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 $\ensuremath{\kappa}$, and electrical conductivity $\ensuremath{\sigma}$, 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 $30\ifmmode\times\else\texttimes\fi{}30\text{ }{\text{nm}}^{2}$. 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 $\mathbit{P}$-type nanowire cross section is smaller than $8\ifmmode\times\else\texttimes\fi{}10\text{ }{\text{nm}}^{2}$ ($6\ifmmode\times\else\texttimes\fi{}7$ and $5\ifmmode\times\else\texttimes\fi{}5\text{ }{\text{nm}}^{2}$) at the excess hole concentration ${p}_{\text{ex}}=2\ifmmode\times\else\texttimes\fi{}{10}^{18}\text{ }{\text{cm}}^{\ensuremath{-}3}$ (${p}_{\text{ex}}=5\ifmmode\times\else\texttimes\fi{}{10}^{18}\text{ }{\text{cm}}^{\ensuremath{-}3}$ and ${p}_{\text{ex}}=1\ifmmode\times\else\texttimes\fi{}{10}^{19}\text{ }{\text{cm}}^{\ensuremath{-}3}$ 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 $\mathbit{P}$-type nanowire is equal to 1.4, 1.6, and 2.8, correspondingly, at temperatures 310, 390, and 480 K and the cross sections $30\ifmmode\times\else\texttimes\fi{}30$, $15\ifmmode\times\else\texttimes\fi{}15$, and $7\ifmmode\times\else\texttimes\fi{}7\text{ }{\text{nm}}^{2}$ $({p}_{\text{ex}}=5\ifmmode\times\else\texttimes\fi{}{10}^{18}\text{ }{\text{cm}}^{\ensuremath{-}3})$. At room temperature, the figure of merit equals 1.2, 1.3, and 1.7, respectively.