Thermoelectric properties of ultrathin silicon nanowires

We calculate the room-temperature thermoelectric properties of highly doped ultrathin silicon nanowires (SiNW) of square cross section (3 $\times$ 3 to 8 $\times$ 8 nm${}^2$) by solving the Boltzmann transport equations for electrons and phonons on an equal footing, using the ensemble Monte Carlo technique for each. We account for the two-dimensional confinement of both electrons and phonons and all the relevant scattering mechanisms, and present data for the dependence of electrical conductivity, the electronic and phononic thermal conductivities, the electronic and phonon-drag Seebeck coefficients, as well as the thermoelectric figure of merit ($ZT$) on the SiNW rms roughness and thickness. $ZT$ in ultrascaled SiNWs does not increase as drastically with decreasing wire cross section as suggested by earlier studies. The reason is surface roughness, which (beneficially) degrades thermal conductivity, but also (adversely) degrades electrical conductivity and offsets the Seebeck coefficient enhancement that comes from confinement. Overall, room-temperature $ZT$ of ultrathin SiNWs varies slowly with thickness, having a soft maximum of about 0.4 at the nanowire thickness of 4 nm.

Figure 1

Electrical conductivity in square SiNWs as a function of the wire cross section for different interface roughness rms heights $\Delta$ ($\Lambda =2.5$ nm). The rapid drop in conductivity for the wires of cross section below 6 $\times$ 6 nm${}^2$ occurs because of a significant increase in surface-roughness scattering.

Figure 2

An exponentially correlated surface generated using FFT for $\Delta$ $=$ 0.25 nm and $\Lambda$ $=$ 5 nm. (Inset) A sample trajectory of a phonon hitting a groove in the surface and spending some time bouncing around before getting out.

Figure 3

Transient temperature profile along the wire for diffusive and ballistic phonon transport. (a) In the diffusive transport regime, a linear temperature profile is established along the wire when the system reaches a steady state. (b) In the ballistic regime, the steady-state temperature is constant along the wire, with discontinuities at the two end cells.

Figure 4

Lattice thermal conductivity as a function of the SiNW cross section for different rms roughness $\Delta$ at the Si/SiO${}_2$ interface. Thermal conductivity in SiNWs is more than an order of magnitude smaller than that in bulk silicon because of strong phonon-boundary scattering.

Figure 5

Electronic contribution to thermal conductivity ${\kappa }_e$ as a function of the SiNW cross section for different rms roughness $\Delta$ at the Si/SiO${}_2$ interface. ${\kappa }_e$ follows the same trend as the $\sigma$ variation with cross section and roughness. As in bulk silicon, ${\kappa }_e$ is much smaller than ${\kappa }_l$.

Figure 6

(a) Electronic and (b) phononic (i.e., phonon-drag) components of the Seebeck coefficient as a function of the SiNW cross section and rms roughness $\Delta$. The electronic component of the Seebeck coefficient is more than an order of magnitude greater than the phonon-drag component. The phonon-drag component of $S$ decreases with increasing roughness at the interface because of the decrease in the mean-free path of phonons.

Figure 7

Density of states (DOS) of a $5\times 5$ nm${}^2$ SiNW (black solid curve) and the effective DOS in the presence of appreciable surface-roughness scattering, for $\Delta =0.14$ nm (red dashed-dotted curve) and $\Delta =0.4$ nm (blue dashed curve). Roughness smears the high peaks in the DOS, thereby negating potential benefits that quantum confinement could have on the Seebeck coefficient (Ref. 71).

Figure 8

Variation of $ZT$ with the SiNW cross section. $ZT$ incorporating thermal conductivity from phonon Monte Carlo with real-space roughness is presented for rms roughness $\Delta =0.2$ nm (red solid curve) and $\Delta =0.5$ nm (dotted black curve). $ZT$ obtained from the RTA assuming completely diffuse boundary scattering for phonons (specularity parameter equal to zero; solid blue curve) overestimates $ZT$.

Figure 9

Variation of $ZT$ with the doping density for different SiNW thicknesses. $\Delta =0.5$ nm.