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<p>This study employs density functional theory (<i>DFT</i>) and time-dependent <i>DFT</i> (<i>TD-DFT</i>) to explore the nonlinear optical (<i>NLO</i>) responses of negatively charged and copper (<i>Cu</i>)-doped germanium (<i>Ge</i>) clusters (<i>G1–G6</i>). Structural analysis reveals <i>Ge–Ge</i> bond lengths ranging from 2.32–2.54&#xa0;Å and <i>Ge</i>–<i>Cu</i> bonds between 2.39–2.49&#xa0;Å, confirming stable interactions upon doping. The HOMO–LUMO gaps varied between 0.16–1.08&#xa0;eV, with <i>Cu</i> doping leading to charge localization on Cu atoms and significant modification of electronic properties. Energy decomposition analysis shows that stabilization is highly dependent on atomic arrangement, with ΔE<sup>tot</sup> values spanning –74 to –26,561&#xa0;kcal<i>/mol</i>. Importantly, the study demonstrates that <i>Cu</i> doping and cluster size growth markedly increase <i>NLO</i> responses as linear polarizability (<Emphasis Type="Underline">α</Emphasis>) rose from 0.84–126.06 <i>a.u</i>., while first-order hyperpolarizability (<i>β</i>) expanded dramatically, with <i>G4</i> and <i>G5</i> emerging as the most promising candidates. Transition density matrix and hole–electron overlap analyses revealed enhanced delocalization and recombination rates in larger, Cu-doped clusters, while global reactivity parameters (hardness 0.08–0.54&#xa0;eV, softness 0.93–6.17, electrophilicity 5.67–13.23) confirmed tunable electronic behavior. Exciton binding energies ranged from 3.38–8.27&#xa0;eV, underscoring strong Coulombic interactions. Overall, the study shows that increasing cluster size and introducing <i>Cu</i> doping significantly enhance stability, charge transfer, and <i>NLO</i> performance, positioning <i>Ge</i> clusters as promising materials for optoelectronic and photonic applications.</p> Graphical abstract <p></p>

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Enhancement of nonlinear optical properties in germanium clusters via copper doping and charge modulation: a density functional theory study

  • 期刊论文

  • 发表日期：

    2026年6月18日

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  • Javed Akram,
  • Kanwal Ranian,
  • Saeed A. Asiri,
  • Islam H. El Azab

摘要

This study employs density functional theory (DFT) and time-dependent DFT (TD-DFT) to explore the nonlinear optical (NLO) responses of negatively charged and copper (Cu)-doped germanium (Ge) clusters (G1–G6). Structural analysis reveals Ge–Ge bond lengths ranging from 2.32–2.54 Å and Ge–Cu bonds between 2.39–2.49 Å, confirming stable interactions upon doping. The HOMO–LUMO gaps varied between 0.16–1.08 eV, with Cu doping leading to charge localization on Cu atoms and significant modification of electronic properties. Energy decomposition analysis shows that stabilization is highly dependent on atomic arrangement, with ΔEtot values spanning –74 to –26,561 kcal/mol. Importantly, the study demonstrates that Cu doping and cluster size growth markedly increase NLO responses as linear polarizability (α) rose from 0.84–126.06 a.u., while first-order hyperpolarizability (β) expanded dramatically, with G4 and G5 emerging as the most promising candidates. Transition density matrix and hole–electron overlap analyses revealed enhanced delocalization and recombination rates in larger, Cu-doped clusters, while global reactivity parameters (hardness 0.08–0.54 eV, softness 0.93–6.17, electrophilicity 5.67–13.23) confirmed tunable electronic behavior. Exciton binding energies ranged from 3.38–8.27 eV, underscoring strong Coulombic interactions. Overall, the study shows that increasing cluster size and introducing Cu doping significantly enhance stability, charge transfer, and NLO performance, positioning Ge clusters as promising materials for optoelectronic and photonic applications.

Graphical abstract