Resistivity of non-crystalline carbon in the 1–100 eV range

For the electron–ion collisions we have made use of the approach to scattering via the quantum Lenard–Balescu (QLB) equation [29]. For a dynamically screened Couloumb potential this results in the e–i collision time being given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{\mathrm{ei}}}=\displaystyle \frac{{Z}^{\star 2}{e}^{4}{n}_{i}{m}_{e}}{4\pi {\varepsilon }_{0}^{2}{p}^{3}}{\displaystyle \int }_{0}^{2\pi /{\hslash }}\displaystyle \frac{{k}^{3}}{{({k}^{2}+{\kappa }_{s}^{2})}^{2}}{\rm{d}}k,\end{eqnarray} \tag{ 7 }$$

where n_i is the ion density, k is the electron wavenumber ($p={\hslash }k$), and ${\kappa }_{s}$ is the screening wavenumber and is related to the screening length, l_s, via ${\kappa }_{s}=1/{l}_{s}$. This expression can be evaluated analytically (e.g., by means of substituting a hyperbolic function), to yield

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{\mathrm{ei}}}=\displaystyle \frac{{Z}^{\star 2}{e}^{4}{n}_{i}{m}_{e}}{4\pi {\varepsilon }_{0}^{2}{p}^{3}}(\mathrm{log}(\sqrt{1+\displaystyle \frac{{\phi }^{2}}{{\kappa }_{s}^{2}}})-\displaystyle \frac{1}{2}\displaystyle \frac{\displaystyle \frac{{\phi }^{2}}{{\kappa }_{s}^{2}}}{1+\displaystyle \frac{{\phi }^{2}}{{\kappa }_{s}^{2}}}),\end{eqnarray} \tag{ 8 }$$

where $\phi =2\pi /{\hslash }$. This is a distinctly different approach from the standard Lee–More model [24] which uses a Landau–Spitzer (LS) approach which results in one obtaining Coulomb logarithms which need to be artificially capped. The QLB approach has also been used to address the problem of electron–ion equilibriation in WDM [30, 31].

In order to account for the effect of electron–electron collisions we have used the approach described by Ebeling and co-workers [32], in which the e–i collision time is reduced by a certain factor. The reduction factor is calculated as follows. First we define a reduction factor in the non-degenerate limit, ${\gamma }_{\mathrm{nd}}$, using the results of Van Odenhoven and Schram [33]

$$\begin{eqnarray}&&{\gamma }_{\mathrm{nd}}=\displaystyle \frac{3\pi }{32}(1+\displaystyle \frac{153{Z}^{2}+509Z}{64{Z}^{2}+345Z+288}),\end{eqnarray} \tag{ 9 }$$

and we then re-scale this according to the prescription of Bespalov and Polishchuk [34] to account for degeneracy

$$\begin{eqnarray}&&\gamma ={\gamma }_{\mathrm{nd}}+(1-{\gamma }_{\mathrm{nd}})*\displaystyle \frac{{T}_{F}}{{T}_{F}+3\;T/2},\end{eqnarray} \tag{ 10 }$$

where T_F is the Fermi temperature, and T is the temperature. The e–i collision time is then multiplied by this factor.

At low temperatures it is possible that ${Z}^{\star }\lt 1$ i.e. not all atoms have undergone the first ionization. This means that we must account for the elastic scattering of electrons from neutral carbon atoms. At any given energy we obtain a cross-section for e–n scattering from Desjarlais' fit [25] to a cross-section that is calculated in the Born approximation using screened potentials [35, 36]. Below we give the expression used from Desjarlais [25]

$$\begin{eqnarray}&&{\sigma }_{\mathrm{en}}=\displaystyle \frac{{\pi }^{3}{({\alpha }_{D}/2{r}_{0}{a}_{B})}^{2}}{{A}^{2}+3B{({{kr}}_{0})+7.5C({{kr}}_{0})}^{2}-3.4D{({{kr}}_{0})}^{3}+10.6668E{({{kr}}_{0})}^{4}},\end{eqnarray} \tag{ 11 }$$

where all lengths are in units of a_B (the Bohr radius), ${\alpha }_{D}$ is the dipole polarizability (in units of a_B³), k is the wavenumber of the incident electron, and ${r}_{0}=\sqrt[4]{{\alpha }_{D}{a}_{B}/2{Z}^{1/3}}$ is a cut-off radius. The coefficients in the denominator are given by

