1. Introduction

The production of graphene nanosheets from bulk graphite with consistent properties and high quality is among the challenges for the production of low-cost carbon based nanomaterials [1]. This desire to extract low-dimension nanomaterials from bulk is driven by the availability of graphite as a resource, the general reduction in energy used in processing, and reduced complexity required for synthesis when compared to the construction of the graphene sheets using bottom-up methodologies [2]. Various top-down approaches and exfoliation strategies have been developed over the years for extracting graphite and graphene nanosheets from bulk sources, including electrochemical [3–5], oxidation routes such as the Hummers method and its variations [6–8], and liquid phase exfoliation (LPE) where application of fluid dynamic forces is used to separate the laminar components in layered crystals. LPE is of interest from a manufacturing perspective because it is potentially scalable [9]. In comparison to graphene oxide and reduced graphene oxide (rGO) flakes, LPE is expected to produce exfoliated flakes with a lower oxygen content and less aromatic sp²-bonding disruption through the use of weaker exfoliation solution, such as water, ethanol, and isopropanol [10–12].

Materials resulting from LPE have potential in electronic [13,14] and photonic applications, such as resonator arrays [15] and optical sensors [16], where optimizing the electrical conductivity of the flakes can improve device performance. Consequently, understanding the relationship between electrical conductivity and the morphology arising from LPE processing aids this goal. To date, electrical properties arising from ballistic transport have been identified to have a size or length dependence in individual nano-graphite and graphene flakes at room temperature [17,18], while the quantum Hall effect has been observed in graphene monolayers at a range of temperatures [19–21]. However, LPE creates distributions of flake sizes and morphologies, begging the question: what are the electrical properties of the flakes produced by LPE compared to those produced by mechanical exfoliation approaches? Application of a contactless, high-frequency approach [22] can provide electrical information about the distributions of flakes in large-area LPE films [23] for comparison against the exfoliation parameters. In particular, terahertz (THz) radiation closely matches typical scattering rates of charge carriers in many solids, making it appropriate for conductivity measurements [24]. Terahertz spectroscopy has been applied to characterize the properties for a variety of material systems. For example, the change in conductivity and charge carrier dynamics in silver nanowires due to oxidation has been shown using THz Time-Domain Spectroscopy [25], the temperature dependent optical and electrical properties of doped silicon have been investigated using THz Spectroscopy [26], and the photocarrier dynamics have been measured using an optical pump and THz probe for SnS₂ flakes [27]. For graphene and graphene composites, THz Time-Domain Spectroscopy (THz-TDS) has been applied to estimate the frequency dependent conductivity and refractive indices in rGO films fabricated from spray coating [28], determine the frequency dependent conductivity and permittivity in 3D graphene networks compared to their 2D counterparts [29], measure the broad band wavelength-dependent absorption coefficient for graphene flakes embedded in polymers [30], extract the electronic properties of PTFE-polymer composites with few-layer graphene platelets [31], and quantify the reduction in charge-carrier mobility in water-dispersed high-quality graphene nanosheets with polydopamine compared to high-mobility graphene nanosheets [32]. Moreover, THz transmission is observed to correlate with the thickness derived from changing the number of mechanical exfoliation rounds for extracting graphene flakes from bulk graphite [33]. Despite this vast literature focusing on how the conductivity of composites depends on the graphene/graphite inclusions, studies do not include the intrinsic electrical properties of the flakes or the role of flake morphology resulting from LPE processing on the intrinsic electrical conductivity.

In this work, THz-TDS is used to measure the electrical conductivity of LPE graphene and nano-graphite as a function of average size, separated through cascade centrifugation. The complex AC conductivity for the flakes is calculated and fitted with conductivity models to extract parameters to describe the charge-carrier transport properties. These values are correlated with flake morphologies determined from scanning electron microscopy, Raman spectroscopy and atomic force microscopy/optical profilometry. It is found that morphology and conductivity are linked, demonstrated by an increase in carrier concentration and reduction of scattering time (or mobility) with decreasing flake sizes. For smaller flakes, an increase in mid-gap states originating from increased edge defects may lead to increased carrier concentration through self-doping. The confinement of carriers in smaller flakes results in a decrease in scattering times at higher charge-carrier concentrations through increased carrier-carrier interactions in addition to an increase in charge-carrier scattering at flake edges through a reduced mean free path. Overall, small flake and low-density samples are studied to reduce the effect of inter-flake transport by percolation.

