Optimal thermoelectric figure of merit in Bi${}_2$Te${}_3$/Sb${}_2$Te${}_3$ quantum dot nanocomposites

Following the idea that there exists an optimal bandwidth for maximizing the thermoelectric figure of merit ($ZT$), we conduct detailed calculations in this paper to search for the optimal $ZT$ in Bi${}_2$Te${}_3$/Sb${}_2$Te${}_3$ quantum dot (QD) nanocomposites (NCs) with Bi${}_2$Te${}_3$ QDs uniformly embedded in Sb${}_2$Te${}_3$ matrix where electron minibands are formed. The two-channel transport model, which considers both the miniband transport by the quantum-confined carriers and the background transport by the bulklike carriers, is used for electrical transport, while the lattice thermal conductivity is modeled using the modified effective medium approximation. Simultaneous decrease of the lattice thermal conductivity and the Lorenz number leads to an enhanced $ZT$ in QD NCs when the Seebeck coefficient is not dramatically decreased. The optimal structural parameters that result in optimal electronic structure for maximizing $ZT$ are found, with the consideration of realistic carrier scattering physics including phonon bottleneck effect. The optimal QD size is found to be ∼$6\mathrm{nm}$, and the optimal interdot distance depends on the QD size and the doping concentration. For a given QD size, the maximum $ZT$ is determined by the minimum of Lorenz number, which occurs when the quantum-confined carrier transport overwhelms the bulklike carrier transport.

Figure 1

Schematic diagram of the Bi${}_2$Te${}_3$/ Sb${}_2$Te${}_3$ QD NCs with spherical Bi${}_2$Te${}_3$ QDs with size $a$ uniformly embedded in Sb${}_2$Te${}_3$ matrix material with interdot distance $D$.

Figure 2

(a) Electrical conductivity and (b) Seebeck coefficient, of $p$-type Bi${}_2$Te${}_3$ and $p$-type Sb${}_2$Te${}_3$ bulk materials plotted as a function of temperature in comparison with the experimental data reported in Ref. 43. The doping concentration of Bi${}_2$Te${}_3$ is $p=1.1p_0$ and that of Sb${}_2$Te${}_3$ is $p=8.7p_0$, where $p_0={10}^{19}/{\mathrm{cm}}^3$.

Figure 3

(a) The DOS (left $Y$ axis) of electronic minibands in QD NCs with different sizes of QDs and different interdot distances, and the DOS of bulk Sb${}_2$Te${}_3$ material. The relationship of the doping concentration ($p$, right $Y$ axis) and the chemical potential ($\mu$) is also plotted for comparison. (b) Bandwidth of quantum-confined carrier miniband ($W_b$) plotted as a function of interdot distance for different sizes of QDs with $a=6$, 7, and $8\mathrm{nm}$. (c) Relaxation time of the lowest miniband of quantum-confined carriers (${\tau }_{1,\mathrm{QD}}$, left $Y$ axis) and bulklike holes (${\tau }_{h,\mathrm{M}}$, right $Y$ axis) in QD NCs with $(a,D)=(6\mathrm{nm},8\mathrm{nm})$, $(6\mathrm{nm},8.5\mathrm{nm})$, and $(7\mathrm{nm},9\mathrm{nm})$, in comparison with the carrier relaxation time in bulk Sb${}_2$Te${}_3$ bulk material.

Figure 4

Temperature dependence of (a) electrical conductivity, (b) Seebeck coefficient, (c) Lorenz number, (d) lattice thermal conductivity, and (e) $ZT$, for Bi${}_2$Te${}_3$/Sb${}_2$Te${}_3$ QD NCs with different QD size $a$ and interdot distance $D$ when $p=0.5p_0$. Calculated transport properties for $p$-type Sb${}_2$Te${}_3$ bulk material are also presented for comparison.

Figure 5

The dependence of (a) electrical conductivity, (b) Seebeck coefficient, (c) Lorenz number, and (d) $ZT$, on the doping concentration for Sb${}_2$Te${}_3$/Bi${}_2$Te${}_3$ QD NCs with $a=6\mathrm{nm}$ and different interdot distance $D$ at $T=300\mathrm{K}$. Calculated transport properties for $p$-type Sb${}_2$Te${}_3$ bulk material are also presented for comparison.

Figure 6

(a) Lorenz number ($L$, left $Y$ axis) and $L+{\kappa }_p/\sigma T$ (right $Y$ axis) plotted as a function of the electronic miniband width for QD NCs with different size $a$ of QDs in comparison with the bulk Sb${}_2$Te${}_3$ value when $T=300\mathrm{K}$ and $p=0.5p_0$. (b) Bandwidth dependence and (c) $D-a$ dependence of $ZT$ for QD NCs with different sizes of QDs and different doping concentrations at $T=300\mathrm{K}$. (d) The dependence of $ZT$ on the chemical potential for QD NCs with $a=6\mathrm{nm}$ and different interdot distances at $T=300\mathrm{K}$.