided.

limiting or crucial factor. Also, in some cases the direction of transport is out-of-plane rather than lateral within the network. However, one application, in which the charge transport in a nanotube network also plays an important role is thermoelectrics.
In thermoelectric devices a temperature gradient is turned into a voltage (Seebeck effect) as majority carriers (holes or electrons) move from the hot to the cold side. By connecting pdoped and n-doped semiconducting blocks thermally in parallel and electrically in series an overall current can be source and, e.g., waste heat can be turned into electricity. Numerous reviews are available on the details of novel thermoelectrics, which can be based on a variety of inorganic, organic, composite and nanostructured materials. 70-74 However, with regard to charge transport in carbon nanotube networks it is important to keep in mind that the figure of merit for thermoelectric materials, the zT value (unitless), is directly proportional to the electrical conductivity σ (S·cm −1 ) and the square of the Seebeck coefficient α (μV·K −1 ) but inversely proportional to the thermal conductivity κ (W·m −1 ·K −1 ).
For maximum power generation from a certain temperature gradient the power factor σα² has to be maximized. The electrical conductivity obviously depends on the carrier concentration and mobility of the active layer just as in a transistor, although the type and number of mobile carriers is typically determined by chemical doping (p-type or n-type) 75,76 and only in some cases by electrochemical doping (e.g., electrolyte gating). 77,78 The efficiency of the required doping of the nanotube network strongly depends on the combination of nanotube diameter (i.e., bandgap), applied dopants and processing. 76,79,80 The thermal conductivity is governed by lattice phonons and electronic contributions and hence also increases with carrier concentration while the Seebeck coefficient decreases. Furthermore, the thermal conductivity of a single nanotube is exceptionally high with about 3500 W·m -1 ·K -1 , 81 but the additional phonon

A. Percolation in nanotube networks
The overall conductance of any network of conducting wires with a certain length depends on the density and connectivity of these wires as described in general by percolation theory. 89 Below the so-called percolation threshold, the conductance is negligible and around it the conductance increases by orders of magnitude before reaching a nearly constant value.
Percolation models (in 3D and 2D) have been especially important for the application of singlewalled and multiwalled carbon nanotubes as conductive fillers in insulating polymers 90 but also for transparent conductive electrodes. 91,92 Due to their high aspect ratio the percolation limit for nanotubes in thin films or networks is reached at extremely low concentrations with the precise values depending on the average nanotube length in relation to the electrode distance. [93][94][95] With respect to 2D networks of semiconducting nanotubes in field-effect transistors, percolation models were mainly useful for describing the impact of residual metallic nanotubes and the resulting low on/off current ratios. 96-98 Using a simple stick-percolation-based transport model already enabled a better understanding of the influence of nanotube alignment within the channel and the prediction of preferred current pathways. 99-101 For example, Monte Carlo simulations of networks with different levels of nanotube alignment indicated that partially aligned rather than strongly aligned nanotube films should show the lowest resistivity, with the minimum depending on the nanotube length and density as well as channel length. 102 This behaviour was explained with the decreased number of possible junctions and thus conduction pathways for almost perfect alignment (i.e., parallel nanotubes).
Practical nanotube networks are typically far above the percolation threshold, even reaching multilayer thicknesses. Networks with linear densities above 10 SWNTs per micrometer already show no more dependence of the effective carrier mobility on the network density in experiments and thus deviate significantly from the simple 2D percolation model. 23 Nevertheless, more realistic 3D percolation models that could capture the real morphology of a network are still rare 103 and the inclusion of junctions 104 as well as the intrinsic charge carrier density-dependent resistance of different nanotubes poses a significant challenge.

B. Intra-versus inter-nanotube transport
Assuming the nanotube network is dense enough, such that percolation is not a limiting factor anymore, the question of hole and electron transport in a thin film of only semiconducting carbon nanotubes of certain diameters, lengths etc. comes to the fore. Fig. 4 visualizes the various factors contributing to conductance or resistance in a dense nanotube network: the intrinsic charge transport along a stretch of nanotube, nanotube lengths, nanotube-nanotube junctions (between SWNTs of the same chirality or with different bandgaps), the energetic landscape created by a distribution of different diameters of nanotubes as well as the surrounding dielectric (e.g., trap states or dipolar disorder), scattering due to lattice defects (intentional or unintentional) and field screening in bundles or aggregates. Clearly, the interplay of all of these factors is complex and no analytical model or even numerical simulation exists that can take all of them into account. However, with certain simplifications at least some general trends can be explained.
Let us first consider the intrinsic transport within individual semiconducting carbon nanotubes depending on their specific diameter, band structure and DOS. As shown experimentally and theoretically, individual SWNTs that are longer than the mean free path of electrons (< 1 µm) 15 exhibit diffusive band transport limited by scattering at phonons or possibly defects. This also means that the maximum conductance and charge carrier mobility increase with decreasing temperature due to a lower phonon scattering rate. 105, 106 Perebeinos et al. found that the impact of electron-phonon scattering on the carrier mobility µ of an individual (defect-free) nanotube depending on its diameter (d) and temperature (T) can be described empirically by 105 with 0 = 12 000 cm²V -1 s -1 and β = 2.26. This relation would yield an intrinsic charge carrier mobility of ~6400 cm²V -1 s -1 for a single (6,5) SWNT (d = 0.757 nm) at room temperature, but a ~3 times higher mobility for a (13,5) SWNT (d = 1.278 nm), highlighting the strong impact of the tube diameter on its charge transport properties. Measurements on nanotubes grown by chemical vapour deposition found lower effective mobilities but nearly the same diameter and temperature dependence. 106 The one-dimensional DOS of nanotubes also results in a limited quantum capacitance 24 and a charge carrier density-dependent mobility with a clear maximum (see Fig. 2c). The decrease in carrier mobility after the maximum is due to filling of the first subband, which has been shown theoretically and experimentally for individual nanotubes and nanotube networks. 24,107,108 For even higher carrier concentrations the second subband is filled, as demonstrated for electrolyte-gated small-bandgap nanotubes. 107

