Interplay of long-range and short-range Coulomb interactions in an Anderson-Mott insulator

In this paper, we tackle the complexity of coexisting disorder and Coulomb electron-electron interactions (CEEIs) in solids by addressing a strongly disordered system with intricate CEEIs and a screening that changes both with charge carrier doping level $Q$ and temperature $T$. We report on an experimental comparative study of the $T$ dependencies of the electrical conductivity $\sigma$ and magnetic susceptibility $\chi$ of polyaniline pellets doped with dodecylbenzenesulfonic acid over a wide range. This material is special within the class of doped polyaniline by exhibiting in the electronic transport a crossover between a low-$T$ variable range hopping (VRH) and a high-$T$ nearest-neighbor hopping (NNH) well below room temperature. Moreover, there is evidence of a soft Coulomb gap ${\mathrm{\Delta }}_{\mathrm{C}}$ in the disorder band, which implies the existence of a long-range CEEI. Simultaneously, there is an onsite CEEI manifested as a Hubbard gap $U$ and originating in the electronic structure of doped polyaniline, which consists of localized electron states with dynamically varying occupancy. Therefore, our samples represent an Anderson-Mott insulator in which long-range and short-range CEEIs coexist. The main result of the study is the presence of a crossover between low- and high-$T$ regimes not only in $\sigma \left(T\right)$ but also in $\chi \left(T\right)$, the crossover temperature $T^{*}$ being essentially the same for both observables over the entire doping range. The relatively large electron localization length along the polymer chains results in $U$ being small, between 12 and 20 meV for the high and low $Q$, respectively. Therefore, the thermal energy at $T^{*}$ is sufficiently large to lead to an effective closing of the Hubbard gap and the consequent appearance of NNH in the electronic transport within the disorder band. ${\mathrm{\Delta }}_{\mathrm{C}}$ is considerably larger than $U$, decreasing from 190 to 30 meV as $Q$ increases, and plays the role of an activation energy in the NNH.

Figure 1

(a) EB (undoped PANI). The dots represent electron lone pairs on the N atoms. (b) Fully protonated PANI-${\mathrm{H}}^{+}{\mathrm{A}}^{-}$, obtained by a reaction of EB with an acid ${\mathrm{H}}^{+}{\mathrm{A}}^{-}$. The “+” symbols on the protonated N atoms depict holes. (c) DBSA molecule. When DBSA protonates PANI, the ${\mathrm{C}}_{12}{\mathrm{H}}_{25}$ tail points outwards from the PANI backbone.

Figure 2

(a) Experimental $log\sigma$ vs $T^{-1/2}$ for $Q=0.38$, exhibiting a linear behavior for $T$ below $T^{*}$ (marked by an arrow) but not when $T>T^{*}$. (b) $log\sigma$ vs $T^{-2/5}$ for $Q=3.39$ is also linear only for $T<T^{*}$. Insets to (a) and (b): zoom into the vicinity of $T^{*}$. (c) Plots of $log\sigma$ vs $T^{-1}$ for the samples from (a) and (b). These plots are linear for $T>T^{*}$ regardless of the low-$T$ value of $\alpha$. Inset to (c): expanded view of the linear behavior at $T>T^{*}$ for $Q=0.38$. Experimental data are in all the plots shown by symbols, and the linear dependencies are outlined by solid lines.

Figure 3

(a) Experimental log $\mathrm{\Sigma }=\partial ln\sigma /\partial lnT$ vs $logT$ for $Q=0.38$, 1.10, 2.09, and 3.39 (symbols). At $T=T^{*}$, the exponent $\alpha$ which is given by the negative slope of the linear dependence (solid lines) [see Eq. (1)] changes from $\sim 0.4$ or $\sim 0.5$ (at $T<T^{*}$) to $\sim 1$ (at $T>T^{*}$), indicating different transport mechanisms below (VRH) and above (NNH) $T^{*}$. (b) Calculated values of $\alpha$ (symbols) plotted against $Q$. Below $T^{*}, \alpha$ is close to $\frac{1}{2}$ at low $Q$ and to $\frac{2}{5}$ at high $Q$. For $T>T^{*}, \alpha$ is close to 1 irrespective of $Q$.

Figure 4

Coulomb gap in PANI-DBSA, determined by two independent methods as explained in the text.

Figure 5

$\chi T$ vs $T$ for different $Q$. Note that the intercept for the $Q=0$ sample (black solid circles) is zero, i.e., there is no paramagnetism originating in localized spins.

Figure 6

Experimental ${\chi }_{\mathrm{p}}T$ vs $T$ (symbols) for $Q=0.38$ and 1.10 (where $\alpha =\frac{1}{2}$ in VRH), and $Q=2.09$ and 3.39 (where $\alpha =\frac{2}{5}$ in VRH). For clarity, the $Q>0.38$ plots are offset vertically (each by 2 emuK/mol from the previous one). The arrows point to the positions of the crossover temperatures $T^{*}$, whereas the straight lines represent fits in the ranges of linearity. The cusp at $T\sim 50\mathrm{K}$ for $Q=0.38$ is of little relevance for the current topic and has been addressed elsewhere [47, 48]. The physical parameters determining the linear fits are discussed in the text and listed in Table 1.

Figure 7

Crossover temperature $T^{*}$ as a function of $Q$, determined by three independent methods as explained in the text. Error bars are of the order of the symbol size.