The ability to control carrier concentration based on the extent of Cu solubility in the $mathrm{Cu_{2x}Hg_{2-x}GeTe_4}$ alloy compound (where 0 $leq$ x $leq$ 1) makes $mathrm{Cu_{2x}Hg_{2-x}GeTe_4}$ 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 $mathrm{Cu_{2x}Hg_{2-x}GeTe_4}$ 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 $mathrm{Cu_{2x}Hg_{2-x}GeTe_4}$ structure. REXD across the $mathrm{Cu_k}$ edge facilitates the characterization of Cu incorporation in the $mathrm{Cu_{2x}Hg_{2-x}GeTe_4}$ alloy and enables direct quantification of anti-site 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 reveal 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 $mathrm{Cu_{Hg}}$ anti-site defects quantified by REXD was found to be directly proportional to the experimentally measured hole concentration, indicating that the $mathrm{Cu_{Hg}}$ defects are the driving force for tuning carrier concentration in the $mathrm{Cu_{2x}Hg_{2-x}GeTe_4}$ alloy. The link uncovered here between crystal structure, or more specifically anti-site 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.
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