FIG. 4.
Illustration of different contributions to conductance or resistance in dense networks of purely semiconducting SWNTs: Charge carrier transport across nanotube-nanotube junctions (inter-nanotube) and along nanotube segments (intra-nanotube), bandgap differences, dipolar environment (e.g., at the semiconductordielectric interface), dielectric environment (e.g., due to residual wrapping polymer from the nanotube selection process), as well as scattering at unintentional defects (e.g., from growth or processing) and intentional sp 3 defects.
In SWNT networks, charge carriers do not only move through segments of individual nanotubes but also have to 'hop' across junctions between different nanotubes. These junctions are associated with resistances on the order of 10 2 -10 5 kΩ and have been shown to also depend on the presence of bundles and the electronic type of the crossing nanotubes. 109-111 Tunnelling through a junction is more likely between two nanotubes of large and similar diameter and also depends on the angle between the nanotubes. 112 In a field-effect transistor, realistic junctions are further complicated by the limited gating of a nanotube on top of another or in a bundle due to field screening. A typical assumption is that the junctions are the main bottleneck for transport. A nanotube network can thus be seen as a type of disordered semiconductor or more generally, a disordered material with large conducting regions whose macroscopic conductivity is limited by carrier hopping between the conducting elements. With this simplification one may arrive at an analytical transport model that describes the temperature dependence and ideally also the carrier concentration dependence of the conductivity or apparent carrier mobility. Several models, which originally had been developed to describe charge transport in disordered inorganic or organic semiconductors, have been applied to networks of SWNTs. 113 Most commonly encountered among these models are the variable range hopping (VRH) model 114 and the fluctuation-induced tunnelling (FIT) model. 115 The VRH model describes the hopping of charges between localized states close to the Fermi level, with the hopping probability depending on the energetic and spatial separation of these states. 20,114 In the framework of the VRH model, the temperature-dependent conductivity ( ) is expressed as with the prefactor 0 , the dimensionality of the system , and 0 as the hopping parameter.
Note that hopping is a thermally activated process and the conductivity and carrier mobility increase with temperature in contrast to the inverse temperature dependence of the intrinsic carrier mobility of SWNTs. The FIT model, which was originally developed for conducting polymers, is based on the fluctuation-induced tunnelling of charge carriers through barriers between conductive segments. 115,118 According to the FIT model, the conductivity follows the form described in Eq. 6: where 0 is a temperature-independent prefactor, 1 is the activation energy required to promote a charge carrier over the insulating barrier, and 0 is the temperature above which as well as the inter-nanotube junctions. 104 They found that charge transport in networks is strongly influenced by the density, length and relative orientation of SWNTs, and that phonon scattering is reduced in dense nanotube networks, leading to higher currents.

A. Temperature-dependent transport measurements
The proposed charge transport models for nanotube networks (see above) can be tested against temperature-dependent measurements of charge carrier mobilities and conductivities in fieldeffect transistors or doped thin films. Measuring the change of carrier mobility with temperature (usually from room temperature to cryogenic temperatures) is indeed a commonly used tool to investigate charge transport in semiconductors and, for example, enables the distinction between band or band-like transport and various types of thermally activated transport in disordered systems. Determining the temperature dependence of charge transport in novel semiconductors such as organic single crystals, [119][120][121][122] polymers, [123][124][125][126] and inorganic nanomaterials [127][128][129] has helped to understand their underlying properties and to improve them.

