Optimal Bandwidth for High Efficiency Thermoelectrics

The thermoelectric figure of merit ($ZT$) in narrow conduction bands of different material dimensionalities is investigated for different carrier scattering models. When the bandwidth is zero, the transport distribution function (TDF) is finite, not infinite as previously speculated by Mahan and Sofo [Proc. Natl. Acad. Sci. U.S.A. 93, 7436 (1996)], even though the carrier density of states goes to infinity. Such a finite TDF results in a zero electrical conductivity and thus a zero $ZT$. We point out that the optimal $ZT$ cannot be found in an extremely narrow conduction band. The existence of an optimal bandwidth for a maximal $ZT$ depends strongly on the scattering models and the dimensionality of the material. A nonzero optimal bandwidth for maximizing $ZT$ also depends on the lattice thermal conductivity. A larger maximum $ZT$ can be obtained for materials with a smaller lattice thermal conductivity.

Figure 1

$ZT$ plotted as functions of the chemical potential with respect to the upper band edge $\mu -W_{\alpha }/2$, and of the bandwidth $W_{\alpha }$ in (a) the 1D system when ${\gamma }_0=0.06$, (b) the 2D system when ${\gamma }_0=0.06$, (c) the 3D system when ${\gamma }_0=0.06$, and (d) the 3D system when ${\gamma }_0=0.1$.

Figure 2

$ZT$ plotted as a function of the bandwidth $W_{\alpha }$ in (a) the 1D system and (b) the 3D system when ${\gamma }_0=0.06$, 0.1, 0.14, and 0.18. The calculations use $\mu -W_{1\mathrm{D}}/2=2k_{B}T$ for the 1D and $\mu -W_{3\mathrm{D}}/2=k_{B}T$ for the 3D system.

Figure 3

$Z{T}_{3\mathrm{D}}$ plotted as a function of the bandwidth $W_{3\mathrm{D}}$ in the 3D system for different carrier scattering models as shown in Table : constant relaxation time ${\tau }_{3\mathrm{D},0}=0.1\mathrm{ps}$ (solid curve); relaxation time inversely proportional to DOS where $C_{3\mathrm{D}}={10}^{34}\mathrm{s}/\mathrm{J}{\mathrm{m}}^3$ (dashed curve); and constant carrier MFP $l_{3\mathrm{D}}=10\mathrm{nm}$ (dotted curve). The curve with square dots is the case when a constant background TDF ${\Xi }_{\mathrm{bg}}=0.045{\Xi }_{3\mathrm{D}}(0)$ is superimposed to the narrow band TDF for the constant MFP model. The calculations use $T=300\mathrm{K}$, ${\kappa }_p=0.2\mathrm{W}/\mathrm{mK}$, $a=1\mathrm{nm}$, and $\mu -W_{3\mathrm{D}}/2=k_{B}T$.