Nanoscale mechanisms for the reduction of heat transport in bismuth

Hand-on routes to reduce lattice thermal conductivity (LTC) in bismuth have been explored by employing a combination of Boltzmann's transport equation and ab initio calculations of phonon-phonon interaction within the density functional perturbation theory. We have first obtained the temperature dependence of the bulk LTC in excellent agreement with available experiments. A very accurate microscopic description of heat transport has been achieved and the electronic contribution to thermal conductivity has been determined. By controlling the interplay between phonon-phonon interaction and phonon scattering by sample boundaries, we predict the effect of size reduction for various temperatures and nanostructure shapes. The largest heat transport reduction is obtained in polycrystals with grain sizes smaller than 100 nm.

Figure 1

Bulk bismuth. Lattice [panels (a) and (b)] and electronic [panels (c) and (d)] thermal conductivity (TC) as functions of temperature. Left panels: binary direction. Right panels: trigonal direction. Solid lines: calculation with experimentally observed AOPI. Dashed lines: ab initio calculations. Green symbols: TC from expt. of Refs. [1, 26, 34]. Red dash-dotted line: previous ab initio calculation from Ref. [28]. Our electronic contribution was obtained as a difference between the experimentally observed total TC of Ref. [26] [panel (c)] or Ref. [35] [panel (d)] and our calculated LTC.

Figure 2

Bulk bismuth. Left panel: phonon frequencies along some directions of high symmetry in the BZ. Solid line: the theoretical dispersion used in this work. Dashed line: optical phonon branches obtained with the ab initio calculation. Right panel: joint density of states (JDOS) available for the phonon-phonon interaction. The JDOS for the acoustic-optical phonon interaction (AOPI) is marked with an arrow. Green and blue circles: expt. from Refs. [39, 40].

Figure 3

Bi thin films. Lattice thermal conductivity in the binary direction (solid lines) as a function of the thin-film thickness (nm) (bottom abscissa axis) or of the grain size (top abscissa axis; data from Ref. [18]) at 100 K, 200 K, and 300 K (top, center, and bottom panels). Red filled diamonds: expt. total thermal conductivity ${\kappa }_{\mathrm{TOT}}$ from Ref. [18]. Empty red squares: expt. ${\kappa }_{\mathrm{TOT}}$ from Ref. [19].

Figure 4

Lattice thermal conductivity (LTC) for different geometries of bismuth nanostructures. Solid lines: LTC in the binary direction for monocrystalline thin films as a function of the film thickness. Dotted lines: LTC along the binary axis of monocrystalline nanowires. Dashed lines: spherical grain.

Figure 5

Nanostructuring-reduced lattice thermal conductivity (LTC). Abacus of the reduction factor as a function of the nanostructure size and temperature, at 300 K (black), 200 K (red), 100 K (green), 50 K (blue), and 20 K (violet). Solid lines: single-crystalline thin films, LTC along the binary direction; dashed lines: spherical grain geometry.

Figure 6

Nanostructuring-reduced total thermal conductivity in the binary direction of single-crystalline thin films. Solid lines: semiconducting thin films (same as Fig. 5). Dash-dotted lines: conducting thin films. Same color legend as in Fig. 5.

Figure 7

Bulk (thin lines) and nanostructured (thick lines) Bi: relative contributions of different phonon modes to the lattice thermal conductivity in the binary direction. Nanostructured Bi with 100 nm grain geometry. Solid lines: TAs phonons. Dashed lines: TAf phonons. Dash-dotted lines: LA phonons. Dotted lines: optical phonons. We define TAf phonons, where “f” stands for fast, as those having a larger sound velocity than the one of TAs phonons, where “s” stands for slow.

Figure 8

Bulk (thick line) and nanostructured bismuth (thin lines): Brillouin zone distribution of the LTC as a function of $\left|\mathbf{q}\right|$ of the initial phonon, in the binary direction and at 300 K. Nanostructured bismuth in the grain geometry of diameter 100 nm (dashed line), 50 nm (dash-dotted line), and 10 nm (dotted line). Inset panel: the density of $\mathbf{q}$-phonon states.

Figure 9

Illustration of different sample geometries: spherical grain, wire, and thin film.

Figure 10

Bulk Bi. Brillouin zone distribution of the acoustic phonon contributions (thin lines) to the LTC (thick lines) as a function of the modulus of the phonon vector of incoming phonon $\left|\mathbf{q}\right|$ in the binary (left) and trigonal (right) directions at 300 K (top) and 5 K (bottom). Thin solid lines: TAs branch. Thin dashed lines: TAf branch. Thin dash-dotted lines: LA branch. See caption of Fig. 7 for the definition of TAs and TAf.