Thin film conductivity and field-effect transistors
The temperature and diameter dependence of the transport properties of individual semiconducting SWNTs in back-gated field-effect transistors were studied early on by Zhou et al. who found a linear increase in peak mobilities and on-state conductance with 1/T in the temperature range between 50 K and 300 K. 106  CoMoCAT SWNTs (nearly monochiral (6,5) SWNTs, diameter 0.76 nm) to compare the different network properties and possibly distinguish intra-and inter-nanotube transport. By using a gFPP device layout, the authors were able to extract the contact resistances and the intrinsic field-effect mobilities for temperatures between 100 K and 300 K. A strong dependence of the contact resistance on temperature was found, which underlined the necessity of gFPP method especially for devices with large-bandgap nanotubes.
Decreasing on-currents and on-voltage shifts to higher absolute values for holes and electrons with decreasing temperature were observed for all transistors in this study (see The temperature dependence of the contact resistance-corrected carrier mobilities for all types of networks in this study was best described by the FIT model (see Fig. 5c) 35 , although with varying quality and reliability of the fits. The HiPco SWNT network with the largest spread in diameters (bandgaps) exhibited the strongest temperature dependence. However, the relative temperature-dependence of the plasma torch SWNT network consisting of large-diameter nanotubes with a narrow bandgap distribution proved to be consistently lower than for the monochiral (6,5) SWNT network, and the absolute mobilities were higher. While this study focussed on both hole and electron mobilities, the same diameter-dependent trends were later confirmed for doped n-type FETs of the same polymer-sorted nanotube networks. 151 These data indicate that either the junction resistance between nanotubes and its temperature dependence strongly vary with the SWNT diameter or the overall SWNT network mobility is not only governed by the nanotube junctions, but also by the intrinsic temperature (µ ~ 1/T) and diameter dependence (µ ~ d²) of the intra-nanotube transport. This question was further examined in a subsequent study by Brohmann et al. where the authors created semiconducting SWNT networks with tailored compositions by mixing large-bandgap, monochiral (6,5) SWNTs and small-bandgap but polydisperse plasma torch SWNTs in different ratios. 137 Temperature-dependent transport measurements revealed the strongest temperature dependence of the contact resistance-corrected carrier mobilities for the (6,5) SWNT network. FETs based on pure plasma torch nanotube networks showed the weakest temperature dependence, but overall behaved quite similar to networks with equal shares of both stock dispersions, which suggested that the latter already contained a substantial number of transport pathways with only large-diameter nanotubes. This systematic difference in temperature-dependence for the different network compositions could again be explained as a superposition of different contributions, namely the thermally activated inter-tube transport across junctions, the dependence of intra-nanotube transport on the inverse temperature, and its dependence on the square of the nanotube diameter.
For many organic semiconductors a strong dependence of the carrier mobility on energetic disorder induced by the randomly oriented dipoles of the gate dielectric reflected in its dielectric constant (k) can be found, leading to higher mobilities in FETs with low-k dielectrics such as CYTOP TM . 119,153,154 For individual carbon nanotubes the impact of the gate dielectric or polar substrate such as SiO2 has been described theoretically with surface polar phonon scattering, which reduces the intrinsic mobility (limited by acoustic phonon scattering) by an order of magnitude and should indeed be even worse for high-k dielectrics such as ZrO2 or HfO2. 155  CYTOP TM (k = 1.8), and a high-k ferroelectric relaxor P(VDF-TrFE-CTFE) (k = 14.2) as gate dielectrics. They confirmed thermally activated transport in all cases and from fits according to the FIT model, a higher activation energy for transport was obtained for the high-permittivity ferroelectric polymer (see Fig. 5d). Correspondingly, PMMA and CYTOP TM exhibited lower activation energies. Similar trends were obtained with Arrhenius fits. However, further systematic studies, especially on monochiral SWNT networks and networks with larger diameters (and hence lower contact resistance), with a wider range of dielectrics will be needed to fully understand their impact on the intrinsic transport properties of nanotube networks.
The studies discussed above do not explicitly consider defects in the employed nanotube networks that may be introduced during growth or processing. Recently however, the controlled low-level chemical functionalization of SWNTs has attracted considerable interest due to the possibility to tune and enhance their emission properties. By covalently grafting, e.g., small aryl groups onto the nanotube sidewalls, so-called sp 3 defects are created, which act as efficient exciton traps and lead to red-shifted emission. 164,165 Measurements on singlenanotube devices indicated a decrease of the SWNT conductivity upon sp 3 functionalization, 166,167 however, the impact of sp 3 defects on charge transport in random nanotube networks was only recently investigated for pristine and functionalized (6,5) SWNT network FETs. 152 Both hole and electron mobilities decreased with increasing sp 3 defect density. Furthermore, a stronger temperature dependence of the carrier mobilities was found for functionalized nanotube networks compared to the unfunctionalized reference transistors (see Fig. 5e), with the differences being most pronounced in the low-temperature range. Due to the low number of sp 3 defects, it was assumed that they predominantly affected the intrananotube conductance within the networks rather than the inter-nanotube junctions, which may indicate that the variations of intrinsic nanotube mobility play a non-negligible role even in a network. 152

