Duan and Stampfl's first-principles study of defect complexes and cluster doping in wurtzite InN (Phys. Rev. B 79, 035207 (2009)) predicted, at the LDA level, that oxygen–indium-vacancy complexes have very high equilibrium formation energies (a conclusion the authors restricted to thermal equilibrium), that Mg–O clusters become increasingly bound with size and could both compensate Mg acceptor doping when oxygen is present as a contaminant and, for off-stoichiometric compositions, provide more efficient p- (or n-) type doping than single dopants, and that Mg_In–V_N complexes are only moderately bound, while explicitly acknowledging that the quantitative transition levels were conditioned on the exchange–correlation approximation — suggesting that methods yielding a more accurate description of the band gap, such as GW, exact exchange, or SIC, would be valuable — and that ~0.2 eV supercell and methodological errors left marginal cases ambiguous. Hybrid-functional calculations have since resolved the beyond-LDA picture for native point defects and for carbon impurities, in both cases substantially revising the LDA-era conclusions; positron-annihilation experiments have shown that indium-vacancy complexes exist in highly n-type InN at concentrations far above equilibrium predictions, demonstrating non-equilibrium formation but leaving the microscopic decoration identity unresolved. What remains open, with evidence, is the coherent core: quantitative hybrid/GW-level formation energies, binding energies, and charge-transition levels for the specific impurity–vacancy and dopant-cluster complexes (ONVIn\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}}, 3ONVIn3\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}}, MgInVN\mathrm{Mg}_{\mathrm{In}}V_{\mathrm{N}}, MgmOn\mathrm{Mg}_m\mathrm{O}_n, SiiCj\mathrm{Si}_i\mathrm{C}_j); whether and how these complexes actually form under real non-equilibrium InN growth; and, as supporting rather than constitutive evidence, their direct experimental identification through correlated optical and electrical signatures.

Background

Indium nitride (InN) is the narrowest-gap member of the III–nitride semiconductor family, with a band gap of about 0.65 eV, and is of interest for infrared optoelectronics, high-speed electronics, and terahertz devices. Two doping problems dominate its materials physics: as-grown InN is almost always unintentionally n-type with electron concentrations that are difficult to control, and p-type doping (by magnesium, the standard acceptor in the related semiconductor GaN) has been hard to reproduce and stabilize. The microscopic origin of both problems is believed to lie in point defects and defect complexes. Within density-functional theory (DFT), a defect's relevance is judged by its formation energy EfE^f, which depends on the atomic chemical potentials (the reservoirs from which atoms are exchanged, e.g. bulk Mg or the secondary phase Mg3N2\mathrm{Mg}_3\mathrm{N}_2 for a Mg dopant) and on the Fermi level, the electron chemical potential that positions the charge state of the defect; a defect is likely to form when its formation energy is low under the actual growth and doping conditions. Charge-transition levels ε(q/q′)\varepsilon(q/q') are the Fermi-level positions at which the stable charge state of a defect switches, and they determine whether a defect is a shallow dopant (ionizes at room temperature) or a deep compensating center (traps carriers and neutralizes intended doping). Defect complexes are bound aggregates of individual point defects (e.g. an oxygen atom on a nitrogen site next to an indium vacancy, ONVIn\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}}); 'cluster doping' or 'codoping' strategies exploit the fact that a complex can have a lower formation energy than its separated constituents, so binding can either enhance desirable doping or create compensating centers. A key methodological caveat is the band-gap problem of local and semi-local exchange–correlation approximations such as the local-density approximation (LDA): they severely underestimate band gaps and misplace defect levels. Duan and Stampfl treated the In 4d4d electrons with a nonlinear core correction (a scheme that keeps the 4d4d states chemically in the core while approximately including their density in the exchange–correlation potential, which opens the calculated InN gap toward the experimental value), but acknowledged that quantitative defect-level placement required beyond-LDA methods such as GWGW quasiparticle corrections (a many-body perturbative treatment of electron self-energy) or hybrid functionals (functionals mixing a fraction of exact exchange with LDA-type exchange, e.g. HSE, which largely cure the gap underestimate for these materials). Their 2009 study, using LDA calculations in supercells (large repeated computational cells that model a defect in an otherwise perfect crystal), found that ON\mathrm{O}_{\mathrm{N}} and SiIn\mathrm{Si}_{\mathrm{In}} are shallow donors with low formation energies; MgIn\mathrm{Mg}_{\mathrm{In}} and CN\mathrm{C}_{\mathrm{N}} are acceptors; ONVIn\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}} and 3ONVIn3\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}} have very high equilibrium formation energies (about 7.3 and 4.0 eV) despite large binding energies (up to ~4 eV); MgInVN\mathrm{Mg}_{\mathrm{In}}V_{\mathrm{N}} is only moderately bound; and MgmOn\mathrm{Mg}_m\mathrm{O}_n clusters become increasingly bound with size, so that equal-ratio clusters could play a role in the compensation of Mg acceptor doping when oxygen is present as a contaminant, while off-stoichiometric MgmOn\mathrm{Mg}_m\mathrm{O}_n complexes could provide more efficient p- (or n-) type doping than single Mg (or single Si and O) dopants through tuning of the relative concentrations of Mg and O. The authors themselves acknowledged the limitations that frame this problem: the exchange–correlation and In-4d4d-treatment sensitivity of the transition levels (noting that 'for a more quantitative picture, methods which yield a more accurate description of the band gap, such as GW, exact exchange, or SIC, would be valuable'), ~0.2 eV supercell and methodological error bars, the restriction of their high-formation-energy conclusions to thermal equilibrium — which leaves the relevance of these complexes under non-equilibrium (kinetically trapped) growth open — and the sensitivity of all conclusions to the assumed impurity chemical potentials and possible competing second phases.

