Electron transport mechanisms in polymer-carbon sphere composites

Carbon compounds exist in various forms, some of which are electrical insulators, some are electrical conductors, and some are semiconducting. These materials exhibit a range of chemical, physical, and mechanical properties depending on the bonding between carbon atoms and the resulting crystal structure. Common forms of carbon-based materials that have been studied before include spheres, tubes, sheets, and fibers. Over the past two decades, the field of nanoscience and nanotechnology has focused on carbon based materials like fullerenes,^1,2 carbon nanotubes,³ and graphene⁴ due to their promise to revolutionize the electronics industry. Recently, however, there has been renewed interest in carbon microspheres (CSs) for use in super strength carbon based objects and passive electronic components such as supercapacitors.^5,6 Various techniques have been discovered to make these spheres which include chemical vapor deposition, templates, pyrolysis of carbon rich sources, and hydrothermal carbonization (HTC) of large sugar molecules.^7–14 This interest results from the easy one pot hydrothermal method of preparation that yields in large quantities uniform CSs that exhibit high conductivity via post treatment and are chemically very stable under standard atmospheric conditions. In this work, we present our results on the CSs conductivity $(σ)$ as a function of temperature in the range $80 K−300 K$⁠. The electrical conductivity of a pressed pellet composed of CS and Poly(ethylene) Oxide (PEO) was $0.22 S/cm$ at room temperature and dropped to $0.18 S/cm$ at $80 K$⁠, passing through a maximum of $0.24 S/cm$ at $258 K$⁠. The PEO was used as a plasticizer in the pellet preparation.

We analyze possible contributions of various transport mechanisms to most plausibly interpret experimental results. Carrying on this analysis, we pay a special attention to thermally induced electron tunneling between adjacent CSs. This transport mechanism was first suggested by Sheng¹⁵ to describe the tunneling conduction in disordered systems and later employed to analyze electron transport in various polymer/carbon composites. We show that this transport mechanism is mostly responsible for the maximum in the temperature dependence of observed CS/PEO sample conductivity, whereas the thermally activated transport predominates at lower temperatures. A unique feature of the presented model is that it covers the entire temperature range from $80 K$ to $300 K$⁠.

A schematic of the preparation method of the CSs is shown in Fig. 1. A $0.8M$ aqueous sucrose solution in distilled water was transferred into a $50 ml$ stainless-steel autoclave. The solution was heated at $200 °C$ for 4 h and then cooled down to room temperature. The sucrose molecules underwent hydrolysis giving rise to monosaccharides such as glucose.¹⁶ Subsequent dehydration of the monosaccharides promoted nuclei-oligomers' formation within spherical micelles. Further polymerization led to the increment in the spheres' diameters determining the spheres' size. Additional carbonization has little effect on the CSs morphology, as demonstrated in Ref. 7. After cooling the solution, the black precipitate was collected and washed with water and ethanol to remove residues from the sucrose decomposition and then dried at $70 °C$ for 12 h. The as-prepared spheres were thermally annealed in a tube furnace under flowing nitrogen at $800 °C$⁠. The sample was heated at a rate of $10 °C/min$ until $800 °C$⁠. This temperature was held for 1 h and then allowed to cool down. A JEOL JSM-6360 Scanning Electron Microscopy (SEM) was used to characterize the morphologies and size distributions of the spheres. Energy Dispersive Spectroscopy (EDS) was used to analyze the chemical composition. The structural characterization was performed by X-Ray Diffraction (XRD) on a Bruker D2-Phaser XRD with a $CuKα=0.154 nm$ source. To fabricate the pellet, we mixed CS and PEO (MW 200 000, Sigma Aldrich) in a $1:1$ mass ratio and ground the mixture in a mortar and pestle. The pellet consists of nearly homogeneous dispersion of CSs and PEO which was used as the adhesive agent. However, our studies of the pellet electrical conductivity give grounds to suggest that some CSs are probably arranged in chains extended through the sample. This arrangement does not destroy the general dispersion homogeneity provided that a relatively small amount of CSs are included into such chains.

