Comprehensive Model for the Thermoelectric Properties of Two-Dimensional Carbon Nanotube Networks

Aditya Dash, Dorothea Scheunemann, and Martijn Kemerink

Phys. Rev. Applied 18, 064022 – Published 8 December 2022

Abstract

Networks of semiconducting single-walled carbon nanotubes (SWCNTs) are interesting thermoelectric materials due to the interplay between CNT and network properties. Here, we present a unified model to explain charge and energy transport in SWCNT networks. We use the steady-state master equation for the random resistor network containing both intra- and intertube resistances, as defined through their one-dimensional density of states that are modulated by static Gaussian disorder. The tube-resistance dependence on the carrier density and disorder is described through the Landauer formalism. Electrical and thermoelectric properties of the network are obtained by solving Kirchhoff’s laws through a modified nodal analysis, where we use the Boltzmann-transport formalism to obtain the conductivity, Seebeck coefficient, and electronic contribution to the thermal conductivity. The model provides a consistent description of a wide range of previously published experimental data for temperature and charge-carrier-density-dependent conductivities and Seebeck coefficients, with energetic disorder being the main factor to explain the experimentally observed mobility upswing with carrier concentration. Moreover, we show that, for lower disorder energies, the Lorentz factor obtained from the simulation is in accordance with the Wiedemann-Franz law for degenerate-band semiconductors. At higher disorder, deviations from simple band behavior are found. Suppressed disorder energy and lattice thermal conductivity can be the key to higher thermoelectric figures of merit, zT, in SWCNT networks, possibly approaching or even exceeding zT = 1. The general understanding of transport phenomena will help the selection of chirality, composition, and charge-carrier density of SWCNT networks to improve their efficiency of thermoelectric energy conversion.

Received 3 August 2022
Revised 25 October 2022
Accepted 26 October 2022

DOI:https://doi.org/10.1103/PhysRevApplied.18.064022

Physics Subject Headings (PhySH)

Electrical conductivity Network flow Thermal conductivity Thermoelectrics

Carbon-based materials Nanotubes

Boltzmann theory Landauer formula

Condensed Matter, Materials & Applied PhysicsEnergy Science & Technology

Authors & Affiliations

Aditya Dash, Dorothea Scheunemann , and Martijn Kemerink ^*

Institute for Molecular Systems Engineering and Advanced Materials, Heidelberg University, Im Neuenheimer Feld 225, 69120 Heidelberg, Germany

^*martijn.kemerink@cam.uni-heidelberg.de

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Vol. 18, Iss. 6 — December 2022

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Images

Figure 1
Outline of the numerical model. (a) 2D SWCNT network consisting of randomly dispersed 1D tubes of linear density $l_{d}$ and mean length $l_{0}$ . Fat horizontal lines are source and drain contacts. (b) Magnification of a single junction, which divides tubes i and j into two segments each, leading to (c) resistor (sub)network. (d) Aggregated SWCNTs may form bundles in the network, which is accounted for by a bundling factor, $β$ .
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Figure 2
Calculated normalized mobility using mobility maxima (μ_max) for (6,5) chiral SWCNT networks: (a) versus Fermi energy, parametric in energetic disorder (inset is a magnification of the development of the mobility peak at higher disorder), $l_{0} = (1.5 \pm 0.6) μ m$ , $l_{d} = 7 μ m^{- 1}$ ; (b) versus 2D charge-carrier density, compared with experimental data from Ref. [47] measured in field-effect-transistor geometry (symbols) and the best fit from Ref. [47] (blue line). Black line is for the full model; the value for $ρ_{t}$ in the legend is the approximate numerical value for the tube resistivity; the red line is for $ρ_{t} = 0$ . Linear density of the network is $l_{d} = 30 μ m^{- 1}$ with mean tube length $l_{0} = 1 μ m$ , dispersion $σ_{l} = 0.3 μ m$ , and $β = 1$ .
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Figure 3
Current distribution in the SWCNT network with varying disorder at constant Fermi energy, $E_{F} = 0.5 eV$ . Cutoff $I_{cut} = 0.05 \times I_{\max}$ , where $I_{\max}$ is the maximum current carried by a single tube segment in the network, is applied to prevent the low-current-carrying tubes in the network from dominating the image. Disorder $σ_{DOS} = 0, 90, and 180 meV$ from left to right; otherwise, same parameters as in Fig. 2 are used.
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Figure 4
Simultaneous fit to the temperature dependence of mobility (a) and Seebeck coefficient (b) of (6,5) chiral SWCNT networks. Experimental data are measured in field-effect-transistor geometry and are taken from Statz et al. [17]. Linear density of the network is $l_{d} = 12.6 μ m^{- 1}$ with mean tube length $l_{0} = (0.95 \pm 0.4) μ m$ . (6,5) SWCNT networks are treated with 1,2,4,5-tetrakis(tetramethylguanidino)benzene (TTMGB), which leads to pure electron transport. Key simulation parameters are given in the legends. Moreover, $σ_{DOS} = 110 meV$ , $β = 4.3$ . (c) Fit to S versus T only for $σ_{DOS} = 92 meV$ and $β = 7$ . Carrier density corresponding to $E_{F}$ mentioned in the caption, n = 1.29 × 10¹², 1.83 × 10¹², and 2.54 × 10¹² cm⁻², going from lower to higher $E_{F}$ . (μ_300,max refers to the mobility maxima at 300 K.)
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Figure 5
Electronic contribution to thermal conductivity from the numerical model for various energetic disorder values versus (a) carrier concentration and (b) conductivity. Inset in (b) shows the effective Lorentz factor, L, as defined through the Wiedemann-Franz (WF) law. Parameters used are $β = 4.3$ , $l_{d} = 12.6 μ m^{- 1}$ , and $l_{0} = (0.95 \pm 0.4) μ m$ .
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Figure 6
Calculated thermoelectric figure of merit versus charge density (a), parametric in energetic disorder, and lattice thermal conductivity (b). Calculation parameters are $β = 4.3$ , $l_{d} = 12.6 μ m^{- 1}$ , and $l_{0} = (0.95 \pm 0.4) μ m$ .
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