$$\begin{eqnarray}&&A=1+2{\kappa }_{s}{r}_{0}+\displaystyle \frac{7}{{\pi }^{2}}{({\kappa }_{s}{r}_{0})}^{2}+\displaystyle \frac{\pi }{7}{({\kappa }_{s}{r}_{0})}^{3},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&B=\mathrm{exp}(-18{\kappa }_{s}{r}_{0}),\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&C=\displaystyle \frac{1+22{\kappa }_{s}{r}_{0}-11.3{({\kappa }_{s}{r}_{0})}^{2}+33{({\kappa }_{s}{r}_{0})}^{4}}{1+6{\kappa }_{s}{r}_{0}+4.7{({\kappa }_{s}{r}_{0})}^{2}+2{({\kappa }_{s}{r}_{0})}^{4}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&D=\displaystyle \frac{1+28{\kappa }_{s}{r}_{0}+13.8{({\kappa }_{s}{r}_{0})}^{2}+3.2{({\kappa }_{s}{r}_{0})}^{3}}{1+8{\kappa }_{s}{r}_{0}+10{({\kappa }_{s}{r}_{0})}^{2}+{({\kappa }_{s}{r}_{0})}^{3}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&E=1+0.1{\kappa }_{s}{r}_{0}+0.3665{({\kappa }_{s}{r}_{0})}^{2}.\end{eqnarray} \tag{ 16 }$$

The cross-section that this is a fit to is itself obtained using the 'polarization potential' [35, 36]. This is a model potential [37] that has often been used for electron–neutral scattering, of the general form ${V}_{\mathrm{PP}}\propto {-\alpha }_{D}/{({r}^{2}+{r}_{0}^{2})}^{2}$. At large radii this potential corresponds to a dipolar potential, as one would physically expect. In the polarized atom the potential is not singular at the origin, and the cut-off, r₀ ensures that this does not occur in the model potential. The dipole polarizability used for neutral carbon atoms was taken to be $10.403{a}_{B}^{3}$ based on the results of Miller and Kelly [38]. Once the cross section has been obtained the electron–neutral scattering time is determined from

$$\begin{eqnarray}&&{\tau }_{\mathrm{en}}=\displaystyle \frac{1}{{n}_{{\rm{C}}}{\sigma }_{\mathrm{en}}{v}_{e}},\end{eqnarray} \tag{ 17 }$$

where n_C is the number density of neutral carbon atoms, and v_e is the electron velocity magnitude.

At moderately low temperatures the partially ionized state of warm dense carbon means that electrons can collide with atomic states of multi-electron carbon ions inelastically resulting in electrons being promoted to excited states. This is contingent on the 'randium' assumption for the carbon ions (that they are highly disordered), as only in this case is the assumption of localized atomic-like states a good one [26]. Having assumed this, we then take the cross-sections for electron-impact excitation from Itikawa et al's [39] compilation of electron-impact excitation cross-sections for isolated carbon atoms. These will neglect any effects of being immersed in dense plasmas, so the effect of this on the cross-sections is neglected. Excitation processes were only considered for C⁺–C³⁺ as the excitation energies for the highest ionization states is so large as to have negligible effect on the resistivity. The population of the electronic states was determined from the Boltzmann distribution and the ion density obtained from the ionization model.

In the case of ${{\rm{C}}}^{3+}$ we only considered the 2s²S and 2p ²P states and electron impact excitation from the former to the latter. The energy for this transition is 8 eV. The next transition is the 2p ²P to 3 s ²S transition with an energy of 29.55 eV. This excitation was found to be negligible so this and all other possible electronic excitations were neglected.

For ${{\rm{C}}}^{2+}$ we considered the excitations from the 2s^{2 1}S to the 2s2p¹P state and the excitation from the 2s2p ³P to the 2p^{2 3}P state. The former excitation has an energy of 12.69 eV and the latter has an energy of 10.54 eV. The cross-section for excitation from the 2s^{2 1}S to the 2s2p ³P state (energy of 6.5 eV) is sufficiently small for this to play a negligible role.