Without PEO, the pressed pellet was difficult to manipulate as it would easily crack making it impossible to measure its conductivity. We aimed at preparation of a manageable sample with high concentration of CSs. We have found that a pellet characterized by $1:1$ mass ratio is sufficiently hard, whereas pressed pellets with higher CSs concentrations appeared too brittle to handle. Therefore, we proceeded with experiments using this ratio. The corresponding volume concentration of PEO in the sample was estimated to be about 40%. Since the sample was electrically conducting, we believe that the content of carbon spheres was above the percolation threshold. The CS/PEO sample was compressed using a Carver Press (Wabash, IN) under a load of 20 000 lbs and cut into a rectangular shape. Silver glue was used to fabricate two metallic electrodes attached to the sample to perform electrical measurements. The current-voltage $(I−V)$ characteristics of the pellet were measured at $300 K$ using a Keithley $6517 A$ electrometer and a Keithley 6487 picoammeter/voltage source in a vacuum of $∼10−2 Torr$⁠. A closed cycle helium refrigerator was used to lower the sample temperature which was controlled by a CryoCon 32 B temperature controller. The pellet was mounted on a sapphire substrate (⁠$Al2O3$ with $〈0001〉$ orientation and 0.5 mm thickness). Sapphire is an electrical insulator and a good thermal conductor. The pellet was attached to the sapphire substrate using Apiezon N grease, and the substrate was attached to the cold finger of the helium refrigerator with silver epoxy. The current in the pellet was measured at a fixed voltage of $0.05 V$ in the range of $300 K$ to $80 K$ from which the conductivity was later calculated.

The SEM images in Figs. 2(a) and 2(b) show low and high magnification images of typical CSs produced with HTC and thermal annealing. The CSs have spherical shapes with smooth surfaces and diameters ranging from $0.125 μm$ to $10 μm$⁠. Further control of the sphere diameters can be obtained by varying the sucrose concentration in the precursor, as well as the temperatures and times for HTC and thermal annealing.¹³ The EDS spectra in Fig. 2(c) confirm that the spheres are composed of 99.5% carbon. The XRD pattern in Fig. 2(d) has characteristic peaks at $22.5°$ and $44°$⁠, corresponding to reflections of 002 and 101 planes of graphitized carbon.¹⁰ 

The thermal annealing process eliminates functional groups and increases the crystallinity of the CSs in agreement with the reports in literature.¹⁷ These physical characteristics greatly influence the conductivity of the carbon spheres. Fig. 3(a) shows plots of the current-voltage (I-V) curves of the CS/PEO pellet taken at $300 K$ and $80 K$⁠. The schematic diagram in the inset of the figure demonstrates Ag electrodes used for the electrical characterization and a SEM image of a section of the pellet. The spheres retain their shape even under compression. The I-V curves indicate an Ohmic response and the absence of Schottky contacts between the Ag electrodes and the CS/PEO. The conductivity of the CS/PEO was calculated from Fig. 3(a) to be $0.22 S/cm$ at room temperature and $0.18 S/cm$ at $80 K$⁠. The temperature dependence of the conductivity shown in Fig. 3(b) shows a drop to $0.18 S/cm$ when the temperature decreases from room temperature to $80 K$⁠.

Our results also indicate a change in the slope, with the conductivity increasing as temperature is lowered from $300 K$ to $258 K$ and then decreasing as temperature is further lowered down to $80 K$⁠. Similar behavior of temperature-dependent conductivity was previously reported for some polymer/carbon nanotube composites,^18–21 polymer/carbon fibers composites,^22–24 and polymer/carbon black composites.²⁵ In each case, the specifics of conductivity/resistivity temperature dependence are governed by the interplay between several different transport mechanisms. Relevant transport mechanisms include thermally activated electron transport of the same kind as that predominating in intrinsic semiconductors, variable range electron hopping (VRH), and electron tunneling. In Section III, we analyze contributions of these mechanisms to electron transport in CS/PEO composites based on the above described experimental results.

