One-dimensionality of thermoelectric properties of semiconducting nanomaterials

Thermoelectric conversion, which is the generation of electricity from waste heat, can play an important role in renewable energy use. Lowering the dimensionality of semiconductor thermoelectric materials is a promising approach for improving thermoelectric performance, and ultimately one-dimensional (1D) semiconductor materials have the potential to exhibit maximized performance because of the presence of a 1D electronic structure, such as the van Hove singularity (vHs) in the density of states. However, experimentally verifying the effect of the 1D nature on the thermoelectric performance in semiconductor nanomaterials has been difficult because we cannot observe any traces of the 1D electronic structure in terms of conventional thermoelectric parameters, such as the Seebeck coefficient or power factor. Here, we show that a thermoelectric parameter, the thermoelectrical conductivity ($L_{12}$), is strongly correlated with the electronic structure and exhibits a unique 1D trace with single-walled carbon nanotubes (SWCNTs). We experimentally clarify that the $L_{12}$ of high-purity semiconducting SWCNTs has a peak structure with a chemical potential in the vicinity of the vHs. For comparison, the $L_{12}$ of monolayer molybdenum disulfides and graphene, which are chosen as 2D models, shows a different behavior, simply exhibiting constant values. Furthermore, we find that theoretical calculations support these $L_{12}$ behaviors, which are consistent with the expected behaviors of 1D and 2D electronic structures. Our results demonstrate that $L_{12}$ is a very good parameter for evaluating the traces of dimensionalities, thereby advancing the elucidation of the fundamental thermoelectric properties necessary for the development of low-dimensional materials.

Figure 1

Thermoelectric properties for homogeneous semiconductors calculated using the Boltzmann transport theory within an effective-mass approximation. The top panels show the energy dependence of the density of states (DOS) for (a) 3D, (b) 2D, and (c) 1D structures. Underneath the top row and from top to bottom, the panels show the Seebeck coefficient ($S$), power factor ($P$), and thermoelectrical conductivity ($L_{12}$) as a function of chemical potential ($\mu$) for an effective mass $m^{*}=9.11\times {10}^{-31}$ kg, a relaxation time $\tau =100$ fs, and a unit-cell length $a=1$ nm at $T=300$ K. The vertical axes of (b) and (c) are the same as those in (a). The band edges of the valence and conduction bands for all semiconductors are set to $\pm 0.5$ eV (dotted lines). Details are provided in Table S1 of the SM.

Figure 2

Thermoelectric parameters as a function of electrical conductivity ($\sigma$), calculated using the Boltzmann transport theory within an effective-mass approximation. The relationship is shown between the Seebeck coefficient ($S$) and $\sigma$ for the (a) 3D, (b) 2D, and (c) 1D structures. $\sigma$ at the band edge is described as ${\sigma }_{\mathrm{b}}$ (yellow gradient color). The vertical axes of (b) and (c) are the same as those in (a). The relationship is shown between thermoelectrical conductivity ($L_{12}$) and $\sigma$ for the (d) 3D, (e) 2D, and (f) 1D structures. The vertical axes of (e) and (f) are the same as those in (d).

Figure 3

Experimental results of the thermoelectrical conductivity ($L_{12}$) as a function of electrical conductivity ($\sigma$). (a) $\left|L_{12}\right|$ for (6,5) SWCNTs. $\sigma$ at the band edge is described as ${\sigma }_{\mathrm{b}}$, which is obtained by analyzing the relationship between the power factor and $\sigma$ (see Fig. S3 of the SM). A solid line presents the result of the theoretical calculation by the Kubo-Lüttinger theory. Details about the calculations are provided in Sec. 4 of this paper and in Fig. S4 of the SM. The relative error in $L_{12}$ is $\sim 10\%$, calculated from the Seebeck coefficient measurement errors of the devices. (b) $\left|L_{12}\right|$ for monolayer $\mathrm{Mo}{\mathrm{S}}_2$ as a representative 2D semiconducting material. The data are reanalyzed from Ref. [15] for $\mathrm{Mo}{\mathrm{S}}_2$. The result as a function of gate voltage is shown in Fig. S5 of the SM.

Figure 4

Chirality dependence of the thermoelectrical conductivity ($L_{12}$). Experimental results of $\left|L_{12}\right|$ in the $p$-type regions for (a) (9,4), (b) (10,3), and (c) Semi SWCNTs as a function of electrical conductivity ($\sigma$). $\sigma$ at the band edge is described as ${\sigma }_{\mathrm{b}}$, which is obtained by analyzing the relationship between the power factor and $\sigma$. The relative error in $L_{12}$ is $\sim 10\%$, calculated from the Seebeck coefficient measurement errors of the devices.