Understanding Cu incorporation in the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ structure using resonant x-ray diffraction

The ability to control carrier concentration based on the extent of Cu solubility in the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ alloy compound (where $0\le x\le 1$) makes ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ an interesting case study in the field of thermoelectrics. While Cu clearly plays a role in this process, it is unknown exactly how Cu incorporates into the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ crystal structure and how this affects the carrier concentration. In this work, we use a combination of resonant energy x-ray diffraction (REXD) experiments and density functional theory (DFT) calculations to elucidate the nature of Cu incorporation into the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ structure. REXD across the ${\mathrm{Cu}}_k$ edge facilitates the characterization of Cu incorporation in the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ alloy and enables direct quantification of antisite defects. We find that Cu substitutes for Hg at a 2:1 ratio, wherein Cu annihilates a vacancy and swaps with a Hg atom. DFT calculations confirm this result and further indicate that the incorporation of Cu occurs preferentially on one of the $z=1/4$ or $z=3/4$ planes before filling the other plane. Furthermore, the amount of ${\mathrm{Cu}}_{\mathrm{Hg}}$ antisite defects quantified by REXD was found to be directly proportional to the experimentally measured hole concentration, indicating that the ${\mathrm{Cu}}_{\mathrm{Hg}}$ defects are the driving force for tuning carrier concentration in the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ alloy. The link uncovered here between crystal structure, or more specifically antisite defects, and carrier concentration can be extended to similar cation-disordered material systems and will aid the development of improved thermoelectric and other functional materials through defect engineering.

Figure 1

The ternary endpoint, $\left[\mathrm{Va}\right]{\mathrm{Hg}}_2{\mathrm{GeTe}}_4$, of the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ alloy crystallizes in the defect chalcopyrite structure and the quaternary endpoint, ${\mathrm{Cu}}_2{\mathrm{HgGeTe}}_4$, crystallizes in the stannite structure. The unit cell for these two compounds is shown here using the $I\overline{4}$ space group. The locations of specific atomic sites, as defined by the $I\overline{4}$ space group, are denoted numerically for each endpoint on the $z=0$, 1/4, 1/2, 3/4, and 1 planes and are colored based on the atom occupying that site for each composition—where vacancies are beige, Cu atoms are orange, Hg atoms are green, Ge atoms are blue, and Te atoms are gray. Each schematic plane represents a $2\times 2$ unit cell, which is used later in DFT simulations, and the atoms contained in the original unit cell are highlighted within the yellow boxes on each plane.

Figure 2

(a) A sample REXD pattern collected for the sample at the ${\mathrm{Cu}}_k$ edge (8979 eV). The (2,1,1) peak is pointed out here with an arrow. (b) REXD simulations of the (2,1,1) peak for the quaternary sample demonstrating the difference between a sample with no antisite defects and a sample with 5% ${\mathrm{Cu}}_{\mathrm{Hg}}$ antisite defects. The sample with 5% antisite defects demonstrates a sudden decrease in intensity across the ${\mathrm{Cu}}_k$ edge that is not present for the sample without antisite defects.

Figure 3

(a–c) The Cu and Hg occupancies are plotted here for sites 1, 2, and 3. All occupancies were obtained via Rietveld refinement of REXD data on the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ alloys at room temperature. (d) An example unit cell for the ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ alloy composition is shown here with sites 1, 2, and 3 indicated by the legend.

Figure 4

(a) DFT total energies for approximately 50 permutations (gray dots) of the intermediate ${\mathrm{CuHg}}_{1.5}{\mathrm{GeTe}}_4$ alloy. The energies are referenced to the lowest energy configuration, shown by the light-green dot. (b) The lowest energy permutation of each of the three possible scenarios from REXD results is demonstrated schematically. Scenario 3, where Cu occupies sites 1 and 2 exclusively on the $z=1/4$ plane, has the lowest energy and is therefore the most favorable. Te atoms that sit above and below the $z=1/4$ and 3/4 planes are shown in light gray and dark gray circles, respectively.

Figure 5

The experimentally measured hole concentration, taken from Ref. [21] (with a standard error of 10%), is plotted here as a function of the site 3 ${\mathrm{Cu}}_{\mathrm{Hg}}$ concentration from the current study. A line with a slope of 1 is also drawn to illustrate the strong correlation, which indicates that ${\mathrm{Cu}}_{\mathrm{Hg}}$ antisite defects are responsible for controlling the carrier concentration of ${\mathrm{Cu}}_{2x}{\mathrm{Hg}}_{2-x}{\mathrm{GeTe}}_4$ alloys.