  We present a theoretical analysis of electronic structure evolution in the
group-IV alloy Ge$_{1-x}$Pb$_{x}$ based on density functional theory. For
ordered alloy supercells we demonstrate the emergence of a singlet conduction
band (CB) edge state, suggesting the emergence of a direct band gap for Pb
compositions as low as $x \approx 1$%. However, application of hydrostatic
pressure reveals Pb-induced hybridisation, with the CB edge state in a
Ge$_{63}$Pb$_{1}$ ($x = 1.56$%) supercell retaining primarily indirect (Ge
L$_{6c}$) character. For an ordered Ge$_{15}$Pb$_{1}$ ($x = 6.25$%) supercell
we find that the CB edge has acquired primarily direct (Ge $\Gamma_{7c}$)
character, confirming the presence of an indirect- to direct-gap transition.
The importance of alloy disorder is highlighted by investigating the impact on
the electronic structure of the formation of a nearest-neighbour Pb-Pb pair.
Having established the importance of short-range disorder, we analyse the
electronic structure evolution as a function of $x$ using a series of 128-atom
special quasi-random structures (SQSs). Our calculations reveal a strong
reduction (increase) of the band gap (spin-orbit splitting energy), by $\approx
100$ meV ($\approx 40$ meV) per % Pb replacing Ge. We find an indirect- to
direct-gap transition occurring in a narrow composition range centred about $x
\approx 7$%, close to which composition we calculate that the alloy becomes
semimetallic. Further analysis suggests that long-range order introduced by
Born von Karman (supercell) boundary conditions leads to overestimated energy
splitting of the Ge L$_{6c}$-derived CB states in the 128-atom SQSs. Accounting
for these finite-size effects, we expect a direct band gap to emerge in
Ge$_{1-x}$Pb$_{x}$ for $x \gtrsim 3 - 4$%.