For C⁺ we considered the following transitions : 2s²2p ²P to 2s2p^{2 2}D, 2s²2p ²P to 2s2p^{2 2}P, 2 s²2p ²P to 2s2p^{2 4}P, 2s2p^{2 4}P to 2s2p^{2 2}D, 2s2p^{2 2}D to 2s2p^{2 2}S, 2s2p^{2 2}D to 2s2p^{2 2}P, 2s2p^{2 4}P to 2s2p^{2 2}S, and 2s2p^{2 4}P to 2s2p^{2 2}P. The energies relative to the ground state (2s²2p ²P) are : 5.33 eV (2s2p^{2 4}P), 9.29 eV (2s2p^{2 2}D), 11.96 eV (2s2p^{2 2}S), and 13.71 eV (2s2p^{2 2}P).

The excitation time for each excitation at a given electron momentum was then determined via $\tau =1/n\sigma {v}_{e}$, where n is the density of carbon atoms in the initial state of the excitation process.

Electron impact ionization can potentially also affect the resistivity. In order to examine the significance of electron impact ionization we included it by estimating the cross section from the Lotz formula [40]

$$\begin{eqnarray}&&{\sigma }_{Z}=4\times {10}^{-14}\epsilon \displaystyle \frac{\mathrm{log}(\varepsilon /{U}_{Z})}{\varepsilon {U}_{Z}},\end{eqnarray} \tag{ 18 }$$

where is the incident electron energy in eV, U_Z is the ionization energy in eV (from the Zth ionization state), and is the number of equivalent electron in the shell from which the ionization occurs. In order to account for continuum lowering, the ionization energy is reduced by $\Delta {E}_{c}$, where we use the approximation

$$\begin{eqnarray}&&\Delta {E}_{c}=\displaystyle \frac{3{Z}^{\star }{e}^{2}}{8\pi {\varepsilon }_{0}{r}_{ii}},\end{eqnarray} \tag{ 19 }$$

where ${r}_{ii}$ is the inter-ionic distance, this being the simplest estimate of the continuum lowering [2]. Note that the inter-ionic distance is obtained from ${r}_{ii}=\sqrt[3]{3/4\pi {n}_{i}}$, and for vitreous carbon is 0.147 nm. The electron impact ionization time is then determined via $\tau =1/{n}_{Z}{\sigma }_{Z}{v}_{e}$. We have assessed the role of electron impact ionization under both the assumption that all ions are in the ground electronic state, and for a Boltzmann distribution that includes both the ground state and the first excited state. In both cases we found that ionization processes made only a very minor contribution to the resistivity. We therefore did not include electron-impact ionization in the final model.

We have calculated a resistivity curve for vitreous, or glassy, carbon assuming a density of $\rho =1500$ kg m⁻³. The results for the full model is plotted in figure 1.

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These results are plotted against results from DF-QMD calculations, and against a curve calculated when only electron–ion and electron–electron scattering are included, as well as a curve calculated when only electron–ion, electron–electron, and electron–neutral scattering are included. This plot shows that the resistivity obtained from the DF-QMD calculation in the 2–20 eV range is much higher than the resistivity that is calculated when only e–i, e–e and e–n scattering is accounted for by more than a factor of 4. The plot shows that when we attempt to account for the effects of electron-impact excitation the resistivity curve that we obtain is much closer (albeit still not in perfect agreement) to the resistivity obtained from the DF-QMD calculation.

The comparison between the model with and without excitation can be done more quantitatively by calculating the percentage of the resistivity due to excitation (according to the model) from the resistivity curves obtained with and without excitation. The results of this calculation are shown in figure 2.

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From figure 2 we find that in the 2–20 eV range, excitation processes account for over 50% of the resistivity, and this is up to 70% at some points. It is therefore clear that excitation is at least as equally important as standard scattering processes in this temperature range (for solid vitreous carbon), and in some cases excitation process make the dominant contribution to the resistivity.

In addition to the contribution of the excitation processes, we also find, however, that the resistivity model seems to over-predict the resistivity at very low temperatures compared to the DF-QMD calculation. There could be a number of reasons for this, which stem from the present model not treating the solid-state regime well, for example : in the solid state, both amorphous and glassy carbon are better characterized as poor conductors rather than poor insulators. The conduction mechanism has been described as a hopping mechanism between localized states. Such a mechanism is not described by this model, so it is not entirely surprising that the present model is deficient at very low temperatures.