Thermoelectric measurements
In the field of organic electronics, the study of thermoelectric properties and in particular the determination of the temperature-and carrier density-dependent Seebeck coefficients has led to remarkable new insights. The Seebeck coefficient can, for example, help to assess the relevance of electron-phonon coupling in molecular semiconductors. It is also a measure of the entropy transported by thermally excited charge carriers. 126,168,169 Doped semiconducting nanotube networks have been investigated for a while as possible thermoelectric materials (see above) and correlations between residual polymer, nanotube diameter and diameter distribution were made. 76,84,170 However, a deeper look may provide even more fundamental insights.
In a detailed study, Blackburn et al. investigated the impact of charge transfer doping on the temperature-dependent thermoelectric properties of semiconducting SWNT networks. 170 The electrical conductivity was found to be thermally activated, reflecting the inter-nanotube junctions as bottlenecks to charge transport especially in the low-doping regime. In contrast to that, these limitations to charge transport only had a minor impact on the Seebeck coefficient, which increased with temperature (100 K -300 K) irrespective of the doping level of the SWNT network, exhibiting a behaviour similar to that of a highly conductive material. From the dependence of the Seebeck coefficient and consequently the power factor on the electrical conductivity, a transition between two different transport regimes was identified (see Fig. 6a).  . 6b), which would indicate a low-disorder system. Thermoelectric transport in the nanotube networks was further described according to the Boltzmann transport formalism, incorporating the framework of charge transport in heterogeneous media and the fluctuationinduced tunneling as corroborated by the extracted temperature-dependent carrier mobilities.
Simulations according to this model suggested that one-dimensional scattering with acoustic and optical phonons is not sufficient to reproduce carrier scattering in the SWNT networks.
Instead, scattering at inter-nanotube junctions needs to be included. Based on these results, trap-free SWNT networks with a narrow DOS distribution were proposed for optimized performance of electrical and thermoelectric devices.
mechanisms, such as the interplay between intra-and inter-nanotube transport in networks of different semiconducting SWNTs. 35,118,137 The nanotube junctions act as barriers to charge transport through the networks, and the activation energy required to promote a carrier across these junctions overall results in a thermally activated conduction mechanism in semiconducting SWNT networks as demonstrated in several studies. The temperature dependence of carrier mobilities in mixed semiconducting small-bandgap SWNT networks was lower than that in monochiral, large-bandgap nanotube networks. 137 Such insights are also important for the application of SWNT network transistors in electronic circuits with optimized performances. Indeed, for short channel transistors (channel length ~2 µm) with nanotube networks, the opposing trends of phonon scattering and thermally assisted hopping actually help to create ring-oscillators with a weak overall temperature dependence of the mobility and maximum oscillation frequency. 173 Unfortunately, only the averaged overall network properties such as conductivities or mobilities can be extracted from temperature-dependent electrical measurements. Due to the lack of spatial resolution, the formation of certain transport pathways can only be assumed, but not visualized. This especially concerns networks consisting of multiple semiconducting SWNT species, in which preferential charge transport through large-diameter (i.e., smallbandgap) nanotubes would be expected. To distinguish between different nanotube species, spectroscopic techniques can be utilized and will be discussed next.

B. Spectroscopic techniques
SWNTs exhibit strong and narrow absorption and emission features that are not only very specific to their structure but also highly sensitive to their environment and doping. The presence of charge carriers leads to substantial changes in the optical spectra of SWNTs. This property has enabled a wide range of methods utilizing the nanotubes' chirality-specific optical  [188][189][190] and can be efficiently quenched through interaction with structural lattice defects including nanotube ends, 191 other excitons, 192 and charge carriers. 193 A detailed description and overview of the photophysics of SWNTs can be found in recent reviews on the topic. 183,[194][195][196] Importantly, the spectral changes of absorption, photoluminescence and electroluminescence of SWNT networks can be employed to investigate the distribution of charge carriers and predominant transport paths in thin film devices even with some limited spatial resolution.

Absorption spectroscopy
Charge accumulation on SWNTs (e.g., by electrostatic, electrochemical or chemical doping) is associated with the effective bleaching of absorption, which has found practical application The CMS spectrum of a monochiral (6,5) SWNT layer 201 showed gate voltage-dependent bleaching of the main E11 absorption peak and red-shifted charge-induced absorption features (trions or polarons) as shown in Fig. 7b. 208,209 The correlation of the CMS signals of a multichiral network of five different semiconducting nanotubes (for absorption spectrum and DOS see Fig. 7c,d) with increasing offset voltage showed that at low carrier densities charges preferentially move through small-bandgap SWNTs (here (8,7) and (9,7)), irrespective of their comparatively low abundance in the network. With increasing total carrier concentration, the contribution of large-bandgap SWNTs (here (7,5)) to the overall charge transport increases (Fig. 7e).

