Transition into a phonon glass in crystalline thermoelectric ${\left(\mathrm{S}{\mathrm{b}}_{1-x}\mathrm{B}{\mathrm{i}}_x\right)}_2\mathrm{T}{\mathrm{e}}_3$ films

Gaining fundamental insights into phonon interactions is of high importance for the improvement of thermoelectric materials. The particular challenge is to enable a phonon glass state with low thermal conductivity in crystalline materials with high electron conductivity. We present here the relation between atomic structure and $c$-axis lattice thermal conductivity ${\kappa }_{c,l}\left(T\right)$ of epitaxially grown nearly single crystalline ${\left(\mathrm{S}{\mathrm{b}}_{1-x}\mathrm{B}{\mathrm{i}}_x\right)}_2\mathrm{T}{\mathrm{e}}_3$ thin films. Aberration corrected high-resolution transmission electron microscopy shows a highly ordered crystalline lattice, with doping dependent statistical occupation of the Sb sublattice by Bi atoms and a very low density of planar defects. The observed strong decrease of ${\kappa }_{c,l}\left(T\right)$ with doping is due to an increase both of the Rayleigh scattering rate at point defects as well as of the phonon-phonon scattering rate. For $x=0.24$ a transition to a low, almost temperature independent ${\kappa }_{c,l}\left(T\right)$ is observed, indicating a transition to a phonon glass state. The theoretical calculations reveal that the phonon mean free path is reduced below the phonon wavelength for the majority of phonon frequencies, suggesting a breakdown of the phonon approximation to heat transport due to the strongly anharmonic lattice.

Figure 1

HRSTEM images of $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3/\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ interface and top surface of the $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ film. The interface reconstruction and twinned areas (2–3 u.c.) close to the $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3/\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ interface is sometimes observed. STEM imaging reveals excellent crystallinity of the films, furthermore supported by XRD and local selected area electron diffraction (SAED) patterns shown in the Supplemental Material [47].

Figure 2

Experimental HRSTEM HAADF images (zone axis [010]) of $\mathrm{S}{\mathrm{b}}_{0.88}\mathrm{B}{\mathrm{i}}_{0.12}\mathrm{T}{\mathrm{e}}_3$ (a) and $\mathrm{S}{\mathrm{b}}_{0.76}\mathrm{B}{\mathrm{i}}_{0.24}\mathrm{T}{\mathrm{e}}_3$ (c) as well as simulated images (b) and (d), respectively. A random substitution Bi atoms is used in the simulated image of $\mathrm{S}{\mathrm{b}}_{0.88}\mathrm{B}{\mathrm{i}}_{0.12}\mathrm{T}{\mathrm{e}}_3$ films, leading to random appearance of individual bright spots due to atomic columns with higher Bi occupation. The occupation of Sb sites by Bi atoms gives rise to an increased intensity of the $Pn$ sites in the quintuple $\text{−}\left[\mathrm{Te}\left(\mathrm{I}\right)\text{−}Pn\text{−}\mathrm{Te}\left(\mathrm{II}\right)\text{−}Pn\text{−}\mathrm{Te}\left(\mathrm{I}\right)\right]$ layers in (c) and (d) (Sb: $Z=51$, Te: $Z=52$, Bi: $Z=83)$. Some examples are marked with the white arrows.

Figure 3

Electron diffraction with dominant Bragg diffraction in the $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ film (a) and a mixture of Bragg and diffuse scattering in the $\mathrm{S}{\mathrm{b}}_{0.76}\mathrm{B}{\mathrm{i}}_{0.24}\mathrm{T}{\mathrm{e}}_3$ film (b). Line profiles are shown in (c). For $x=0.24$, additional weak spots between the 300 and 3015 reflections indicate an enlargement of the unit cell in the $c$ direction due to Bi/Sb point disorder on $Pn$ sites. In addition, the pronounced diffuse scattering streaks reflects thermal disorder in the $c$-lattice parameter. Italic indexing refers to spots of the film, whereas others are diffraction spots from the substrate. (a) and (b) are cutouts from SAED patterns shown in Figs. S6(a) and S6(c) in the Supplemental Material [47].

Figure 4

Thermal conductivity of epitaxial ${\left(\mathrm{S}{\mathrm{b}}_{1-x}\mathrm{B}{\mathrm{i}}_x\right)}_2\mathrm{T}{\mathrm{e}}_3$ films along the $c$ axis, obtained from 3ω measurements. (a) Total thermal conductivity ${\kappa }_c\left(T\right)$. (b) Lattice thermal conductivity ${\kappa }_{c,l}\left(T\right)$ after subtraction of the electronic contribution ${\kappa }_{c,e}\left(T\right)$.

Figure 5

(a) Comparison of experimental and theoretical temperature dependence of ${\kappa }_{c,l}\left(T\right)$, including the calculated minimum lattice thermal conductivity of $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ in the $c$ direction. (b) and (c) Show the obtained doping dependence of the scattering strength for umklapp scattering, $A^{-1}$ [Eq. (6)]), and Rayleigh scattering, $B^{-1}$ [Eq. (7)], respectively. Error bars stem from estimated error in the mode dependent density of states used in Eq. (1).

Figure 6

Lattice thermal conductivity as a function of the upper integration boundary ω at $T=100$, 200, and 300 K, for (a) $x=0$ and (b) $x=0.24$. The transition to saturation gives the upper frequency limit of the phonons contributing to thermal conductivity.

Figure 7

Experimental lattice thermal conductivity of the epitaxial film with Bi content $x=0.24$. Blue: Calculated curve within the Callaway model, fitted to the low temperature values of ${\kappa }_{c,l}\left(T\right)$. Grey $\left(\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3\right)$ and orange $\left(\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3\right)$: Minimum thermal conductivity model for disordered systems with $l_{mfp}=\frac{\lambda }{2}$.

Figure 8

Phonon mean free path $l_{mfp}$ versus the angular frequency for (a) 100 K and (b) 300 K, as calculated from Eqs. (5), (6), and (7) and using the parameters $A^{-1}$ and $B^{-1}$ in Fig. 5. Additionally, the half phonon wavelength is plotted. At frequencies where $l_{mfp}\le \frac{1}{2}\lambda$ the approximation of phonon quasiparticles as eigenstates of a harmonic periodic lattice break down.

Figure 9

Comparison of the lattice thermal conductivity in the $c$ direction at $T=300\mathrm{K}$ with literature values for solid solutions [10, 11] and for a $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3/\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3$ SLs of different bilayer and single layer thicknesses (10/50, 30/60, 20/30, and 30/30 Å) as published by Venkatasubramanian [11].