2. Experimental

Graphite powder (325 mesh) was acquired from Sigma-Aldrich. Ethylcellulose (EC, 48% ethoxy content) and polyvinylpyrrolidone (PVP, 8000 MW) were obtained from Acros Organics. Ethanol (EtOH, 200 proof) was purchased from Fisher Scientific, and deionized water (DI water) was obtained from an in-house filtration system with a resistivity of 18.2 $\mu $S/$\textrm{c}{\textrm{m}^2}$. Double sided polished sapphire wafers (50.8 mm diameter, 300 $\mu $m thickness) were purchased from University Wafers (MA, USA) and single sided polished sapphire wafers (50.8 mm diameter, 430 $\mu $m thickness) were purchased from Cryscore Optoelectronic Limited (Henan, China).

Figure 1 shows the process steps for exfoliation, size separation, and deposition of nano-graphite/graphene coatings from the graphite powder precursor. First, exfoliation was performed in solution using a bath sonicator (Branson 2510, ∼40 W output) or an overhead shear mixer (40 L Digital High Speed Shear Emulsification Machine, 300 W output power, max rotation speed 13,000 rpm, 70 mm mixing head diameter, Model AE300L-H). For both EtOH and water-based solutions, a 1% wt/vol (i.e. 10 mg/mL) solution was made by combining EC with EtOH and PVP with water respectively. Following complete dissolution, graphite was added into a falcon centrifuge tube containing 20 mL of the exfoliation solvent at a concentration of 20 mg/mL and sonicated for either 2 or 4 hrs. The shear-exfoliated samples were processed at a rotor speed of 5000 rpm for 3 hrs. Following sonication or shear mixing, the exfoliated graphite in EC/EtOH or PVP/water were size separated by a cascade centrifugation technique [34–36] using an Eppendorf 5702 or 5415D with maximum relative centrifuge force (RCF) of 3000 g or 16,000 g.

 

Fig. 1. Process schematic diagram of the exfoliation from the precursor graphite, size-selecting flakes by cascade centrifugation, and deposition of the size sorted flakes using a dip-coating procedure.

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The sediment was separated into centrifuge binning ranges based on the g-time product; that is, the centrifugal force multiplied by the duration of the centrifuging time in minutes. The sedimented flakes for each centrifuge binning range were collected by draining the supernatant to isolate the sediment. Further washing was performed to reduce the polymer/cellulose on the flakes by adding either EtOH or water, sonicating to redisperse, then centrifuging at the upper collection speed and draining the supernatant again. This process was then repeated for an additional time to further purify the flakes of each centrifuge binning range. Finally, a modified Langmuir-Blodgett assembly method was employed to deposit the cleaned flakes onto sapphire substrates by redispersing the flakes in pure EtOH and depositing onto the surface of a DI water subphase [37]. A 3D printed wafer holder with a ${\approx} $34° tilt was used with a dip coater to control the wafers retraction speed through the flakes assembled on the water subphase at 25 mm/min. Following this process, the samples were left to dry on a hot plate at 60°C. The post processing temperature was kept low to isolate the influence of heating on the material properties of the exfoliated flakes.

Figure 2 shows representative scanning electron micrographs using a JOEL JSM-7600F of the (a) precursor graphite and (b) exfoliated graphite extracted using a centrifuge range of 90-966 kg-min. These images give a visual measure of morphology, which is characterized by flake size and layer thickness. For corroboration of morphology parameters, atomic force microscopy (using an Asylum MFP-3D) was performed on smaller flakes and optical profilometry [using a Bruker ContourGT KO Optical Profiler (VSI mode, 10× objective)] was performed on thicker layer. Example images from both instruments are shown in Section 1 of the Supplement 1. Figure 2(c) shows the comparison of the extracted flake size and layer height for each of the centrifuge binning ranges, determined in the size-sorting process above.