Photoluminescence spectroscopy
Besides absorption bleaching, injected charges in SWNTs mediate non-radiative Auger recombination of excitons and hence PL quenching. 193,210 Owing to the high exciton and charge carrier mobilities in individual SWNTs but also in dense networks, 211 . 7d). 117 However, as quenching of the PL intensity can arise from either trapped or mobile charge carriers, it cannot be correlated unambiguously with the actual charge transport properties of the network.
To address this issue, Zorn et al. combined voltage-dependent PL spectroscopy with modulation spectroscopy to perform charge-modulated PL measurements (CMPL, similar setup to CMS but with PL excitation and detection). 201 In these experiments, the differential PL quenching (ΔPL) induced by mobile charge carriers is recorded. Analogous to the CMS experiments in transmission, trapped charges do not contribute to the detected signal as they cannot follow the modulation frequency. From the fraction of quenched PL, which reflects the share of mobile charges on a given nanotube species, the chirality-dependent contributions within the SWNT network were assessed quantitatively (Fig. 8c). The obtained values were in good agreement with previously reported numerical simulations, 108 further corroborating the idea that at low carrier concentrations (e.g., close to the subthreshold region) charge transport is dominated by the nanotubes with small bandgaps. Incidentally, these also exhibit the lowest injection barriers for both holes and electrons when using gold electrodes. As the carrier density increases, more of the larger-bandgap nanotubes start to contribute to charge transport until the entire network is involved at high gate voltages close to the maximum mobility. Importantly, these data suggest that the effective network density changes with applied gate voltage and that a monochiral network is preferable for more controlled and uniform device performance, e.g.
with regard to a steeper and reproducible subthreshold slope. 213

Electroluminescence imaging and spectroscopy
Excitons can also be created electrically in SWNT FETs by impact excitation with strongly accelerated carriers (unipolar operation), 214,215 or by injection of carriers with opposite polarity and subsequent electron-hole capture (ambipolar operation, for a schematic illustration see Fig.   9a) 216 218 They also found a red-shift of the emission maximum (~1950 nm) of the EL and PL spectra as well as spectral narrowing compared to the expected emission distribution (see Fig. 9b), which they assigned to predominant current flow through large-diameter nanotubes and exciton transfer from large-bandgap to small-bandgap SWNTs.
These initial studies suggested that EL spectra might be a good indicator for preferential charge transport in nanotube networks but that energy transfer between nanotubes may play an important role as well as seen for the PL spectra. 211,212 Networks with fewer nanotube species and with well-distinguishable emission features were clearly desirable to improve interpretation together with possible spatial resolution of emission.
Semi-aligned SWNT networks with only five distinct nanotube species selected via polymerwrapping were later studied by Rother et al. to determine preferential EL compared to PL. 117 By recording EL spectra at different gate voltages and calculating the share of emission for each chirality, the chirality and carrier concentration dependence of charge transport in such multichiral semiconducting SWNT networks was analyzed quantitatively (Fig. 9c,d). In agreement with CMS and CMPL data (see above), the nanotubes with the smallest bandgaps were found to contribute predominantly to the EL and thus current, even though these species only accounted for a small portion of all nanotubes within the network. This was evident from their large EL shares, which were significantly higher than what would be expected from their nominal abundance and PL spectra recorded from the same channel region. Nanotubes with large bandgaps barely contributed to charge transport at low gate voltages and only started to show EL at higher carrier densities. This notion was further corroborated by numerical simulations of current and EL distribution depending on gate voltage in a numerically simulated network with the same composition. 108 Similar to single nanotube devices, 175 Using very low nanotube concentrations close to the percolation threshold, they were able to map only a few distinct percolation paths, whereas the majority of SWNTs did not participate in charge transport as shown by comparison with PL images. Remarkably, the visualization of single current pathways revealed that these conductive paths are far from uniform and can even fork into several trails. With increasing SWNT concentrations, the number of transport pathways also increased but remained clustered rather than being homogeneously distributed.
This non-uniformity can be attributed to inhomogeneities arising from the network deposition, but also to the presence of some minority species with smaller bandgaps that should appear brighter in EL images.
photoswitchable spiropyran molecules into the polymeric gate dielectric layer of their (6,5) SWNT network FETs. Upon UV irradiation, the spiropyran undergoes a ring opening to form its merocyanine isomer, which was found to significantly reduce both hole and electron mobilities but did not affect the PL yield. Without any treatment the composite EL image of the channel was very homogeneous, representative of the uniform charge transport through a dense network of monochiral nanotubes. After UV light exposure through photolithography masks, regions of low (i.e., exposed) and high (i.e., not exposed) carrier mobilities were formed within the SWNT network. This patterning was confirmed by composite EL images that reproduced the mask structures (see Fig. 10b The investigation of SWNT networks with techniques combining spectral and spatial resolution remains a challenging task, but is highly desirable to obtain further insights into the relationship of charge transport properties and microscopic network structure. Although EL mapping has enabled the visualization of preferential current pathways and individual percolation paths, the achievable spatial resolution on the order of ~1 µm prevents studies on a single nanotube level. Furthermore, the role of diameter-dependent junction resistance versus intra-nanotube transport cannot be resolved by these measurements. For understanding transport on a microscopic level, scanning probe techniques should be considered.

