Thermoelectric properties of bismuth telluride nanowires in the constant relaxation-time approximation

Electronic structure of bismuth telluride nanowires with the growth directions [110] and [015] is studied in the framework of the anisotropic effective-mass method using the parabolic band approximation. The components of the electron and hole effective-mass tensors for six valleys are calculated for both growth directions. For a square nanowire, in the temperature range from 77 to 500 K, the dependence of the Seebeck coefficient $S$, the thermal $\kappa$, and electrical conductivity $\sigma$, as well as the figure of merit $ZT$ on the nanowire thickness and on the excess hole concentration $p_{ex}$, are investigated in the constant relaxation-time approximation. The carrier confinement is shown to play essential role for nanowires with cross section less than $30\times 30{\text{nm}}^2$. In contrast to the excess holes (impurities), the confinement decreases both the carrier concentration and the thermal conductivity but increases the maximum value of the Seebeck coefficient. The confinement effect is stronger for the direction [015] than for the direction [110] due to the carrier mass difference for these directions. In the restricted temperature range, the size quantum limit is valid when the $\mathit{P}$-type nanowire cross section is smaller than $8\times 10{\text{nm}}^2$ ($6\times 7$ and $5\times 5{\text{nm}}^2$) at the excess hole concentration $p_{\text{ex}}=2\times {10}^{18}{\text{cm}}^{-3}$ ($p_{\text{ex}}=5\times {10}^{18}{\text{cm}}^{-3}$ and $p_{\text{ex}}=1\times {10}^{19}{\text{cm}}^{-3}$ correspondingly). The carrier confinement increases the maximum value of $ZT$ and shifts it toward high temperatures. For the growth direction [110], the maximum value of the figure of merit for the $\mathit{P}$-type nanowire is equal to 1.4, 1.6, and 2.8, correspondingly, at temperatures 310, 390, and 480 K and the cross sections $30\times 30$, $15\times 15$, and $7\times 7{\text{nm}}^2$ $\left(p_{\text{ex}}=5\times {10}^{18}{\text{cm}}^{-3}\right)$. At room temperature, the figure of merit equals 1.2, 1.3, and 1.7, respectively.

Figure 1

(Color online) Electron and hole energy spectrum for valleys 1 and 4 in the intrinsic nanowire $\left(a_x=a_z=15\text{nm}\right)$ with growth direction [110]. The solid lines correspond to quantum numbers $n=1$ and 2, and $l=1$, dotted lines to $l=2$ and dashed lines to $l=3$.

Figure 2

(Color online) Temperature dependence of the reduced energy separation between the hole subbands (1, 1) and (2, 1) in the square nanowire, with thickness $a=30\text{nm}$ (dashed-dotted line), 15 nm (dashed line), and 7 nm (solid line) for the first and fourth valley.

Figure 3

(Color online) The temperature dependence of (a) the Fermi energy and (b) the carrier concentration in the square intrinsic ${\text{Bi}}_{0.37}{\text{Te}}_{0.63}$ nanowire with thickness 7 nm (solid line), 15 nm (dashed line) and the growth direction [110]. Dashed-dotted and dotted lines correspond to thickness 15 and 7 nm for the growth direction [015]. Filled squares show the Fermi energy and the carrier concentration in the bulk.

Figure 4

(Color online) Temperature dependence of (a) the highest hole subband (filled squares) and the Fermi energy for the square $\mathit{P}$-type ${\text{Bi}}_2{\text{Te}}_3$ nanowire with thickness 15 (dashed line) and 7 nm (solid line) at $p_{\text{ex}}=5\times {10}^{18}$ and $1\times {10}^{19}{\text{cm}}^{-3}$ (dotted line) for the growth direction [110], (b) the hole concentration in the bulk (filled squares) and in the nanowire with thickness 7 (solid line) and 15 nm (dashed line) for the growth direction [110] at $p_{\text{ex}}=5\times {10}^{18}{\text{cm}}^{-3}$. Dashed-dotted (dotted) line shows the Fermi energy (hole concentration) in the 7 nm (15 nm) thick nanowire with the growth direction [015] at $p_{\text{ex}}=5\times {10}^{18}{\text{cm}}^{-3}$.

Figure 5

(Color online) Temperature dependence of the Seebeck coefficient of (a) the intrinsic and (b) the $\mathit{P}$-type bismuth telluride compound (filled squares) and the square nanowire with thickness 7 (solid line) and 15 nm for the growth direction [110] (dashed line) and [015] (dashed-dotted line) at $p_{\text{ex}}=5\times {10}^{18}{\text{cm}}^{-3}$. The dotted line corresponds to nanowire thickness 7 nm and the growth direction [110] at $p_{\text{ex}}=1\times {10}^{19}{\text{cm}}^{-3}$.

Figure 6

(Color online) Temperature dependence of the thermal conductivity in (a) the bulk material (filled squares) (Ref. 13), the intrinsic square nanowire and (b) the $\mathit{P}$-type square nanowire with thickness 7 (solid line) and 15 nm for the growth direction [110] (dashed line) and [015] (dashed-dotted line). The excess hole concentration is $p_{\text{ex}}=5\times {10}^{18}{\text{cm}}^{-3}$. For the growth direction [110], a dotted line corresponds to the nanowire with thickness 7 nm at $p_{\text{ex}}=1\times {10}^{19}{\text{cm}}^{-3}$.

Figure 7

(Color online) Temperature dependence of the figure of merit of (a) the bismuth telluride compound (filled squares), the intrinsic square nanowire, and (b) the $\mathit{P}$-type square nanowire with thickness 7 (solid line) and 15 nm with the [110] (dashed line) and [015] (dashed-dotted line) growth direction at $p_{\text{ex}}=5\times {10}^{18}{\text{cm}}^{-3}$. Dotted line corresponds to nanowire with thickness 7 nm grown along direction [110] at $p_{\text{ex}}=1\times {10}^{19}{\text{cm}}^{-3}$.