 

Fig. 2. Scanning electron micrographs of flake coatings on sapphire for the (a) precursor graphite and (b) exfoliated graphite extracted using a centrifuge range of 90-966 kg-min, each with a 50-$\mathrm{\mu }$m scale bar. (c) Flake surface areas (bars) and thicknesses (scatter points) measured at different centrifuge binning ranges and two exfoliation solvents.

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Raman spectroscopy is performed using a Renishaw inVia Raman microscope with a 532-nm laser with a spot diameter of ${\approx} $1 µm at 10% power from a 50× objective. Raman spectra were evaluated with focus on the spectral features that are common to graphene and graphite respectively.

Terahertz Time-Domain Spectroscopy (THz-TDS) was conducted using a home-built setup with ${\approx} $100-fs laser pulses from a 1-kHz regenerative amplifier laser (Coherent Libra) centered at 800 nm [38]. The output is split into two paths. The first path is used to generate THz radiation by optical rectification of the pulse in a 500-$\mu $m thick (110)-cut GaP or (110)-cut CdSiP₂ nonlinear crystal, which is collected and refocused onto the sample by two off-axis parabolic mirrors. The THz transmission is then collected and refocused with two more off-axis parabolic mirrors onto a 300-$\mu $m thick (110) ZnTe crystal for electro-optic sampling, which is gated by the second path from the laser. Adjusting the relative delay time between the generation and gating pulse allows for determination of the THz transient. The entire THz path is enclosed in a box with sapphire windows for optical access and filled with flowing dry air (or N₂) to reduce the effects of water vapor absorption. Finally, the electro-optic signal is measured on a balanced photodiode pair, with a differential voltage that is fed into a lock-in amplifier and referenced to a mechanical chopper in the THz generation path. The differential voltage can be converted into the THz electric-field ${E_{THz}} = ({2c/{\omega_g}n_0^3{r_{41}}d} )\,({\mathrm{\Delta }V/{V_{max}}} )$, where ${V_{max}} = 150$ mV is the average max voltage on the photodetector from only the gate pulse, c is the vacuum light speed, ${\omega _g} = 374.74$ THz is the frequency of the gate pulse, and ${r_{41}} = 4$ pm/V, ${n_0} = 2.85$, and $d = 300$ µm are the electro-optic coefficient, linear refractive index and thickness of the ZnTe electro-optic crystal [39,40].

3. Results and discussion

3.1 Raman spectra

As seen from the electron micrographs, sonication-based exfoliation with cascaded centrifuge steps results in a range of flake sizes and thicknesses from the precursor graphite. Increasing the centrifuge spin speed decreases the flake area and thickness, from particle clumps and larger flakes for the lowest binning to small flakes for the highest binning. Flake area decreases more than flake thickness with an increase in centrifuge speed, yet solvent makes little difference. These observations can be confirmed by Raman spectra.

Figure 3 shows representative Raman spectra for the precursor graphite and exfoliated flakes. The data shown is normalized to the intensity of the G peak, which for all the samples was the feature with the largest intensity. For each spectrum, the exfoliation time, solvent system, and centrifuge range used to collect the flakes are noted. The spectra are sorted by the centrifuge binning speed, with the flake area decreasing from the bottom to top, with graphite as the largest and 3-hr shear mixed graphite in PVP/water as the smallest. Raman spectra are very useful for characterizing graphite and graphene [41,42]. Most of the visible Raman features are present in graphite and are denoted on the graphite spectrum (bottom, black line). However, a couple of additional peaks visibly arise only in the exfoliation samples and are denoted there. The G peak arises from in-plane stretch mode of the carbon atoms, resulting in phonons about the $\mathrm{\Gamma }$-point of the first Brillouin zone and therefore Raman processes that occur only on the same side of the Dirac cone in a single valley of graphene-based systems. The disorder, or D, peak arises from the symmetry breaking of the breathing modes of sp²-atoms in rings, resulting in phonon mixing between opposite valleys or intervalley scattering. In the exfoliated samples only, the $D^{\prime}$ peak is associated with further symmetry breaking of the $\mathrm{\Gamma }$-point phonons that results in additional intervalley scattering [42]. Similarly, the $D + D^{\prime}$ peak is present in exfoliated samples and results from mixing of the transition pathways that lead to the D and $D^{\prime}$ peaks and produces intervalley scattering. In contrast, the $D + D^{\prime\prime}$ peak is present even in pristine graphene, and is due to mixing of the intervalley scattering pathways that lead to the D and $D^{\prime\prime}$ peaks, the $D^{\prime\prime}$ originating as a lower wavenumber phonon from the LA branch [41]. Additionally, the $D^{\prime\prime}$ peak originates from the presence of amorphous carbon [43]. Processes that emit phonons with opposite wavevectors to conserve momentum do not require defects to exist, although they lead to overtones of their counterpart single-phonon processes. Here, these include the $2D$ (sometimes referred as $G^{\prime}$) peak and the $2D^{\prime}$ peak. The presence of the small D peak in the graphite suggests a low amount of disorder or defects, with the low intensity possibly resulting from flake edges probed in the sample.