C. Scanning probe techniques
Studying charge transport processes in SWNT networks on a microscopic level requires the use of methods that are capable of providing extremely high spatial resolution. In this context, scanning probe techniques such as conductive atomic force microscopy (c-AFM) or Kelvin probe force microscopy (KPFM) have proven to be very useful, as they are able to resolve current pathways on a nanometer to micrometer scale. Key parameters such as the conductance and resistance can be determined locally, i.e., along individual nanotubes or at specific nanotube-nanotube junctions.

Conductive atomic force microscopy (c-AFM)
Conductive AFM is a frequently used technique to investigate conduction paths in organic semiconductors such as polymers, 227 small molecules, 228,229 or blends thereof. 230 Usually, the semiconductor is deposited on an insulating substrate but in electrical contact with an electrode (vertically or laterally). A conductive AFM tip (usually Pt-or Pt/Ir-coated) is used to raster scan over the sample in contact or intermittent contact mode while a fixed bias is applied between the sample and the tip. This enables the simultaneous measurement of the surface topography and current flow between electrode and tip at each position. The experimental setup for c-AFM measurements is schematically illustrated in Fig. 11a including a current map of a network of polymer-sorted (6,5) nanotubes in contact with a gold electrode measured in air.
Note that reproducible c-AFM images of nanotubes and their networks are very challenging due to the need to be in contact with the sample while scanning without altering the surface (e.g., moving the nanotubes) nor damaging the cantilever tip (abrasion of the metal coating).
Scanning with an intermittent contact mode is more suitable for such samples. 231,232 Early

KPFM and related techniques
Another prominent method among the scanning probe techniques is KPFM, which allows the surface potential or work function of a surface to be mapped with high resolution. 243 In KPFM, a metallic cantilever is scanned over the (semi-)conducting sample under investigation and the potential difference is recorded. The method was successfully employed to elucidate charge injection and charge transport in organic thin film transistors. [244][245][246][247] In studies on individual SWNTs, this method has provided detailed insights into local electrostatic properties and conduction mechanisms. 248,249 For example, Fuller et al. used modified KPFM to obtain the potential profiles along individual nanotubes in transistor structures for different source-drain voltages. 250 The same technique was used in a subsequent study to determine the mean free path of carriers in SWNTs. 251 Using KPFM on SWNT networks enables spatial mapping of the potential. Okigawa et al.
recorded the potential across the channel in FETs based on dense, CVD-grown nanotube networks. 252 In the subthreshold regime, they found a constant potential at zero source-drain bias, whereas the potential profile for a small source-drain voltage exhibited step-like drops. A non-uniform distribution of regions with higher and lower resistance appeared in the corresponding resistance distribution maps acquired by c-AFM. The data indicated that the current flow was restricted to only a few conductive paths connecting areas of low resistance, meaning only few nanotubes contributed to the charge transport in the subthreshold regime. In Closely related to KPFM is electrostatic force microscopy (EFM), which directly records the electrostatic force acting on the cantilever. Topinka et al. used EFM to study SWNT transistors with mixed metallic/semiconducting CVD-grown nanotube networks. 253 They observed pronounced differences between individual devices. In some samples the network resistance was relatively low and the voltage dropped smoothly between the source and drain electrodes, for others the network resistance was high with abrupt voltage drops at specific points within the channel (see Fig. 11c). Simulations of current flow in random nanotube networks indicated that the presence or absence of steep voltage drops could be the result of single percolation paths at certain points in the film. Substantial variations in device performances probably resulted from differences in the ratio of semiconducting to metallic nanotubes. Consequently, the authors noted that enrichment of semiconducting SWNTs was a key step towards reliable device measurements. conductivity. 109,110,241 In networks consisting of metallic and semiconducting nanotubes, the ratio between these different types led to the formation of very different conducting paths as predicted by percolation theory. 253 Overall, these reports suggest that charge transport in SWNT networks is strongly dominated by the junctions and any residual metallic nanotubes.
Yet, the discussed techniques require specific knowledge about the system as they can neither distinguish between different nanotube species nor clearly between metallic and semiconducting SWNTs (except MIM). Most studies used networks of as-grown nanotubes, which are very clean (no surfactants) and contain few bundles but also include nanotubes with a range of diameters and about 30% metallic nanotubes. Hence, the results are only of limited applicability to state-of-the-art SWNT networks with only semiconducting or even monochiral nanotubes. 34 Unfortunately, many questions regarding microscopic charge transport in purely semiconducting nanotube networks such as their junction versus intra-nanotube resistance depending on doping remain unanswered.

