Enhanced thermoelectric performance by van Hove singularities in the density of states of type-II nodal-line semimetals

The effects of the unique density of states (DOS) of a topological type-II nodal-line semimetal (NLS) on its thermoelectric (TE) transport properties are investigated through a combination of semianalytical and first-principles methods with “spinless ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$” as the artificial material. The DOS in such a type-II NLS possesses two van Hove singularities near the energy of the nodal line that leads to a large $S$ value compared to the normal metals. Combined with the linear band at the nodal line that gives high electrical conductivity $\sigma$, the type-II NLS can exhibit a relatively high TE power factor ($\mathrm{PF}=S^2\sigma$) at the nodal line. In particular, we find $\text{PF}\sim 60$ $\text{μ}\mathrm{W}\text{/}{\mathrm{cm}\mathrm{K}}^2$ at 300 K for the $n$-type ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$ by considering the electron-phonon scattering, in which the relaxation time $\tau$ of carriers can be expressed as $\tau \propto {\text{DOS}}^{-1}$ for the type-II NLS. Furthermore, we optimize parameters for the TE power factor of type-II NLSs in general by adopting the two-band model with the DOS-dependent relaxation-time approximation. Our results suggest the type-II NLSs as a potential class of high-performance TE materials among metals and semimetals, which are traditionally considered inadequate TE materials compared to semiconductors.

Figure 1

Type-II nodal-line semimetal (NLS). (a) Two-band model for type-II NLS with a linear band and a Mexican-hat band. The dashed line denotes the nodal line. (b) Total density of states (solid line) of the linear band (green) and Mexican-hat band (red) has two van Hove singularities (vHSs) at $E_0$ and $E_0-E_1$ [see Eq. (3)]. The dashed lines (with shaded areas) in the DOS plot represent the individual contribution of each band.

Figure 2

${\mathrm{Mg}}_3{\mathrm{Bi}}_2$ as an example of type-II NLSs. (a) Crystal structure of ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$. The orange, green, and blue balls indicate Bi, Mg1, and Mg2 atoms, respectively. (b) Coordination polyhedrons of the nonequivalent Mg1 and Mg2 atoms. (c) Electronic band structure with atomic orbital projections in color. The inset shows a tiny gap and a type-II Dirac point along the $\mathrm{\Gamma }\text{−}M$ and $\mathrm{\Gamma }\text{−}K$ directions, respectively. (d) Electronic density of states (DOS) and scattering rate at 300 K as a function of energy. A local increase in DOS occurs near the nodal line at $\sim 0.1$ eV.

Figure 3

(a) Seebeck coefficient $S$, (b) electrical conductivity $\sigma$, (c) thermoelectric power factor PF, and (d) figure of merit $ZT$ of spinless ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$ NLS are plotted as a function of the chemical potential $\mu$ at 300 K. Four scattering models are calculated for the relaxation-time approximation, including the EPA $[\tau \left(E\right)={\tau }^{\mathrm{el}\text{−}\mathrm{ph}}]$, the DRTA ($C/\mathcal{D}$ and $C_0/{\mathcal{D}}_{2b}$), and CRTA (${\tau }_0$). Yellow and blue regions denote $p$ type and $n$ type, respectively. The optimized PF and $ZT$ values are found at the nodal line $\mu \sim 0.1$ eV for $n$-type ${\mathrm{Mg}}_3{\mathrm{Bi}}_2$.