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We present a theoretical analysis of electronic structure evolution in thehighly-mismatched dilute carbide group-IV alloy Ge$_{1-x}$C$_{x}$. For orderedalloy supercells, we demonstrate that C incorporation strongly perturbs theconduction band (CB) structure by driving hybridisation of $A_{1}$-symmetriclinear combinations of Ge states lying close in energy to the CB edge. Thisleads, in the ultra-dilute limit, to the alloy CB edge being formed primarilyof an $A_{1}$-symmetric linear combination of the L-point CB edge states of theGe host matrix semiconductor. Our calculations describe the emergence of a"quasi-direct" alloy band gap, which retains a significant admixture ofindirect Ge L-point CB edge character. We then analyse the evolution of theelectronic structure of realistic (large, disordered) Ge$_{1-x}$C$_{x}$ alloysupercells for C compositions up to $x = 2$%. We show that short-range alloydisorder introduces a distribution of localised states at energies below the GeCB edge, with these states acquiring minimal direct ($\Gamma$) character. Ourcalculations demonstrate strong intrinsic inhomogeneous energy broadening ofthe CB edge Bloch character, driven by hybridisation between Ge host matrix andC-related localised states. The trends identified by our calculations aremarkedly different to those expected based on a recently proposedinterpretation of the CB structure based on the band anti-crossing model. Theimplications of our findings for device applications are discussed.

Electronic structure evolution in dilute carbide Ge$_{1-x}$C$_{x}$ alloys and implications for device applications