 

Fig. 3. Raman Spectra of graphite flakes processed in ethylcellulose/ethanol and polyvinylpyrrolidone/water. Longer processing times and higher centrifuge forces decrease the surface area, which is reflected in spectral changes such as an increase in the D peak, development of the $\mathrm{D^{\prime}}$ shoulder on the G peak, and the change in the 2D (or $\mathrm{G^{\prime}}$) line shape.

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Flake size, thickness, and crystallite size information can be evaluated from the relative strengths of the Raman spectral features. For graphite and the exfoliated samples, the term crystallites refer to the smaller uniform crystal segments that compose the flakes, which are always present in natural graphite [44]. The size of these crystallites can be qualified by a characteristic length, ${L_a}$ [45], determined from the intensity ratio of the D and G peaks [${I_D}$/${I_G}$] extracted from the Raman spectra, and is given by

Figure 4(a) shows the ${I_D}$/${I_G}$ ratio plotted against the area of the flakes estimated from the micrographs. Flake areas are estimated assuming roughly rectangular shapes with major ($a$) and minor ($b$) axes; see inset of Fig. 4(b). As flake area increases, ${I_D}$/${I_G}$ rapidly decreases and is well represented by a power function. The different solvent systems used to exfoliate the flakes are noted by the color of the data point and have little effect on the observed trend. The higher ${I_D}$/${I_G}$ values and the steepest slope occur for flake areas below the estimated spot area of the Raman laser impinging the samples, which is indicated by the vertical dashed green line. In smaller flakes, a larger number of edges are being probed relative to the flake’s basal plane, which may play a role in the trend of ${I_D}$/${I_G}$, rather than an increase in defects or dislocations arising from the exfoliation process. Nanosheet edges provide the required symmetry breaking to activate the D peak [46], and serves as a dominant presence of boundary-like defects in the samples as indicated from the $D$-to-$D^{\prime}$ ratio in Raman spectra, which gives ${I_D}$/${I_{D^{\prime}}} \approx 3.5$ [47]; see Section 2 of the Supplement 1.

 

Fig. 4. (a) Shows Raman $\textrm{D}$-to-$\textrm{G}$-peak [${\textrm{I}_\textrm{D}}$/${\textrm{I}_\textrm{G}}$] ratio versus the estimated flake area for a range of samples exfoliated with the two solvents. The inset shows the average flake length calculated using Eq. (2) compared against the flake length extracted from electron micrographs. (b) Crystallite size within the flakes calculated from ${\textrm{I}_\textrm{D}}$/${\textrm{I}_\textrm{G}}$ compared with the measured flake length (distance ‘a’) and width (distance ‘b’). The crystallite size is observed to decrease as the flake dimensions decreases.

