A first principle hybrid functional calculation of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>Tm</mml:mtext></mml:mrow><mml:mrow><mml:mi>G</mml:mi><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mtext>V</mml:mtext><mml:mrow><mml:mi>G</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>defect complexes in germanium

By means of density functional theory (DFT), using the screened Heyd, Scuseria, and Ernzerhof (HSE06) hybrid functional we present results of the TmGe3+-VGe defect complexes in germanium (Ge). The formation energies of the first (fnn), second (snn), third (tnn) and fourth (ftnn) nearest neighbour configurations of VGe-VGe were examined. The charge state transition levels for all these configurations were examined as well. The TmGe3+-VGe complexes were found to have a positive binding energies for the neutral charge state in the fnn and ftnn configurations. The thermodynamic transition levels revealed that the TmGe3+-VGe induced shallow levels in the band gap for the fnn, tnn and ftnn configurations and deep level for the tnn configuration. The snn configuration showed no charge state transition level, the − 2 charge state was stable for all Fermi energies in the band gap. The TmGe3+-VGe displayed evidence of a single donor level (+1/0) and an acceptor level (−1/−2) within the band gap. Charge state controlled metastability was exhibited by the TmGe3+-VGe.