Thermoelectric phonon-glass electron-crystal via ion beam patterning of silicon

Ion beam irradiation has recently emerged as a versatile approach to functional materials design. We show in this work that patterned defective regions generated by ion beam irradiation of silicon can create a phonon-glass electron-crystal (PGEC), a long-standing goal of thermoelectrics. By controlling the effective diameter of and spacing between the defective regions, molecular dynamics simulations suggest a reduction of the thermal conductivity by a factor of $\sim 20$ is achievable. Boltzmann theory shows that the thermoelectric power factor remains largely intact in the damaged material. To facilitate the Boltzmann theory, we derive an analytical model for electron scattering with cylindrical defective regions based on partial-wave analysis. Together we predict a figure of merit of $ZT\approx 0.5$ or more at room temperature for optimally patterned geometries of these silicon metamaterials. These findings indicate that nanostructuring of patterned defective regions in crystalline materials is a viable approach to realize a PGEC, and ion beam irradiation could be a promising fabrication strategy.

Figure 1

(a) Schematic of silicon metamaterial for thermoelectric applications, where the dark areas denote damaged domains patterned by ion beam irradiation. (b) The rationale for the patterned system, in which electrical properties are expected to increase towards the bulk values faster than thermal properties with increasing feature size as a result of the different phonon and electron mean free paths. If the feature size is larger than the electron mean free path ${\mathrm{\Lambda }}_e$ and smaller than the phonon mean free path ${\mathrm{\Lambda }}_p$, $ZT$ can be enhanced. (c) A representative supercell for (a) in atomic view, with disordered regions generated by ion beam irradiation, obtained with molecular dynamics simulations. (d) The radial distribution function for the specimens shown in (a) and (c). New peaks and fine shifts can be observed as the degree of disorder $\rho$ increases.

Figure 2

(a) Schematic of the scattering model for calculating the scattering cross section and scattering time with cylindrical defective area. (b) The convergence of the exact solution to the infinite scattering barrier with the number of partial waves $N_l$ considered. For $ka<0.5$ the $S$ wave ($N_l=0,l=0$) is sufficient for accurate descriptions of the scattering cross section.

Figure 3

(a) Thermal conductivity $\kappa$, (b) electrical conductivity $\sigma$, (c) Seebeck coefficient $S$, and (d) figure of merit $ZT$ as a function of $D$ and $L$ for $n$-type silicon doped at a concentration of $3\times {10}^{19}{\mathrm{cm}}^{-3}$ at room temperature. Both $\sigma$ and $\kappa$ are more sensitive to $L$ than $D$. Electron conductivity $\sigma$ grows quickly and saturates sooner than thermal conductivity $\kappa$ with $L$, which allows the patterned metamaterial to be more crystalline for electrons than for phonons. For $T=300\mathrm{K}$ at the given dopant concentration, $ZT$ can be enhanced to around $ZT\approx 0.5$ for optimal $L\approx 11$ nm, $D\approx 5$ nm. The thick blue lines correspond to $L=D$; the regions to the left of the lines are geometrically unphysical.

Figure 4

Scaling of $\zeta =\kappa$ or $\sigma$ as a function of (a) $L$ and (b) $D$ based on kinetic arguments for different intrinsic mean free paths ${\mathrm{\Lambda }}_0$. While ${\mathrm{\Lambda }}_0=10$ nm represents the scaling of electron conductivity, ${\mathrm{\Lambda }}_0=1000$ nm approximates the trend of phonon conductivity. For direct comparison the values of $L$ and $D$ are given with dimensions. The shaded areas signify the parameter ranges considered in this work. (c) and (d) Sensitivity to $L$ and $D$. In the range of $L$, $\kappa$ keeps increasing, and $\sigma$ saturates earlier. Meanwhile, for the considered range of $D$, the conductivities are similarly sensitive to $D$ and $L$. These results are consistent with the trends in Figs. 3 and 3.

Figure 5

The graphical iteration method for the determination of the Fermi level. The charge-neutrality point is indicated by the red circle. This approach gives carrier density and Fermi level simultaneously. In this case, electron density is $7.5\times 15{\mathrm{cm}}^{-3}$, and Fermi level is 34 meV below CBM.