# Variable Range Hopping Model Based on Gaussian Disordered Organic Semiconductor for Seebeck Effect in Thermoelectric Device

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}σT/κ, where S, σ, T, and κ are the Seebeck coefficient, electrical conductivity, absolute temperature, and thermal conductivity, respectively [1,2]. Owing to low thermal conductivity, potential low cost, and ease of low-temperature processing, organic semiconductors (Oss) are now considered promising thermoelectric candidates [3,4]. However, the development of Oss in thermoelectric generators is impeded by their low carrier concentration and poor electrical conductivity. Caused by the much weaker inter-molecular bonding for the van der Waals interaction strength (usually in the order of 100 meV) [5,6], Oss exhibit poor crystallinity, a low degree of intermolecular electronic coupling, and a high disorder degree, which severely impact the electrical performance. Even though the chemical doping can efficiently provide high electric performance of Oss, it also increases the structural and energetic disorder and broadens the density of states (DOS) [3]. The complicated and highly disordered OS system requires insight into more detailed information about charge transport mechanism and thermoelectric property. To further explore the potential of Oss in thermoelectric applications, it is indispensable to have a specific and certified transport model for thermoelectric characterization. However, there is still controversy about the transport models of Oss for predicting Seebeck effect.

## 2. Model Methods

_{t}is the concentration of localized states, ${\delta}^{\prime}=\delta /{k}_{B}T$, and $\delta $ is the width of the Gaussian distribution reflecting the disorder’s degree. The localized states form a discrete array of hopping sites, in which the carriers transport as incoherent hopping. The probability for carrier hopping from an occupied site i to an empty site j is given by [11]:

_{ij}is the hopping distance, ε

_{i}and ε

_{j}are the localized sites’ energy, k

_{B}is Boltzman constant, and T is the temperature. For physical and mathematical simplicity, the Apsley’ hopping space method is employed later to make the energy space and real space equivalent, by defining the reduced coordinates ${R}^{\prime}=2\alpha R$ [12,13], ${\epsilon}^{\prime}=\frac{\epsilon}{{k}_{B}T}$, and the hopping range in the hopping space $R={R}_{ij}^{\prime}+\left({\epsilon}_{j}^{\prime}-{\epsilon}_{i}^{\prime}+/{\epsilon}_{j}^{\prime}-{\epsilon}_{i}^{\prime}/\right)/2$.

_{0}is the attempt-to-escape frequency, and the effective carrier mobility is obtained as $\mu =\frac{{{\displaystyle \int}}_{-\infty}^{+\infty}\sigma \left({\epsilon}_{i}^{\prime}\right)d{\epsilon}_{i}^{\prime}}{nq}$, with $\sigma \left({\epsilon}_{i}^{\prime}\right)$ as the conductivity distribution function $\sigma \left({\epsilon}_{i}^{\prime}\right)=\frac{{e}^{2}}{{k}_{B}T}D\left({\epsilon}_{i}^{\prime}\right)f\left({\epsilon}_{i}^{\prime},{\epsilon}_{f}^{\prime}\right)g\left({\epsilon}_{i}^{\prime}\right)$. The transport energy level ε

_{t}is defined as the energetic position of the most probably conductance ${\sigma}_{\mathrm{max}}$ occurs.

## 3. Results and Discussion

_{t}= 1 × 10

^{22}cm

^{−3}, α = 2 × 10

^{7}cm

^{−1}, T = 300 K, and v

_{0}= 1 × 10

^{13}s

^{−1}. It is observed that the mobility (with Gaussian width δ = 4 k

_{B}T) exhibits strong n-dependent properties when carrier concentration is larger than around n = 1 × 10

^{18}cm

^{−3}, and gradually becomes n-independent, reflecting carriers’ saturated diving process in DOS tail. The result is consistent with Blom’s and Baranovski’s theoretic interpretations of OCC-PPV and P3HT’s data [18,19]. The Seebeck coefficient S increases with the carrier concentration, decreasing when the carrier concentration is high, and is weakly dependent on the degree of disorder. When the carriers concentration drops around n = 1 × 10

^{18}cm

^{−3}, mobility becomes n-independent, while the Seebeck coefficient keeps monotonically decreasing and exhibits n-dependent properties over the whole range of carrier concentration. Physically, the saturation of mobility at low carrier concentration arises from the invariable activation energy E

_{a}that carriers need during the process of hopping from equilibrium level ε

_{∞}to transport level ε

_{t}[20]. However, the Seebeck coefficient is directly linked to the energy that carriers bring, which is defined relative to the Fermi level ε-ε

_{f}, having no relation with the equilibrium level ε

_{∞}.

