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Activated Hopping Transport in Nematic Conducting Aerogels

  • ELECTRONIC PROPERTIES OF SOLID
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Abstract

The transport properties of nematic aerogels, which consist of highly oriented Al2O3⋅SiO2 nanofibers coated with a graphene shell with a large number of defects, are studied. The temperature dependences of the electrical resistivity in the range of 9–40 K strictly follow the formula derived to describe the variable range hopping (VRH) conductivity, in which exponent α changes from 0.4 to 0.9 when the number of layers in the graphene shell decreases from 4–6 to 1–2. The dependence of α on the shell thickness can be explained by a simultaneous change in the dimension of hopping transport and the character of the energy dependence of the density of localized states near the Fermi level. The fact that α approaches unity at the minimum graphene shell thickness indicates a gradual transition from VRH transport to nearest neighbor hopping (NNH) transport. The magnetoresistance measured at T = 4.2 K is negative, increases significantly with decreasing graphene shell thickness, and is approximated by a formula for the case of weak localization with a good accuracy. The phase coherence lengths are in a reasonable relation with the graphene grain sizes. The conducting aerogels under study complement the well-known set of materials that exhibit hopping electron transport at low temperatures, which is characteristic of media with strong carrier localization, and also a negative magnetoresistance, which usually manifests itself under weak localization conditions.

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Notes

  1. The case α = 1/2 is known to be not only a one-dimensional version of Eq. (4), but above all to be the result of appearance of a Coulomb gap near the Fermi level due to electron–electron interaction regardless of the dimensionality (Shklovskii–Efros law [14]).

  2. In this regard, work [20] (mentioned above), where α increased from 0.25 to 1 when the volume fraction of a PANI network in a PMMA matrix decreased, is noteworthy. In addition, according to [23], a certain set of α values was observed in a series of polycrystalline graphene samples.

  3. Another carbon system, namely, so-called carbynes, should be mentioned here; they are supposed to consist of chains of carbon atoms with the sp bonds. Hopping conduction with α= 1/2, 1/3, and 1/4 is observed in carbyne samples depending on the synthesis temperature, and the case of α = 1/2 is explained by a one-dimensional character of hopping transport rather than by the presence of a Coulomb gap (see [43] and Refs. therein).

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Funding

This work was supported by the Russian Science Foundation, project no. RNF-20-42-08004.

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Authors and Affiliations

  1. Lebedev Physical Institute, Russian Academy of Sciences, 119991, Moscow, Russia

    V. I. Tsebro & R. A. Khmel’nitskii

  2. Kapitsa Institute for Physical Problems, Russian Academy of Sciences, 119334, Moscow, Russia

    V. I. Tsebro, E. G. Nikolaev & L. B. Lugansky

  3. Metallurg Engineering Ltd., 11415, Tallinn, Estonia

    M. S. Kutuzov

  4. Prokhorov General Physics Institute, Russian Academy of Sciences, 119991, Moscow, Russia

    A. A. Tonkikh

  5. Institute for High Pressure Physics, Russian Academy of Sciences, 142190, Moscow, Russia

    A. I. Khar’kovskii

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  1. V. I. Tsebro
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  2. E. G. Nikolaev
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  3. L. B. Lugansky
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  4. M. S. Kutuzov
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Correspondence to V. I. Tsebro or E. G. Nikolaev.

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Translated by K. Shakhlevich

APPENDIX

APPENDIX

In the Schnabel method, two measurement methods are possible. In the first Schnabel geometry, an electric current is supplied to contacts A and D and potential difference VBC between contacts B and C is measured (see the designations of the contacts in the left part of Fig. 5). In the second Schnabel geometry, the current flows through contacts A and B and potential difference VCD is measured between contacts C and D. From such measurements, conventional resistances R1 = VBC/IAD and R2 = VCD/IAB are determined. VBC and VCD can be analytically expressed by solving the problem of the electric field potential distribution in the sample volume when current I passes through the current contacts.

In Schnabel’s original works [39, 40], this problem was solved for a sample in the shape of an infinite flat plate, where only two geometric parameters, namely, plate thickness d and the distance between neighboring contacts s, were present, and in the shape of an infinite strip, where another geometric parameter, namely, strip width b, was added.

In our works [38, 58], a solution to this problem was found for samples having the shape of a rectangular parallelepiped of finite dimensions. R1 and R2 were shown to be represented as

$${{R}_{1}} = \frac{\rho }{d}G(a,b,d,s),\quad {{R}_{2}} = \frac{\rho }{d}H(a,b,d,s),$$

where ρ is the electrical resistivity of the conducting medium; a, b, and d are the sample sizes along the principal axes; and s is the distance between contacts AB and CD, respectively. Functions G and H are analytically expressed in the form of double infinite series [38, 58]. In the case of an isotropic sample, resistivity ρ can be found from any one of these measurements (either from R1 or R2).

In an anisotropic case, van der Pauw [59] showed that a simple linear transformation of coordinates can be used to reduce the problem of potential distribution in an anisotropic sample to a similar problem for a hypothetical isotropic sample with different sizes and electrical resistivity. Here, we briefly present the main final calculations concerning the case of the anisotropic nematic aerogel samples studied in this work.