Of course, one could question whether the use of the QLB approach, as opposed to other approaches, has lead us to significantly underestimate the e–i collision time and thus its contribution to the resistivity. To address this question we start by noting that we could instead follow the LS approach in this model by adopting the following expression for the e–i collision time:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{\mathrm{ei}}}=\displaystyle \frac{{Z}^{\star 2}{e}^{4}{n}_{i}{m}_{e}}{4\pi {\varepsilon }_{0}^{2}{p}^{3}}\mathrm{ln}[\displaystyle \frac{{k}_{\mathrm{max}}}{{k}_{\mathrm{min}}}],\end{eqnarray} \tag{ 20 }$$

instead of equation (7). This is very much like the e–i scattering time used by the model of Lee and More. However we must choose values for the arbitrary cut-offs, and for this we will follow Lee and More by taking k_min to be the inverse screening length, but we keep ${k}_{\mathrm{max}}=2p/{\hslash }$. In figure 3 we show the resistivity due to the e–i collisions alone as calculated by the QLB and LS approaches (the version we described above). Figure 3 shows that reverting to the LS approach leads to an increase in the contribution to the resistivity due to e–i collisions compared to the QLB approach. Of course, one can question how physical this is [30, 31], and whether it is correct to believe the LS approach merely because it predicts a higher resistivity. However when we compare both results to the DF-QMD simulation results we see that the difference in the e–i contributions from either approach are small in comparison to the total resistivity seen in the DF-QMD calculations. This strengthens our conclusion that e–i collisions cannot account for the majority of the resistivity seen in the DF-QMD calculations, as pursuing an alternative approach does little to bridge the gap.

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Finally there is the issue of ionic structure. A fundamental assumption of this model is that the ions are highly disordered. In some situations where one would want to apply this model there may be both appreciable ion–ion coupling and sufficient time for ion ordering to develop. The current model can easily be extended to account for this by including the dynamic ion–ion structure factor, S_ii(k), in the integral over k in equation (7) [29]. It is important to understand the consequences of this. The most important consequence is that it will lead to longer e–i scattering times compared to the highly disordered (${S}_{\mathrm{ii}}(k)\;=\;1$) case and thus reduce the contribution of e–i collisions to the resistivity compared to those shown in figure 1. Therefore the presence of some short-range ionic structure (like in a liquid metal [26]) will likely not change the fundamental conclusions about the importance of excitation processes.

Model predictions for hydrocarbon materials were also produced by modifying the model so that it could handle C–H mixtures. The main modification was to use the modified Thomas–Fermi model to determine a mean ionization and electron density based on the average atomic and mass number of an atom in the compound. Then, using the electron density obtained from this, one can use a single ionization Saha model to determine the degree of hydrogen ionization. From these two the carbon ionization state was determined via

$$\begin{eqnarray}&&\bar{{Z}^{\star }}=\displaystyle \frac{{Z}_{{\rm{C}}}^{\star }{n}_{{\rm{C}}}+{Z}_{{\rm{H}}}^{\star }{n}_{{\rm{H}}}}{{n}_{{\rm{C}}}+{n}_{{\rm{H}}}},\end{eqnarray} \tag{ 21 }$$

where $\bar{{Z}^{\star }}$ is the ionization state of the average atom, and ${Z}_{{\rm{C}}/{\rm{H}}}^{\star }$ is the ionization degree of carbon/hydrogen. No electron-impact excitations of hydrogen atoms were included, but electron–neutral and electron–ion scattering from hydrogen atoms and ions were included. With this method we produced resistivity curves for polyethylene (${({\mathrm{CH}}_{2})}_{n}$) at $\rho$ = 1000 kg m⁻³.

As we can see from figure 4, the electron impact excitation of carbon makes a substantial difference to the resistivity in the 3–20 eV range. This shows that solid density hydrocarbon resisitivity in this temperature range is also strongly affected by electron impact excitations, not just pure carbon.

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A resistivity curve for liquid methane ($\rho =$ 422 kg m⁻³) was also generated using the extended model for CH compounds described in section 3.2. The resistivity curve with and without C excitations is shown in figure 5.

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Figure 5 clearly shows that C excitations strongly affect the resistivity in the 3–20 eV range in liquid methane as well as polyethylene and vitreous carbon. In figure 6 we compare the three resistivity curves generated by the full models (i.e. including C excitations) for vitreous carbon, polyethylene and liquid methane.

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This shows that the resistivity of liquid methane follows that of polyethylene very closely, but the resistivity of both behaves substantially different from that of vitreous carbon. The reason for this being the difference in the temperature dependence of the ionization state of carbon in the hydrogen bearing versus the pure carbon materials.