Analyzing electron transport mechanisms in CS/PEO, it is reasonable to assume that some of the spheres are stuck together and arranged in chains extended through the whole sample, thus providing pathways for traveling electrons similar to those observed in self-assemblies of monodisperse starburst CSs.⁵ This assumption agrees with the experimental data shown in Fig. 4. The plot of $ln σ$ versus $100/T$ displayed in the left panel of the figure includes a nearly rectilinear piece corresponding to lower temperatures $(T∼80−110 K)$⁠. So, within this temperature range, the conductivity σ mostly originates from a process of thermal activation typical for semiconducting materials and the effect of other transport mechanisms is negligible. The corresponding conductivity $σ1$ can be expressed by a standard Arrhenius relation

Here, the prefactor $σ10$ does not depend on temperature, $ΔE$ is the activation energy, and k is the Boltzmann constant. Basing on the data presented in Fig. 4, one concludes that for the considered sample $ΔE≈2.34 meV$⁠, which nearly coincides with the estimation made for carbon fibers in polyethylene-carbon fiber composites $(ΔE≈2.4 meV)$⁠, and is reasonably close to the activation energy for networked-nanographite $(ΔE≈4.0 meV)$⁠.²⁴ At higher temperatures, the $ln σ$ vs $100/T$ plot departs from the straight-line profile. This indicates that other mechanisms for electron transport start to noticeably contribute to the net conductivity although thermally activated transport still predominates. The appearance of distinguishable signatures of transport mechanisms different from thermal activation may be explained assuming that in the considered sample most of the spheres are separated from each other, and only the minority is combined in chains free from gaps between the spheres. This assumption agrees with SEM images shown in Figs. 2(a) and 2(b).

Electron transport through the gaps separating adjacent spheres may be controlled by two different transport mechanisms. These are the variable range hopping of electrons (VRH) between the sites on PEO polymeric chains filling the gaps between the spheres and the electron tunnelling through the gaps. The VRH transport theory was first suggested by Mott.²⁶ Mott's original theory was later applied to analyze transport mechanisms in noncrystalline substances, doped semiconductors, and polymers. When the electron transport is governed by the VRH, the resistivity is given by

Here, $ρ0$ is temperature-independent, and the characteristic temperature T_h has the form

As follows from this expression, T_h is inversely proportional to the density of localized electron states on the Fermi level $N(EF)$⁠. Also, it strongly depends on the localization length of electrons $l$⁠. The latter is directly proportional to the mean hopping distance R²⁷ 

Usually, the VRH theory is employed at $T≪Th$ when R significantly exceeds $l$⁠. It is considered as not applicable when T is greater than T_h and R takes on values smaller than the localization length. To consider a possible contribution of VRH into electron transport characteristics observed in the present experiments, we analyze the plot of $lnρ$ versus $T−1/4$ shown in the right panel of Fig. 4. In this plot, one may separate out two nearly rectilinear pieces with the crossover at $T≈100 K$⁠. As discussed before, the thermally activated electron transport through chains of glued together carbon spheres is likely to determine the temperature dependence of conductivity/resistivity of the sample at $T<100 K$⁠. So, the closeness of the corresponding part of the plot to a straight line seems to appear by accident. It probably occurs due to a lower sensitivity of $T−1/4$ plots compared with that of $T−1$ plots. At higher temperatures $(T>100 K)$⁠, the thermally activated conductivity becomes nearly independent of temperature because of the low value of the activation energy, so it cannot determine the temperature dependence of conductivity/resistivity. From the slope of the corresponding straight line, we find the characteristic VRH temperature $Th≈410 K$⁠. Then, we use Eqs. (3) and (4) to estimate the localization length and the mean hopping distance. Assuming that in PEO $N(EF)≈2.4×1021 states/eV cm3$⁠,²⁸ we obtain $l≈5.6 nm$ and $R≈2.2−2.6 nm$ within the temperature range of $100 K<T<200 K$⁠. Since R is significantly smaller than $l$⁠, we may presume that VRH mechanism does not strongly contribute to the sample conductivity. At the same time, a comparatively large value of localization length gives grounds to expect that electron transport between the carbon spheres is determined by electron tunneling through potential barriers.