A. Experimental and theory challenges
While there has been a tremendous amount of progress in the fabrication and application of purified semiconducting nanotube networks to the point where they can be used to produce fully integrated microprocessors, 12,13 there are still a lot of open questions regarding the details of charge transport within such networks. As shown above some aspects can be investigated in more detail now due to better control of the network density and composition. Electrical and thermoelectric measurements, dynamic absorption, photoluminescence and electroluminescence spectroscopies as well as scanning probe microscopies have been successfully applied to gain insight into the charge transport in carbon nanotube networks.
However, a complete picture that enables targeted optimization will require further effort on the experimental and the theory side. Some of these challenges are summarized here.
Describing and understanding charge transport in SWNT networks requires theory and models on different levels. Firstly, the charge transfer between two overlapping semiconducting nanotubes of the same (large or small) or different diameters and with various orientations to each other must be modelled at a higher level of theory than what has been done so far. 112 Given the progress in computing power, even chiral nanotubes with relatively large unit cells should be accessible for modelling direct charge transfer similar to studies carried out for charge transfer between π-conjugated organic molecules. 256 importantly allow for physical insights to be drawn from the fitting parameters.
On the experimental side the challenges might be divided between experiments that would be useful to gain more insight into the transport physics and experiments for further optimization of device performance for applications. The former should of course inform the latter as well.
The question of residual wrapping polymer or surfactant of solution-processed networks and its potential impact on charge transport still remains open and requires comprehensive and careful experiments to be resolved. It is commonly assumed that excess polymer can act as a barrier to charge transport through SWNT networks, however, the problem lies within the many parameters that must be considered and cannot be easily controlled or even determined directly (e.g., precise coverage and wrapping geometry of polymers on nanotubes after deposition).
Furthermore, one has to distinguish between the polymer that is actually wrapped around (and likely strongly bound to) the nanotubes, and free, unbound polymer in solution, which is typically removed after SWNT network deposition simply by rinsing with a suitable solvent.
Recent studies have suggested that the residual unbound polymer of spincoated semiconducting nanotube networks does not reduce the effective carrier mobility, 143 260 In contrast to that, the rhenium salt treatment of random (6,5) SWNT/PFO-BPy networks led to a higher degree of nanotube bundling and in fact a decrease in carrier mobilities for the resulting FETs. 35 These examples highlight the problem of trying to investigate one parameter affecting transport in a nanotube network without also changing another.
A perfectly clean network of well-separated and purely semiconducting nanotubes with known composition and deposited from the gas phase should be a good reference for solutionprocessed networks. Unfortunately, such samples are not yet available. Furthermore, while the role of the surrounding dielectric (metal oxides or polymers) was studied (see above) and showed a direct effect on the charge transport, 136,155,156 it is still unclear how to incorporate these effects into a universal model. For the commercial application of semiconducting nanotubes in circuits, a number of issues need to be resolved. The shelf-life of a nanotube ink is limited by the aggregation of the nanotubes in dispersion over time, leading to non-uniform networks with many bundles, aggregates and reduced device performance. 265 Achieving highly uniform nanotube networks processed from solution without extensive post-processing will be necessary for large-area nanotube circuits to become reality. The composition of a network must be precisely controllable and reproducible from batch to batch, not just in terms of absence of metallic nanotubes but also in terms of the length and diameter distribution of the semiconducting nanotubes. As demonstrated above even small amounts of small-bandgap nanotubes can affect the overall transport especially at lower carrier concentrations (i.e., in the subthreshold region).
Reducing this variability and energetic disorder of the network with a single nanotube chirality would also increase the effective carrier mobility. At the moment the only monochiral nanotube dispersions that are available on a larger scale and have been studied in field-effect transistors are (6,5) SWNTs with a relatively small diameter and hence larger injection barriers and lower intrinsic mobilities than large-diameter nanotubes. 35,151 Networks of large-diameter but still mixed semiconducting nanotubes result in the highest field-effect mobilities to date and are predominantly employed for circuit applications. 4,12,173 Hence, methods to produce large amounts of monochiral and large-diameter nanotubes should be pursued further. Recent progress in aqueous two-phase extraction has already enabled the purification and enrichment of several monochiral nanotube dispersions with diameters of around 1.41 nm. 266 Similar dispersions of large-diameter nanotubes might also be possible by polymer-wrapping. 267 It will be interesting to see where the true limits of carrier mobility and on/off current ratios for largediameter/small-bandgap nanotube networks lie and whether they justify the effort of purification compared to multichiral semiconducting networks.
Small-bandgap nanotubes are more likely to show ambipolar transport even under ambient conditions. For ohmic contacts the maximum on/off current ratio for an ambipolar semiconductor is given by its bandgap. 268 The narrow bandgaps of large-diameter SWNTs would start to limit this ratio irrespective of the absence of metallic nanotubes. In any case, for low-power complementary circuits, controlled and selective hole or electron injection and transport are required. The predominant p-type behavior of many SWNT transistors is based on high work function contacts and electron trapping by oxygen and water. 27,28 For n-type transistors low work function electron-injecting contacts are often used, 8,29 while the controlled application of chemical dopants 30 or specific dielectrics 31 are also viable options. The use of alkaline and strongly reducing dopants may also help to remove water and electron traps that lead to hysteresis and poor subthreshold slopes. 32,151 Importantly, these approaches for creating purely p-type and n-type transistors with high on/off current ratios must be reproducible and stable under operating and ambient conditions.
