Heat and charge transport in bulk semiconductors with interstitial defects

Interstitial defects are inevitably present in doped semiconductors that enable modern-day electronic, optoelectronic, or thermoelectric technologies. Understanding the stability of interstitials and their bonding mechanisms in the silicon lattice was accomplished only recently with the advent of first-principles modeling techniques, supported by powerful experimental methods. However, much less attention has been paid to the effect of different naturally occurring interstitials on the thermal and electrical properties of silicon. In this work, we present a systematic study of the variability of heat and charge transport properties of bulk silicon, in the presence of randomly distributed interstitial defects (Si, Ge, C, and Li). We find through atomistic lattice dynamics and molecular dynamics studies that interstitial defects scatter heat-carrying phonons to suppress thermal transport—1.56% of randomly distributed Ge and Li interstitials reduce the thermal conductivity of silicon by $\sim 30$ and 34 times, respectively. Using first-principles density functional theory and semiclassical Boltzmann transport theory, we compute electronic transport coefficients of bulk Si with 1.56% neutral Ge, C, Si, and Li interstitials, in energetically favorable hexagonal, tetrahedral, split-interstitial, and bond-centered sites. We demonstrate that hexagonal-Si and hexagonal-Ge interstitials minimally impact charge transport. As an illustration of the relevance of this work for practical applications, we predict the thermoelectric property of an experimentally realizable bulk Si sample that contains Ge interstitials in different symmetry sites. Our research establishes a direct relationship between the variability of structures dictated by fabrication processes and heat and charge transport properties of silicon. The relationship provides guidance to accurately estimate performance of Si-based materials for various technological applications.

Figure 1

Representative configurations of bulk Si with Ge interstitials studied using (a) classical MD and (b) first-principles DFT method. The MD configuration consists of 4096 host Si (yellow) and 64 interstitial Ge (green) atoms. The DFT configuration has 64 Si (yellow) and 1 Ge (green) atoms. The concentration of Ge in bulk Si is 1.56% in both configurations.

Figure 2

(c) Conventional unit cell of bulk silicon and commonly studied interstitial sites [(a), (b), (d), and (e)] in the diamond cubic lattice. Yellow and red colors represent host and interstitial atoms, respectively.

Figure 3

Thermal conductivities of bulk silicon at 300 K as a function of the number of atoms in the simulation cell, computed from equilibrium molecular dynamics using the Green-Kubo theorem. The filled squares represent the TCs computed using the 2NN MEAM potential [79]. To facilitate a coherent comparison, corresponding TCs obtained with Tersoff potential [82] are shown with open circles [data extracted from Ref. [82]; converged TC: $196.8\pm 33.3$ W/(m K)] alongside 2NN MEAM values. The dashed line connecting the points represents a guide to the eye. The blue shaded region, extracted from Ref. [82], indicates the range of TCs measured in the experiments where most values lie in the darker shaded region. The converged TC of bulk Si using 2NN MEAM is $122.22\pm 13.81$ W/(m K).

Figure 4

Thermal conductivities of bulk silicon with C (blue squares), Si (orange inverted triangles), Ge (green triangles), and Li (magenta diamonds) interstitials at 300 K as a function of interstitial concentrations. To offer a comparison with the effect of widely employed Ge substitutions on TC of bulk Si, corresponding TC values are displayed with black open circles (data extracted from Ref. [101]). The $x$-axis label corresponds to substitutional concentration of Ge in bulk Si in this case.

Figure 5

Nearest-neighbor environment analysis of Si-$I$ (a) -C, (b) -Si, (c) -Ge, and (d) -Li systems. The neighbor distances are sorted in ascending order over all interstitial atoms in the supercell and displayed against the neighbor count.

Figure 6

Vibrational density of states of bulk silicon with 1.56% C (blue), Si (orange), and Ge (green) interstitials. The population of the phonon modes in Si-$I\text{−}X$ systems is slightly affected in the $\sim 15\mathrm{THz}$ region.

Figure 7

Phonon group velocities of bulk silicon with 1.56% C (blue), Si (orange), and Ge (green) interstitials compared to bulk (black). Phonon group velocities are strongly reduced in the Si-$I\text{−}X$ systems above 3–5 THz and the $v_g$ profile is approximately of the order $\mathrm{Ge}<\mathrm{Si}<\mathrm{C}$. Inset: Phonon group velocities of Si-$I$-Ge systems with uniform (magenta) and random (green) distribution of interstitials. Group velocities are more suppressed in Si-$I$-Ge system with randomly distributed interstitials.

Figure 8

Length-dependent phonon transmission functions of (a) bulk Si and (b) Si-$I$-Ge with 1.56% Ge interstitials at 300 K, as a function of frequency. The lengths of the samples are indicated by the legends. (c) Phonon mean-free paths in bulk Si (red) and bulk silicon with 1.56% Ge interstitials (blue), evaluated from the length-dependent transmission functions in (a). (d) Representative spectral thermal conductivities of bulk Si (solid lines) and 1.56% Si-$I$-Ge (dashed lines). Reduced phonon transmission and MFPs result in strongly reduced TC of Si-$I$-Ge for all $\omega >1.5$ THz. The integrated values of TCs are shown within parentheses, next to the legends.

Figure 9

Calculated Seebeck coefficient, electrical conductivity, electronic thermal conductivity, and power factor of bulk Si and bulk Si-$I\text{−}X$ systems.

Figure 10

Left panel: Band structures of bulk Si and semiconductor-like Si-$I\text{−}X$ systems (Underlined italic values in Table 2). Middle panel: The local density of states (LDOS) of Si-$I$-C with C interstitial in the bond-centered and the hexagonal sites. The Fermi level is chosen to be at 0 eV for all cases. Right panel: Figure of merit $\left(ZT\right)$ of semiconductor-like Si-$I\text{−}X$ systems compared to that of bulk Si.

Figure 11

Band structures of bulk Si and metal-like Si-$I\text{−}X$ systems (nonitalic values in Table 2). The plots in the rightmost panels represent the local density of states (arbitrary units) of (c) Si-$I$-Ge-tetrahedral and (f) Si-$I$-Li-tetrahedral systems. The Fermi level is chosen to be at 0 eV for all cases.

Figure 12

Band structures of Si-$I$-Ge systems with interstitials located in (a) hexagonal, (b) tetrahedral, and (c) both hexagonal and tetrahedral sites. The Fermi level is chosen to be at 0 eV for all cases.

Figure 13

Variability of thermoelectric figure of merit $ZT$ of bulk Si with interstitials, as a function of the guest atom type and its symmetry position in the lattice.