Analyzing electron tunneling conductivity, we remark that the tunnel current may be strongly affected by voltage fluctuations across the gaps as suggested by Sheng.¹⁵ In the considered case, these fluctuations originate from thermally excited motions of ions on the surfaces of adjacent spheres. They distort profiles of energy barriers separating the spheres and cause temperature dependences of tunnel conductivity. The effect of thermally induced voltage fluctuations on the tunneling conduction was taken into account in studies of electron tunnel transport in various polymer-carbon composites (see, e.g., Refs. 18–20, 23, and 24).

Within the Sheng's theory, the effect of thermal fluctuations is described by introducing an extra electric potential V_T associated with fluctuations. A tunneling electron sees a potential barrier whose original rectangular profile is modified by the image-force correction and further distorted by thermal fluctuations. The barrier width is determined by the minimum distance between adjacent spheres d for the tunneling probability exponentially decreases as the tunneling distance rises. The simplest barrier profile is given by the expression

Here, the x-axis is supposed to run across the barrier $(0≤x≤d), u=x/d, ϵ=VT/V0$⁠, and V₀ characterizes the height of the rectangular barrier free from the image-force correction. The maximum height of the barrier $[Vm=V04(1−ϵ)2]$ is reached at $u=12(1−ϵ)$⁠. The expression (5) implies that at low temperatures when one may omit the effect of thermal fluctuations $(ϵ→0)$⁠, the barrier's profile acquires a parabolic lineshape. The parabolic profile distortion by thermal fluctuations is illustrated in Fig. 5. Simmons²⁹ and Sheng¹⁵ showed that the potential barrier's profiles may be modeled by a wide variety of lineshapes. Nevertheless, the chosen parabolic approximation was (and still is) used to explain experimental results concerning transport in carbon-polymer composites (see, e.g., Refs. 18–20 and 23).

The considered composite necessarily contains a large number of tunnel junctions with diverse characteristics, and the effects of voltage fluctuations vary from one junction to another. The set of tunnel junctions may be regarded as random resistor network where resistors are identified with sole junctions. The network conductivity/resistivity can be calculated by the self-consistent effective-medium theory.³⁰ Within this approach, the average effect of random conductances associated with the junctions may be treated by replacing all of them by a single effective conductance, so the complete network is reduced to a single junction. It was suggested¹⁵ to treat the most preferable region for electron tunnelling through the junction as a parallel plate capacitor. In the considered case, when the tunnelling occurs between adjacent CSs, one may identify the separation between the capacitor plates with the minimum distance between the spheres d and approximate the plate area as $πd2/4$⁠. Then, the capacitance of this capacitor may be estimated as $C≈ϵ0κπd, κ$ being the PEO dielectric constant.

The Sheng's model is analytically solvable, provided that the barrier profile is approximated by Eq. (5). Carrying out calculations described in the  Appendix, one obtains the following expression for the thermally induced electrical conductivity:

Here, the prefactor $σ02$

and the tunneling constant $χ$

are temperature-independent, and two characteristic temperatures T₀ and T₁ are determined by the relation

where C is the capacitance of a parallel plate capacitor representing the region where the tunneling between the spheres mostly occurs. As $χd>1$⁠, the temperature T₀ is lower than $T1$⁠. At low temperatures $(T≪T0)$⁠, the conductivity $σ2$ is reduced to a simple tunneling conductivity described by a well known expression: $σ2=σ02 exp[−πdχ/4)$⁠. As the temperature rises, the conductivity $σ2$ increases and it reaches its maximum value at $Tm=23T1−T0$⁠, provided that χd is sufficiently large $(χd>6/π)$⁠. When the temperature grows above $Tm, σ2$ decreases. This conductivity behavior qualitatively agrees with that observed in the experiment.

