First principles analysis of electronic structure evolution and the indirect- to direct-gap transition in Ge$_{1-x}$Pb$_{x}$ group-IV alloys

Christopher A. Broderick; Edmond J. O'Halloran; Eoin P. O'Reilly

paper

First principles analysis of electronic structure evolution and the indirect- to direct-gap transition in Ge $_{1 - x}$ Pb $_{x}$ group-IV alloys

arXiv:1911.05679

Abstract

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 $_{6 c}$ ) character. For an ordered Ge $_{15}$ Pb $_{1}$ ( $x = 6.25$ %) supercell we find that the CB edge has acquired primarily direct (Ge $Γ_{7 c}$ ) 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 $_{6 c}$ -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 $x ≳ 3 - 4$ %.

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First principles analysis of electronic structure evolution and the indirect- to direct-gap transition in Ge1−x​Pbx​ group-IV alloys

Abstract

Cited by in corpus (1)

First principles analysis of electronic structure evolution and the indirect- to direct-gap transition in Ge $_{1 - x}$ Pb $_{x}$ group-IV alloys