The coordinate system is assumed to be chosen so that the edges of the bulk aerogel samples are along the principal axes of the resistivity tensor, taken as axes (x1, x2, x3), and segments AB and CD are assumed to be parallel to axis x1 along aerogel nanofibers (Fig. 5). In this coordinate system, tensor ρik is diagonal and has only three components (ρ1, ρ2, ρ3). The coefficients of the linear coordinate transformation are chosen so that the electrical resistivity ρ* of the isotropic image and sizes a*, b*, of the isotropic image and c* are

$$\rho \text{*} = {{({{\rho }_{1}}{{\rho }_{2}}{{\rho }_{3}})}^{{1/3}}},$$
$$a\text{*} = {{({{\rho }_{1}}{\text{/}}\rho \text{*})}^{{1/2}}}a,\quad b\text{*} = {{({{\rho }_{2}}{\text{/}}\rho \text{*})}^{{1/2}}}b,$$
$$d\text{*} = {{({{\rho }_{3}}{\text{/}}\rho \text{*})}^{{1/2}}}d,\quad s\text{*} = {{({{\rho }_{1}}{\text{/}}\rho \text{*})}^{{1/2}}}.$$

Here superscript * denotes the quantity related to the isotropic image of a real anisotropic sample. In this case, measured resistances R1 and R2 of the anisotropic sample under study are equal to the corresponding resistances of its hypothetical isotropic image, \(R_{1}^{*}\) and \(R_{2}^{*}\).

As is seen from the analytical formulas for functions G and H (see [38, 58]), they actually depend on three rather than four (a, b, d, s) arguments; it is convenient to represent these three arguments as ratios (a/s, b/s, d/s) for the sample under study and (a*/s*, b*/s*, d*/s*) for its isotropic image. Then resistances R1 and R2 to be measured can be written as

$$\begin{gathered} {{R}_{1}} = R_{1}^{*} = \frac{{\rho {\text{*}}}}{{d{\text{*}}}}G\left( {\frac{{a{\text{*}}}}{{s{\text{*}}}},\frac{{b{\text{*}}}}{{s{\text{*}}}},\frac{{d{\text{*}}}}{{s{\text{*}}}}} \right) \\ = \frac{{{{{({{\rho }_{1}}{{\rho }_{2}})}}^{{1/2}}}}}{d}G\left( {\frac{a}{s},\frac{{{{\lambda }_{{21}}}b}}{s},\frac{{{{\lambda }_{{31}}}d}}{s}} \right), \\ \end{gathered} $$
$$\begin{gathered} {{R}_{2}} = R_{2}^{*} = \frac{{\rho {\text{*}}}}{{d{\text{*}}}}H\left( {\frac{{a{\text{*}}}}{{s{\text{*}}}},\frac{{b{\text{*}}}}{{s{\text{*}}}},\frac{{d{\text{*}}}}{{s{\text{*}}}}} \right) \\ = \frac{{{{{({{\rho }_{1}}{{\rho }_{2}})}}^{{1/2}}}}}{d}H\left( {\frac{a}{s},\frac{{{{\lambda }_{{21}}}b}}{s},\frac{{{{\lambda }_{{31}}}d}}{s}} \right), \\ \end{gathered} $$

where the designations λ21 = (ρ2/ρ1)1/2 and λ31 = (ρ3/ρ1)1/2 are introduced.

Only two quantities, namely, R1 and R2, are independent during measurements. Therefore, it is impossible to determine all three values of the resistivity tensor from these measurements. However, if two of the three principal values of the resistivity tensor are the same (as in our case of bulk nematic aerogel samples), then it has only two independent principal values, which can be found by measuring R1 and R2. Note that, in our experiments on bulk aerogel samples, the lines along which the AB and CD probes are located are directed along the highest conductivity direction, which is taken as axis x1 with electrical resistivity ρ1 (see Fig. 5), and the electrical resistivities along other two axes are taken to be the same, i.e., ρ2 = ρ3 and λ21 = λ31.

Knowing the analytical expressions for functions G and H [38, 58], the sample sizes (a, b, d), and the distance between point contacts s, we can construct the ratio

$$\frac{{{{R}_{1}}}}{{{{R}_{2}}}} = \frac{{G(a{\text{/}}s,\lambda b{\text{/}}s,\lambda d{\text{/}}s)}}{{H(a{\text{/}}s,\lambda b{\text{/}}s,\lambda d{\text{/}}s)}}$$

as a function of only one argument λ = λ21 = λ31. For each measurement of R1 and R2, this dependence is used to determine anisotropy parameter λ, and ρ1 is then calculated by the formula

$${{\rho }_{1}} = \frac{{{{R}_{1}}d}}{{\lambda G(a{\text{/}}s,\lambda b{\text{/}}s,\lambda d{\text{/}}s)}}$$

or

$${{\rho }_{1}} = \frac{{{{R}_{2}}d}}{{\lambda H(a{\text{/}}s,\lambda b{\text{/}}s,\lambda d{\text{/}}s)}}.$$

Finally, we find ρ3 = ρ1λ2.

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Tsebro, V.I., Nikolaev, E.G., Lugansky, L.B. et al. Activated Hopping Transport in Nematic Conducting Aerogels. J. Exp. Theor. Phys. 134, 222–234 (2022). https://doi.org/10.1134/S106377612202008X

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