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The inset of Fig. 4(a) shows the average flake length $\langle L\rangle $, calculated from the ${I_D}$/${I_G}$ using Eq. (2), plotted against the major axis of the flake ($a$) measured in the micrographs in Fig. 2. This result links together the length information gathered from different measurement types. Note that there is no reference for source graphite because $\langle L\rangle $ is calculated by a difference with respect to its ${I_D}$/${I_G}$ value. A dotted one-to-one correspondence line is plotted to compare the extracted and measured lengths. Overall, the agreement is generally good, although a little better for smaller flakes. Nevertheless, the agreement confirms that the ${I_D}$/${I_G}$ changes are related to a change in flake size, rather than other possible effects of the exfoliation process such as chemical modification.

Figure 4(b) shows the crystallite size, ${L_a}$, calculated from ${I_D}$/${I_G}$ using Eq. (1), plotted against the major ($a$) and minor ($b$) “distance” value for the flake samples measured experimentally from micrographs. Both distance values are plotted for a single value of ${L_a}$. A one-to-one correspondence line is plotted to determine if the calculated crystallite sizes are larger (above) or smaller (below) than those measured in the micrographs. For the entire range of samples, all crystallite sizes determined from ${I_D}$/${I_G}$ fall below the one-to-one correspondence line consistently by almost an order of magnitude. This result indicates that crystallites are smaller than the flakes, with the observed trend likely due to flake fracturing during the exfoliation. This is in contrast to high temperature processing of graphene and graphene oxide, which resulted in the growth of crystallites [29,45] – although that is not occurring here due to the lack of strong thermal treatments. Overall, the exfoliation process is producing flakes two orders of magnitude smaller than their bulk layers with crystallite sizes another order of magnitude smaller.

3.2 AC conductivity

Figure 5(a) shows ${E_{THz}}$ transients from THz-TDS for a precursor-graphite coated sapphire substrate (black), a blank sapphire substate (red), and for no sample, or freespace (blue). The inset shows the orientation of the sample in the THz beam. Compared to the freespace measurement, the sample and substrate transients are shifted to later delay times largely due to the refractive index of the sapphire substrate. There is also a reduction in the total transmission because of the Fresnel loss on each interface. In addition to these large changes between the freespace and sapphire substrate, smaller differences can be seen between the graphite-coated and uncoated substrate, illustrating that the thin graphite coating affects the overall THz transmission.

 

Fig. 5. (a) Representative transients of a graphite-coated substrate, clean substrate and freespace (no sample). A windowing function (dotted green) eliminates secondary reflections in the EO detection system. The inset schematic shows the general orientation of the substrate and coating layer to the incoming THz radiation. (b) Complex conductivity of the precursor 325-mesh graphite flakes calculated from the THz transient. Drude-Lorentz, Drude-Smith, and Plasmon conductivity models are shown for comparison. Of which, the Drude-Smith model best fits the experimental data.

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From these measurements, the complex AC conductivity spectrum in the THz range can be determined from the complex dielectric function using the analysis provided by Sooriyagoda et al [38] and Joyce et al [48], modified to account for thin nano-graphite films that only partially cover the sapphire substrates. First, the complex transmission spectrum of the coating layer is determined by the Fourier transform of the ratio of the film-on-substrate and the blank substrate, such that

This is related to the sample layer’s dielectric function by

Figure 5(b) shows the real and imaginary parts of ${\tilde{\sigma }_{flakes}}(\omega )$ for the precursor graphite coating after applying effective-medium theory. The error associated with the results were estimated using the standard deviations of the input parameters ${\tilde{T}_{sam}}(\omega )$, ${\tilde{T}_{sub}}(\omega )$, ${d_l}$ and f. The real part of ${\tilde{\sigma }_{flakes}}(\omega )$ starts at a non-zero value at low frequency and slowly increases with increasing frequency, whereas the imaginary part starts near zero at low frequency and decreases up to ${\approx} 2$ THz then rolls over and starts to increase again. This trend is consistent with Drude-Smith conduction model. The bandwidth of usable conductivity spectrum for modelling is approximately $0.25 < \omega /2\pi < 2.3$ THz.