_{f}at higher carrier concentration while gradually stabilizing at ε

_{∞}in spite of the downward shifting Fermi level. Simultaneously, the distribution of conductivity or the transport energy ε

_{t}at various carrier concentration is plotted. Different to mobility edge ε

_{c}, transport energy ε

_{t}would rise gradually with an increase in carrier concentration n, especially at larger values of n. Unlike the drastic fall in the Seebeck coefficient at higher conductivity in Mott’s ME model, our VRH model might predict a slower decrease in the Seebeck coefficient with increasing carriers’ concentration. Experimental data for doped PEDOT-Tos film [21] displayed as symbols in Figure 2a are well fitted by our theoretic model, with input parameters N

_{t}= 1 × 10

^{22}cm

^{−3}, α = 2 × 10

^{7}cm

^{−1}, v

_{0}= 1 × 10

^{13}s

^{−1}, and δ = 6 k

_{B}T.

_{t}= 1 × 10

^{22}cm

^{−3}, α = 2 × 10

^{7}cm

^{−1}, v

_{0}= 1 × 10

^{13}s

^{−1}, and δ = 0.1 eV. At higher carrier concentration, the Seebeck coefficient is predicted to slowly increase with temperature, consistent with the polyacetylene experimental results for carrier concentration n = 7 × 10

^{20}cm

^{−3}, shown in Figure 3 [22,23,24]. In the moderate carrier concentration region, at room temperature and below, the Seebeck coefficient also exhibits weak temperature dependence, consistent with experimental data of a pentacene-based organic FET, in which the fitting parameters of carrier concentration in our model coincides with that actual accumulated in an operated OFET (n~10

^{18}~10

^{19}cm

^{−3}).

_{f}) = const. Considering our model, hopping does not necessarily occur near the Fermi level. The energetic position ε

_{t}of the most probable conductance varies with temperature and carrier concentration; therefore, the density of states is not a constant parameter. As long as a proper DOS is employed together with reasonable hopping parameters, it could ideally model the conductivity–Seebeck coefficient relationship with our VRH model using Apsley’ hopping space method (as Equations (2)–(4) shown).

_{t}= 1 × 10

^{22}cm

^{−3}, α = 2 × 10

^{7}cm

^{−1}, and v

_{0}= 1 × 10

^{13}s

^{−1}for δ = 4, 5, 6 and 10 k

_{B}T, and v

_{0}= 3 × 10

^{13}s

^{−1}for δ = 3 and 1.5 k

_{B}T. The apparent deviation of Mott’s ME model from the experiments can be compared by dashed lines ($s=-\frac{1}{qT}\left[\left({E}_{C}-{E}_{f}\right)+{A}_{c}\right]$, $\sigma ={\sigma}_{ME}exp\left[-\left({E}_{C}-{E}_{f}\right)/{k}_{B}T\right]$) in Figure 4b, where the σ

_{ME}is metallic conductivity at mobility edge ε

_{c}, by setting different values σ

_{ME}= 100, 1, and 0.01 S/cm. It is due to this that the invariance of mobility edge ε

_{c}makes the charge carried thermo-power decrease quickly with Fermi level (ε

_{c}-ε

_{f}) when carrier concentration increases. Meanwhile, in our VRH systems, the transport energy ε

_{t}rises together with ε

_{f}, making the thermo-power (ε

_{t}-ε

_{f}) decrease much more gently with the conductivity.

_{0}. This suggests that the doped organic conducting materials are totally disordered systems, better depicted by hopping limited transport than by multiple trapping and release transport in Mott’s ME system. The evolution of transport energy ε

_{t}is essential for the thermo-power at higher carrier concentrations, which could not be expected when the transport is dominated by ME. For achieving a higher Seebeck coefficient, together with higher conductivity in thermoelectric devices, the hopping-limited transport model featured by transport energy ε