We remark that both thermally activated transport through chains of carbon spheres and VRH transport mechanism are associated with conductivities which smoothly increase as the temperature grows and do not show any maxima/minima at certain values of $T$⁠. This gives grounds to expect the thermally induced electron tunneling between the carbon spheres to significantly contribute to electron transport in the considered system. Also, we remark that the present result for the thermally induced tunnel conductivity is obtained by straightforward computations of all relevant integrals. It is more accurate than the corresponding expressions used in earlier works (see, e.g., Ref. 24) where the power factor $(1+T/T0)−3/2$ was lost. This power factor is responsible for the appearance of maximum in $σ2$ versus T curves, so the difference between our Eq. (6) and earlier results is important and may not be disregarded.

Basing on the above described analysis, we conclude that electron transport in the considered CS/PEO composite is mostly governed by two transport mechanisms, namely, thermally activated transport through chains compiled of adjoining spheres and thermally induced tunneling between the spheres separated by gaps. Correspondingly, the net conductivity may be presented in the form

where $σ1$ and $σ2$ are given by Eqs. (1) and (6), respectively. The net conductivity still shows a maximum at a certain temperature. However, the maximum position is shifted with respect to T_m due to the effect of the term $σ1$⁠.

Using the expression (10), one may reach a reasonably good fit between theoretical and experimental results. The fit may be further improved if one more factor is taken into account. As discussed in Ref. 24, the temperature growth leads to thermal expansion of air voids in PEO and to intensification of mobility of polymeric chains originating from their macro-Brownian motion. As a result, the minimum distances between the spheres increase as the temperature rises. The temperature dependence of d was determined by a computer simulation. The simulation results showed that the best fit of theoretical results to the experimental data could be achieved using a nearly constant value of d at temperatures below 150 K replaced by its rather fast increasing at higher temperatures. This is illustrated in the right panel of Fig. 5. Also, it was shown that the best fitting values for the temperatures T₀ and T₁ equal $200 K$ and $1850 K$⁠, respectively. The temperature dependence of conductivity given by Eq. (10) superimposed over the experimental data is shown in Fig. 6. A good agreement between the theory and the experiment is demonstrated. In plotting of the theoretical curve, it was assumed that d varies with the temperature, as shown in Fig. 5. The behavior of d presented in the figure is responsible for making the maximum in the conductivity more sharp and distinct. The curve plotted in Fig. 5 may be approximately represented by an analytical expression

where $T1=1850 K$⁠. To further examine the applicability of Eq. (10), we estimate some relevant parameters characterizing the thermally induced tunneling electron transport.

As was mentioned before, the diameters of carbon spheres in the sample $(D)$ vary between $0.125 μm$ and $10 μm,$ the mean and median diameters being equal $2.93 μm$ and $2.0 μm$⁠, respectively. Accordingly, in further estimations, we choose $D∼2 μm$⁠. The spheres occupy $∼40%$ of the sample volume. So, one may estimate minimum distances between adjacent spheres $d∼10−100 nm$⁠. The PEO dielectric constant κ equals 2.24. Using these data to estimate the capacitance C and substituting the result into Eq. (9), one approximates V₀ as $0.3−0.1 V$⁠. The temperature dependence of conductivity was explored when the bias voltage $Vb=0.05 V$ was applied across the sample. Therefore, one may reasonably assume that bias voltage applied across a sole tunnel junction is much smaller than $V0$⁠. Also, the parameter ϵ may be estimated as $1V0kT/C$⁠, which follows from the equipartition theorem. In the considered case, ϵ varies between 0.18 and 0.3 when the temperature grows from $80 K$ to $300 K$⁠, so it remains significantly smaller than 1 within the whole temperature range explored in the experiment. Finally, computing the ratio $T1/T$⁠, we find that $χd≈12$⁠. Comparison of two terms in Eq. (10) shows that $σ1$ exceeds $σ2$⁠, especially at low temperatures (see Fig. 6). The contribution from the thermally induced tunneling becomes more significant as the temperature rises. Moreover, as explained above, this contribution is responsible for the appearance of a maximum in the temperature dependence of conductivity.