As shown in the preceding sections, the precise composition of a nanotube network and possible residual minority species can have a large impact on the device characteristics.
Reliable and simple methods and metrics to quantify the nanotube species in and the purity of dispersions and networks are required for characterization and quality control. While simple absorption spectroscopy can provide general information about the distribution of majority species even for samples with many chiralities through spectral fitting, 269 it is not sensitive enough at the very low percentages of minority species. Determining and controlling the composition of a nanotube network below chirality concentrations of 0.1% requires highly sensitive techniques. Photothermal deflection spectroscopy (PDS) on thin films can provide reliable measurements of absorbances several orders of magnitude below those of a good absorption spectrometer. 270 For example, minority nanotube species in thin films of (6,5) nanotubes as well as their Urbach tail could be identified with PDS. 88 However, this technique usually requires a custom setup and sufficiently thick films immersed in a high refractive index inert liquid. 126 Photoluminescence excitation-emission maps can be highly sensitive and give quantitative distributions of the detected nanotube species 271,272 but PL spectroscopy is blind toward residual metallic nanotubes and those with very small bandgaps (< 0.8 eV) and thus beyond the typical detection limits (~1600 nm) for InGaAs detectors. The popular and easily accessible technique of Raman spectroscopy cannot provide quantitative information about the concentration of different SWNT species at all, as it is always limited by the employed laser wavelengths and the chirality-dependent variations of the scattering cross sections. [273][274][275] It can however be used to show the absence of certain species if suitable excitation lasers are used.
More reliable identification techniques such as TEM 276 or electron diffraction 277 cannot be applied to large sample volumes. Hence, most studies will continue to rely on a combination of the above-mentioned techniques, keeping in mind their specific limitations.
To be able to compare different nanotube network devices and hence analyze differences and improvements, common standards for extracting and reporting device parameters especially carrier mobilities must be established. For examples, the effective gate capacitance should be measured directly on the device to ensure correct mobility calculations due to its dependence on the network density especially for very thin dielectrics. 23,26 The carrier mobilities in FETs with carbon nanotube networks strongly depend on the carrier concentration and hence gate voltage, which may lead to different numbers depending on the method of mobility extraction.
The maximum field-effect mobility might be a comparable standard if a clear maximum is reached. However, one must ensure that this is indeed a mobility maximum and not an artefact of gate-modulated contact resistance, which has plagued mobility measurements of organic high-mobility semiconductors. 150,278 Ideally, contact resistance-corrected mobilities as extracted from gated four-point probe 149 or van der Pauw measurements 147,279 should be used.
Given the high carrier mobilities in nanotube networks, even relatively low contact resistances can skew the extracted values and more importantly limit the maximum switching frequency of a transistor and the frequency of a ring-oscillator. 280 Moreover, the uniformity and reproducibility of device parameters is not only crucial for fundamental studies, but it directly determines the applicability in actual circuits. Thus, average mobility values with standard deviations for a statistically relevant number of devices instead of values from "hero devices" should always be reported.
The large surface area of carbon nanotubes and the assumed direct impact of their environment on charge transport provides ample opportunities for chemo-and bio-sensing 64-66,69,167,281 as well as (optical) memory applications. 57,282-284 Functionalization (covalent and non-covalent) of nanotubes has already been applied for a range of devices, although the underlying 58 mechanisms are not always clear. A more fundamental understanding of charge transport in networks and the parameters that determine high or low mobility, threshold shifts etc. would be very helpful to further improve the performance of such sensors. Finally, the peculiarities of charge transport in nanotube networks (especially controlled hysteresis) may also enable their application in future neuromorphic devices. [285][286][287][288]

B. Conclusions
Understanding and controlling charge transport in networks of single-walled carbon nanotubes has vastly improved over the last two decades. At first, researchers had to deal with and were frustrated by poorly controlled networks with many metallic SWNTs or semiconducting nanotubes of unknown diameter. These samples showed poor current modulation, large hysteresis and insufficient reproducibility with hardly any perspective for application in highperformance electronics. Today the synthesis and purification of semiconducting and even monochiral nanotubes enables their use as a solution-processable electronic material with specific and tunable properties that is competitive with or even better than many other semiconductors for thin film transistors and other (opto-)electronic devices. Not only is it now possible to investigate the intrinsic charge transport properties of networks with specific and well-controlled compositions, they can be applied in integrated circuits on an industrial scale.
While there are still many open questions, the groundwork for answering them has been laid by the new SWNT purification and sorting techniques as well as the wide range of available characterization and simulation methods.