Several conduction models are compared to the experimental data to best extract carrier concentrations N and scattering times $\tau $. Three models are shown overlaid on the experimental data in Fig. 5(b). The first conduction models include the Drude-Lorentz model [26,50]

This model was modified by Smith to include variable back-scattering of the charge carriers [52–54], such that for single carrier scattering events the Drude-Lorentz model becomes

As a result of complete back scattering or a strong electrostatic restoring force on the charge carriers, the Drude-Smith model becomes the Plasmon model [55,56]

Figure 6 shows the fitted AC conductivity for both exfoliation solvents and a range of centrifuge parameters. In Fig. 6(a), results for large flakes exfoliated in different solvents with similar centrifuge parameters show how the ethanol and water-based exfoliation result in almost identical AC conductivity. A slightly larger magnitude for the water-based solvent is observed, which is accompanied by a ${\approx} 15$% reduction in flake size in this example. The size dependence is further highlighted in Fig. 6(b), where results for a range of flakes sizes due to different centrifuge parameters in both exfoliation solvents, and clearly showing that smaller flake areas yield a larger magnitude of the AC conductivity. From the micrographs shown in Fig. 2(a) and (b), the sample surface is distinctly rougher for the larger, graphite flakes, and would then be expected to result in a higher amount of THz radiation being scattered, giving an overestimation of the AC conductivity. Despite this, the smaller exfoliated flakes show significantly greater AC conductivity compared to the graphite, supporting the dominant impact of flake size on its material properties compared to differences in THz scattering between samples.

 

Fig. 6. Complex conductivity for coatings of exfoliated graphite flakes. The closed shapes show the real conductivity while open shapes show the associated imaginary conductivity. Drude-smith fits for (a) 7.12 $\mathrm{\mu }$m² (Large EC/EtOH), 6.02 $\mathrm{\mu }$m² (Large PVP/Water) flakes to show the similarity with exfoliation solvent, and (b) 7.12 $\mathrm{\mu }$m² (Large EC/EtOH), 6.02 $\mathrm{\mu }$m² (Large PVP/Water), 0.36 $\mathrm{\mu }$m² (Small EC/EtOH) and 0.19 $\mathrm{\mu }$m² (Small PVP/Water) to show the difference due to centrifuge parameters. The magnitude in the conductivity is seen to increase as the flake area decreases.

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Extracting quantitative conductivity information for the explored range of flake sizes and solvents is performed for all three conduction models. Of these, the Drude-Smith model performed the best over the entire range. Figure 7 shows fitting parameters for the carrier concentration ${N_{DS}}$ and scattering time ${\tau _{DS}}$ compared with (a) the flake area estimated from the micrographs, and (b) the Raman ${I_D}$/${I_G}$ ratio and corresponding average crystallite length ${L_a}$. For the error bars shown in parts (a) and (b), discussion on obtaining these error values is provided in Section 4 of the Supplement 1. In Fig. 7(a), each axis is plotted on a logarithmic scale and fit with a linear function and with 90% confidence bands. Large horizontal error bars present in the areas of the exfoliated flakes are contributed to the cascade centrifuge process; while general size selection is present, contamination between size regions is still possible. For the fitted conductivity parameters, the error increases as the flake size decreases. This is due to thinner samples being made of smaller flakes, leading to a higher sensitivity to variations in the THz measurements. The data generally shows that ${N_{DS}}$ decreases and ${\tau _{DS}}$ increases with increasing flake area. The former dependence indicates that carriers must be more confined in flakes with small area. The data shows that ${\tau _{DS}}$ also decreases with increasing ${N_{DS}}$, which is consistent with increased carrier-carrier interactions as the carrier density increases. The extracted ${N_{DS}} \approx 1.3 \times {10^{18}}$ $\textrm{c}{\textrm{m}^{ - 3}}$ for the graphite precursor is below the lower end of carrier concentrations reported for this room temperature semi-metal (${\approx} 4 \times {10^{18}}$ $\textrm{c}{\textrm{m}^{ - 3}} \ll N \ll 1.1 \times {10^{19}}$ $\textrm{c}{\textrm{m}^{ - 3}}$) [51,59]. This may be due to variations in the coating thickness within the THz beam that can result in an over estimation of the amount of material (or the average thickness); therefore, decreasing estimates of the carrier density. Variations in sample thickness also particularly affect the imaginary part of the conductivity [60]. Moreover, the electrical properties of graphite/graphene are known to be highly sensitive to the defect density [61]. The Drude-Smith model also includes the scattering parameter, ${c_p}$, that might slightly lower the extracted ${N_{DS}}$ values. Nonetheless, ${c_p}$ is somewhat invariant with flake area (or the Raman ${I_D}$/${I_G}$ ratio), suggesting similar backscattering characteristics amongst all the samples; see Section 5 of the Supplement 1. Alternatively, validating the carrier concentration with conventional transport measurements is subject to its own challenges that may not lead to higher values.