Presently, carbon microspheres attract a significant attention due to their potentialities in building up elements for microelectronic devices such as supercapacitors. We prepared a large number of uniform CSs with diverse diameters from an aqueous sucrose solution using hydrothermal carbonization of large sugar molecules. The spheres were mixed with PEO resulting in fabrication of a CS/PEO composite. Electrical characterization of the sample did show an Ohmic response and the absence of Schottky barriers between the sample and the attached silver electrodes. The sample conductivity appeared to be rather high despite the effect of a non-conducting PEO filling gap between CSs.

The conductivity is temperature-dependent. As the temperature is lowered down from 300 K to $80 K$⁠, the conductivity first increases and reaches the maximum value at $T≈258 K$ and then it decreases following the temperature decrease. We have analyzed transport mechanisms which may control electron transport in the CS/PEO sample. Our analysis gives grounds to suggest that there exist some chains of glued together CSs extended through the whole sample. These chains form a network of channels for thermally activated electron transport typical for intrinsic semiconductors. We remark that our estimate for the activation energy is reasonably close to the value obtained for networked-nanographite which confirms the above suggestion. As follows from the experimental results, thermally activated transport predominates within the whole temperature range, especially at lower temperatures.

Another important transport mechanism is electron tunneling between adjacent CSs affected by thermally induced voltage fluctuations distorting profiles of energy barriers separating the spheres. Using a parabolic approximation for the energy barrier separating adjacent CSs, we have derived a novel expression for thermally induced conductivity whose temperature dependence shows a maximum at a certain temperature. On the contrary, thermally activated electron transport as well as VRH leads to conductivity which smoothly decreases following the decrease in temperature. Our analysis shows that thermally induced electron tunneling probably brings a significant contribution to the sample conductivity at temperatures close to/or higher than one corresponding to the conductivity maximum. Also, we have studied possible contribution from VRH transport mechanism. Our analysis showed that this mechanism is unlikely to significantly contribute to the temperature-dependent conductivity of the considered CS/PEO sample. This conclusion by no means disagrees with the importance of VRH, in general.

Finally, we conclude that the behavior of conductivity in the considered CS/PEO composite is mostly determined by the combined effect of thermally activated electron transport and of thermally induced electron tunneling between adjacent spheres. We believe that our results may contribute to further understanding of transport characteristics of polymer-carbon composites.

This work was supported by NSF-DMR-PREM 1523463. We thank A. Melendez for providing us with SEM images and G. M. Zimbovsky for help with the manuscript.

When fluctuating voltage V_T is much stronger than the bias voltage V_b applied across the junction, a partial conductivity may be defined as¹⁵ 

Here, $Eb=VB/d$ is the electric field corresponding to the bias voltage across the junction and j is the tunneling current density.

Fluctuation-induced thermal conductivity $σ2$ should be obtained by averaging Σ_T with the fluctuation probability function $P(VT)$

where $P(VT)$ has the form

In this expression, the capacitance C is associated with a parallel plate capacitor representing the most preferable region for tunneling between two carbon spheres.

When the thermally induced voltage exceeds the bias voltage across the tunnel junction, the tunneling current density is given by¹⁵ 

Here, m is the effective mass of a tunneling electron. It was shown³¹ that the barrier transmission factor $D(E,ϵ)$ for barriers with a slowly varying potential may be approximated as

The form of the function $F(E,ϵ)$ depends on the barrier profile. Using the adopted model of a parabolic barrier, we get

In this expression, χ is the tunneling constant

which presumably takes on greater values than $d−1$⁠. Substituting expressions (A5) and (A6) into Eq. (A4), one may calculate the current density $j(VT)$⁠. As long as ϵ takes on sufficiently small values $(ϵ<1)$⁠, electron tunneling predominates over thermally activated transport above the barrier, and the effect of temperature induced smearing of the Fermi level may be neglected. This brings explicit expressions for the thermally induced current density and the partial conductivity, namely,

Using Eqs. (A2) and (A9), we arrive at the result for the thermally induced tunneling conductivity given by Eq. (6).