_{t}, together with lower energetic disorder, is expected. This needs more systematic investigations in the future, for the two factors are not easy realized simultaneously.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Shi, X.-L.; Zou, J.; Chen, Z.-G. Advanced Thermoelectric Design: From Materials and Structures to Devices. Chem. Rev.
**2020**, 120, 7399–7515. [Google Scholar] [CrossRef] [PubMed] - Snyder, G.J.; Ursell, T.S. Thermoelectric Efficiency and Compatibility. Phys. Rev. Lett.
**2003**, 91, 148301. [Google Scholar] [CrossRef][Green Version] - Zhang, F.; Di, C.-A. Exploring Thermoelectric Materials from High Mobility Organic Semiconductors. Chem. Mater.
**2020**, 32, 2688–2702. [Google Scholar] [CrossRef] - Zhang, F.; Zang, Y.; Huang, D.; Di, C.-A.; Gao, X.; Sirringhaus, H.; Zhu, D. Modulated Thermoelectric Properties of Organic Semiconductors Using Field-Effect Transistors. Adv. Funct. Mater.
**2015**, 25, 3004–3012. [Google Scholar] [CrossRef] - Hannewald, K.; Stojanovic, V.M.; Schellekens, J.M.T.; Bobbert, P.A.; Kresse, G.; Hafner, J. Theory of polaron band-width narrowing in organic molecular crystals. Phys. Rev. B
**2004**, 69, 075211. [Google Scholar] [CrossRef][Green Version] - Troisi, A.; Orlandi, G. Charge-Transport Regime of Crystalline Organic Semiconductors: Diffusion Limited by Thermal Off-Diagonal Electronic Disorder. Phys. Rev. Lett.
**2006**, 96, 086601. [Google Scholar] [CrossRef] [PubMed] - Mott, N.F. Conduction in Non-Crystalline Systems: I. Localized Electronic States in Disordered Systems. Philos. Mag.
**1968**, 17, 1259–1268. [Google Scholar] [CrossRef] - Baranovski, S. Charge Transport in Disordered Solids with Applications in Electronics; John Wiley & Sons: Chichester, UK, 2006. [Google Scholar]
- Kang, S.; Snyder, G. Charge-transport model for conducting polymers. Nat. Mater.
**2017**, 16, 252–257. [Google Scholar] [CrossRef] - Bässler, H. Charge Transport in Disordered Organic Photoconductors a Monte Carlo Simulation Study. Phys. Stat. Sol. B
**1993**, 175, 15. [Google Scholar] [CrossRef] - Lu, N.; Li, L.; Liu, M. Universal carrier thermoelectric-transport model based on percolation theory in organic semi-conductors. Phys. Rev. B
**2015**, 91, 195205. [Google Scholar] [CrossRef] - Li, L.; Lu, N.; Liu, M.; Bässler, H. General Einstein relation model in disordered organic semiconductors under quasiequilibrium. Phys. Rev. B
**2014**, 90, 214107. [Google Scholar] [CrossRef] - Mehraeen, S.; Coropceanu, V.; Brédas, J.-L. Role of band states and trap states in the electrical properties of organic semiconductors: Hopping versus mobility edge model. Phys. Rev. B
**2013**, 87, 195209. [Google Scholar] [CrossRef][Green Version] - Vissenberg, M.C.J.M.; Matters, M. Theory of the field-effect mobility in amorphous organic transistors. Phys. Rev. B
**1998**, 57, 12964–12967. [Google Scholar] [CrossRef][Green Version] - Mott, N.F.; Davis, E.A. Electronic Processes in Non-Crystalline Materials; Clarendon Press: Oxford, UK, 1971. [Google Scholar]
- Fritzsche, H. A general expression for the thermoelectric power. Solid State Commun.
**1971**, 9, 1813–1815. [Google Scholar] [CrossRef] - Arkhipov, V.I.; Heremans, P.; Emelianova, E.V.; Bässler, H. Effect of doping on the density-of-states distribution and carrier hopping in disordered organic semiconductors. Phys. Rev. B
**2005**, 71, 045214. [Google Scholar] [CrossRef] - Tanase, C.; Meijer, E.J.; Blom, P.W.M.; De Leeuw, D.M. Unification of the Hole Transport in Polymeric Field-Effect Transistors and Light-Emitting Diodes. Phys. Rev. Lett.
**2003**, 91, 216601. [Google Scholar] [CrossRef][Green Version] - Oelerich, J.O.; Huemmer, D.; Baranovskii, D. How to find out the density of states in disordered organic semiconductors. Phys. Rev. Lett.
**2012**, 108, 226403. [Google Scholar] [CrossRef] - Li, L.; Meller, G.; Kosina, H. Einstein relation in hopping transport of organic semiconductors. Appl. Phys. Lett.
**2008**, 92, 013307. [Google Scholar] [CrossRef] - Bubnova, O.; Khan, Z.U.; Malti, A.; Braun, S.; Fahlman, M.; Berggren, M.; Crispin, X. Optimization of the thermoelectric figure of merit in the conducting polymer poly (3,4-ethylenedioxythiophene). Nat. Mater.
**2011**, 10, 429. [Google Scholar] [CrossRef] - Gao, X.; Uehara, K.; Klug, D.D.; Patchkovskii, S.; Tse, J.S.; Tritt, T.M. Theoretical studies on the thermopower of semiconductors and low-band-gap crystalline polymers. Phys. Rev. B
**2005**, 72, 125202. [Google Scholar] [CrossRef] - Kaiser, A.B. Thermoelectric power and conductivity of heterogeneous conducting polymers. Phys. Rev. B
**1989**, 40, 2806. [Google Scholar] [CrossRef] [PubMed] - Park, Y. Structure and morphology: Relation to thermopower properties of conductive polymers. Synth. Met.
**1991**, 45, 173. [Google Scholar] [CrossRef] - Glaudell, A.M.; Cochran, J.E.; Patel SNChabinyc, M.L. Impact of the doping method on conductivity and thermopower in semiconducting polythiophenes. Adv. Energy Mater.
**2015**, 5, 1401072. [Google Scholar] [CrossRef] - Kaiser, A.B. Electronic transport properties of conducting polymers and carbon nanotubes. Rep. Prog. Phys.
**2001**, 64, 1–49. [Google Scholar] [CrossRef] - Sun, J.; Yeh, M.L.; Jung, B.J.; Zhang, B.; Feser, J.; Majumdar, A.; Katz, H.E. Simultaneous increase in seebeck coefficient and conductivity in a doped poly (alkylthiophene) blend with defined density of states. Macromolecules
**2010**, 43, 2897. [Google Scholar] [CrossRef] - Zhang, Q.; Sun, Y.M.; Xu, W.; Zhu, D.B. What to expect from conducting polymers on the playground of thermoelectricity: Lessons learned from four high-mobility polymeric semiconductors. Macromolecules
**2014**, 47, 609. [Google Scholar] [CrossRef] - Fan, Z.; Ouyang, J. Thermoelectric Properties of PEDOT:PSS. Adv. Energy Mater.
**2019**, 5, 1800769. [Google Scholar] [CrossRef]