 

Fig. 7. (a) Plot relating the measured flake area to the carrier concentration, N_DS, and carrier scattering time, τ_DS, extracted from the Drude-smith model. (b) Carrier concentration and scattering time are compared to the ${\textrm{I}_\textrm{D}}$/${\textrm{I}_\textrm{G}}$ and ${\textrm{L}_\textrm{a}}$ parameters extracted from the Raman spectra. The carrier concentration is represented by the filled blue squares, while the carrier scattering time is show by the hollow red squares.

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Figure 7(b) shows consistent data but with the information connecting the flake area from the micrographs to the Raman ${I_D}$/${I_G}$ ratio and the average crystallite length ${L_a}$. Again, ${N_{DS}}$ decreases and ${\tau _{DS}}$ increases with increasing ${L_a}$, which characterizes morphological mechanisms arising from liquid exfoliation and the cascaded centrifugation method [62,63]. A decreasing ${\tau _{DS}}$ for smaller ${L_a}$ may result from charge carriers scattering off flake edges more frequently as the effective mean free path is reduced in proportion to the lateral reduction of the average crystallite length. Previous studies on similarly sized graphite flakes observed ballistic transport behavior with measured carrier mean free paths on the order of microns [17,18] and with comparable carrier mobilities [59]. In graphite and graphene flakes of reduced size, an increase in mid-gap states is expected to occur due to breaking of electron-hole symmetry through a higher concentration of edge and boundary defects. This results in the lifting of the Fermi energy, leading to self-doping and thus an increase in the carrier concentration [64,65]. This increase can provide an additional contribution to the decrease in scattering time – or increase in scattering rate – through enhanced carrier-carrier interactions in the flakes.

4. Conclusions

In summary, this work relates the morphology of LPE graphite flakes to their AC conductivity measured via terahertz time-domain spectroscopy. Using the Drude-Smith conductivity model, the carrier concentration was found to increase with a decrease in the flake size due to an increase in flake edges and subsequent self-doping, while the increase in the ratio of the flake edges to basal plane results in a decrease in the charge-carrier scattering time, or an increase in the scattering rate, though shorter mean free paths and increased carrier-carrier interactions. Hence, liquid exfoliation and cascade centrifugation processing result in definite changes to the electrical conductivity and carrier transport behavior are a result of the modification of the graphene morphology when creating nano-flakes by LPE and cascaded centrifugation.

Correlations between flake thickness and flake area arising from sonication-based process may be further disambiguated by incorporating additional centrifuge steps, such as longer overnight centrifuging at low speeds to obtain higher area-to-thickness ratios [63]. Electrochemical liquid exfoliation can also be employed, since consistently thin exfoliated flakes can be created with a range of flake areas by incorporating additional sonication steps. Moreover, electrochemical exfoliation methods also allow for dopants and chemical species to be included, modifying the flake conductivity. Identification of the electrical effects of intentional dopants would benefit from temperature-dependent terahertz spectroscopy to probing the carrier mobility [38].

Overall, these findings confirm changes in conductivity with the liquid exfoliation and cascade centrifugation processing. While the inter-flake connectivity and transport behavior plays a significant role in the overall properties in structures such as coatings and films, the intrinsic material characteristics still greatly contribute to the overall applicability of exfoliated nanoflakes. This work serves as the foundation for further investigation into liquid exfoliated flake and nanosheets to develop strategies that actively modify conduction parameters for tailoring functional properties for improved utility.