## 2 Materials and Methods  
The pyrogallol-formaldehyde/ZrO\({}_{2}\) (PF/ZrO\({}_{2}\)) composite was synthesized using a simple sol-gel method, which is cost-effective, environmentally friendly, and time-efficient, as well as allowing for the production of materials with a high specific surface area and porosity. In this study, zirconium acetate (C\({}_{8}\)H\({}_{12}\)O\({}_{8}\)Zr; Sigma, St. Louis, MO, USA), a solution in dilute acetic acid, was used as a zirconium source for the synthesis of ZrO\({}_{2}\) nanoparticles. Pyrogallol (P) (C\({}_{6}\)H\({}_{3}\)(OH)\({}_{3}\); 97%; Fisher Scientific, Waltham, MA, USA) and formaldehyde (F) (CH\({}_{2}\)O; 99%; Biopharm, Iselin, NJ, USA) were used to form the PF matrix. Distilled water (DW) was used as a solvent and picric acid (C\({}_{6}\)H\({}_{3}\)N\({}_{3}\)O\({}_{7}\); 99%; Scharlau, Barcelona, Spain) was added as a catalyst (C) to promote the gelatinization of the matrix. All the reagents and solvents were analytical-grade and used as received without further purification. In the first step, zirconium oxide (ZrO\({}_{2}\)) aerogel was prepared by a sol-gel method based on the typical synthesis process as described by [PERSON] et al. [15; 16]. The molar ratio of P/F and P/DW were set as 1/3 and 1/6. The weight of ZrO\({}_{2}\) added was 5 wt.%. Subsequently, the wet gel was dried in a humid atmosphere at room temperature for two weeks. Then, the samples were placed into an incubator for drying at 150 \({}^{\circ}\)C, with a heating rate of 10 \({}^{\circ}\)C/24 h. Finally, the xerogels were treated in a tubular furnace with a heating rate of 5 \({}^{\circ}\)C/min under an inert atmosphere (pyrolysis). The furnace held the desired temperature for 2 h to obtain the nanocomposite PF/ZrO\({}_{2}\). For the samples reported in this study, the pyrolysis temperature was 600 \({}^{\circ}\)C. The samples were prepared from the beginning into polypropylene molds to have a cylindrical shape. The samples were completed via two electrical contacts in both faces using conductive silver paint.  
The contacts had an active area of 0.02 cm\({}^{2}\) and samples with a thickness from 0.19 cm to 0.075 cm were used.  
The crystallographic structure of the samples was obtained via X-ray diffraction (XRD) in a Brucker D8 Advance X-ray diffractometer using CuK\(\alpha\) (\(\lambda\) = 1.540598 A) radiation. The morphologies of the obtained nanocomposite were performed via field emission scanning electron microscopy using a LYRA3 TESCAN equipped with an electron-dispersive X-ray (EDX). A detailed morphological and structural characterization of the PF/ZrO\({}_{2}\) samples is provided in our previous work [17].  
Scanning electron microscopy (SEM) images of the PF/ZrO\({}_{2}\) composite are shown in Figure 1a, and the corresponding X-ray diffraction (XRD) diffractograms are depicted in Figure 1b. Current-voltage (I-V) characterization was conducted using a Keithley 487 picoammeter/voltage source (Portland, OR, USA) and an Agilent 4156C semiconductor parameter analyzer (Santa Clara, CA, USA). Temperature-dependent measurements were performed using an Advanced Research Systems liquid helium close cycle (Macungie, PA, USA), covering a temperature range from 80 to 320 K.  
The electrical connections of the sample to the measuring circuit are shown in Figure 1c. The samples were mounted on a thick copper holder that acted as a heat sink to dissipate excess heat generated by the Joule effect and prevent thermal runaway. Pulsed I-V measurements were performed by applying voltage steps that lasted 10 milliseconds and were separated in time by 1 s. All electrical measurements were conducted in a vacuum at a pressure of 10\({}^{-6}\) Torr.