**Figure 1.**The schematic of carriers’ hopping near the transport energy. The blue region means the possible hopping range of carriers.

**Figure 2.**(

**a**) Carrier concentration of mobility (black line) and Seebeck coefficient (red lines) in Gaussian disordered organic semiconductor, with fitting data from PEDOT:Tos film (blue circles). (

**b**) Schematic of the energy position evolution with carrier concentration for carrier density distribution (blue lines) and conductivity distribution (red lines); orange dashed line indicates the position of equilibrium level, blue dashed line stands for the “main source” from which the carriers are activated, and red dashed lines indicate the transport energy level.

**Figure 3.**Temperature dependence of Seebeck coefficient at Gaussian width δ = 0.1 eV; experiment data for fitting are from pentacene based FET (red circles), and polymer film based on polyacetylene (blue square).

**Figure 4.**Relationship between conductivity and Seebeck coefficient: (

**a**) comparison of fitting results between our model and Mott’s VRH/ME model including P3HT, P2TDC

_{17}-FT4, and PBTTT-C

_{14}; (

**b**) fitting between theory in this work and experimental data given by the literature, including P3HT:F4TCNQ mixed P3HTT film (black squares and black circles), polyacetylene (blue stars), PBTTT film (orange pentagons and red triangles), and PEDOT:PSS (magenta pentagons).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhao, Y.; Wang, J.
Variable Range Hopping Model Based on Gaussian Disordered Organic Semiconductor for Seebeck Effect in Thermoelectric Device. *Micromachines* **2022**, *13*, 707.
https://doi.org/10.3390/mi13050707

**AMA Style**

Zhao Y, Wang J.
Variable Range Hopping Model Based on Gaussian Disordered Organic Semiconductor for Seebeck Effect in Thermoelectric Device. *Micromachines*. 2022; 13(5):707.
https://doi.org/10.3390/mi13050707

**Chicago/Turabian Style**

Zhao, Ying, and Jiawei Wang.
2022. "Variable Range Hopping Model Based on Gaussian Disordered Organic Semiconductor for Seebeck Effect in Thermoelectric Device" *Micromachines* 13, no. 5: 707.
https://doi.org/10.3390/mi13050707