Long-range interatomic forces can minimize heat transfer: From slowdown of longitudinal optical phonons to thermal conductivity minimum

We investigate the role of interatomic forces beyond the nearest neighbors on the thermal transport through an atomic junction with a heavy isotope impurity and in a silicon-germanium-like alloy with atomistic calculations. The thermal conductance of the junction incorporating second-nearest-neighbors forces reaches its minimum when the longitudinal optical phonon resonances in the phonon transmission are minimized. We relate the weakening of the optical phonon resonance with the flattening of the longitudinal optical phonon band of the infinite diatomic lattice with second-nearest-neighbors forces, which is the limit of an extended junction. We emphasize that the bypass of the heavy-atom components in the diatomic lattice by long-range interatomic bonds is crucial for the realization of the minimum in bulk thermal conductivity. We highlight the connection between the minimal thermal conductivity of a SiGe-like alloy with the flattening of the longitudinal optical phonon band of the diatomic lattice due to the second-nearest-neighbors forces, in combination with enhanced anharmonic phonon processes and phonon localizations.

Figure 1

(a) Spectral transmission coefficients $\alpha (\omega /2\pi ,\rho )$ predicted by Green's function calculations of a quasi-1D model incorporating second-nearest-neighbor bonds $\rho =C_{11}/C_{12}=C_{22}/C_{12}=0$, 0.08, 0.19, and 0.3. (b) Thermal conductance $G\left(\rho \right)$ at $T=300$ K. Inset: quasi-1D tight-binding model which incorporates the second-nearest-neighbor bonds $C_{11}$ bypassing the nearest-neighbor bonds $C_{12}$ between the host atom with mass $m_1$ and the guest atom with mass $m_2$. We choose $m_1=m_{\mathrm{Si}}=28.086$ g ${\mathrm{mol}}^{-1}$ and $m_2=m_{\mathrm{Ge}}=72.64$ g ${\mathrm{mol}}^{-1}$. The heavy isotope atoms are bonded with a second-NN bond $C_{22}$. The brown and green atoms represent heavy isotopes and host atoms. The host atoms are coupled through the nearest-neighbor bonds $C_{12}$. The region inside the red dashed rectangle is the scattering region.

Figure 2

Evolution of (a) the phonon transmission spectrum corresponding to the optimal second-NN bond ratio ${\rho }_{\mathrm{opt}}$ and (b) the thermal conductance versus the relative second-NN bond ratio $\rho$ for junctions with increasing length. The dashed lines are a guide to the eye. “A” and “B” represent host atom and its heavy isotope, respectively. The “-ABABA-” case corresponds to the model in the inset of Fig. 1. The junctions contain two, three, and four segregated heavy isotope atoms of mass $m_2$, respectively. The mass ratio is the same as in Fig. 1, $m_2/m_1=m_{\mathrm{Ge}}/m_{\mathrm{Si}}=2.58$.

Figure 3

Upper panel: 1D diatomic model of harmonic oscillators incorporating second-NN bonds. The red dashed square denotes the unit cell. The notation of bonds and the color code of the atoms follow the same rule as the model in Fig. 1. Lower panel: Phonon band structure $\omega -\mathbf{k}$ in the $\langle 001\rangle$ direction of a ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ crystal for different relative strengths of the second phonon path: (a) $\rho =0$ and (b) $\rho =0.19$, corresponding to the minimum thermal conductance. The letters “T, L, A, O” represent transverse, longitudinal, acoustic, and optical modes, respectively. For example, “TA” denotes the transverse acoustic phonon branch. Black dotted-dashed curves are phonon dispersion of a diatomic lattice of alternating Si and Ge atoms, where only longitudinal modes are present.

Figure 4

Thermal conductivity $\kappa$ of bulk Si at 300 K (red open circles) and ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ alloy at 100 K (dark red squares) and at 300 K (blue solid circles) versus the relative strength of the second phonon path $\rho$. For the ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ alloy, the thermal conductivities were obtained by averaging five different random atomic distributions. Our first-principles calculations show that natural SiGe alloys have $\rho \approx 5.7\%$ [30].

Figure 5

Mode-dependent Grüneisen parameter ${\mathbf{\gamma }}_{\mathbf{q}}$ of the ordered ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ crystal for (c) $\rho =0$ and (d) $\rho =0.19$, open circles and lines show the two different methods. Experimentally measured Grüneisen parameters on the Brillouin zone edge and center are from [39] for comparison. The letters “T, L, A, O” represent transverse, longitudinal, acoustic, and optical modes, respectively. For example, “TA” denotes the transverse acoustic phonon branch.

Figure 6

Relative phonon participation number (RPPN) ${\mathcal{E}}_i\left({}_{\omega }^{\mathbf{q}}\right)$ of ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ crystals with relative second-NN interactions (a) $\rho =0$, (b) $\rho =0.16$, and ($\mathrm{c})\rho =0.28$. (d) Anharmonic phonon lifetimes $\tau \left(\omega \right)$ in ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ crystal for different relative second-NN interactions $\rho$. The band shift is consistent with the phonon dispersion in Fig. 3.