Problem Statement

Determine which defect complexes control doping and compensation in real wurtzite InN, resolving for the impurity–vacancy and dopant-cluster complexes studied by Duan and Stampfl — ONVIn\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}}, 3ONVIn3\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}}, MgInVN\mathrm{Mg}_{\mathrm{In}}V_{\mathrm{N}}, the MgmOn\mathrm{Mg}_m\mathrm{O}_n clusters, and the SiiCj\mathrm{Si}_i\mathrm{C}_j clusters — the linked questions the 2009 study left open: explicitly, the beyond-LDA accuracy of the complexes' energetics (the authors suggested that methods yielding a more accurate description of the band gap, such as GWGW, exact exchange, or SIC, 'would be valuable'), and, implicitly through the authors' restriction of their high-formation-energy conclusions to thermal equilibrium, whether these complexes matter under real growth. Namely: (1) their quantitative formation energies, binding energies, and charge-transition levels at a beyond-LDA level of theory — methods yielding a more accurate description of the band gap than LDA-nlcc, such as GWGW, exact exchange (hybrid functionals), or SIC, the class suggested by the source authors — with controlled supercell size, image-charge (finite-size electrostatic) corrections, and chemical-potential/second-phase conventions, replacing the LDA-nlcc values whose accuracy the authors themselves questioned; (2) whether these complexes, among which the source found the oxygen–indium-vacancy complexes (ONVIn\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}}, 3ONVIn3\mathrm{O}_{\mathrm{N}}V_{\mathrm{In}}) to have equilibrium formation energies too high for them to be important defects in equilibrium, actually form under real non-equilibrium InN growth conditions — i.e. their expected concentrations as a function of growth temperature, fluxes, and doping level, for which formation-pathway, migration and binding barrier, and trapping-during-growth analysis is a natural but not mandatory route, given that positron-annihilation experiments (which identify open-volume vacancy-type defects from the lifetimes of positrons implanted into the sample) already show indium-vacancy complexes at concentrations far above equilibrium predictions; and, as supporting rather than constitutive evidence, (3) correlation of the complexes' predicted optical and electrical signatures (transition energies, compensation behavior) with measurements on doped InN samples that would directly identify at least the complexes that dominate. An answer that establishes which of these complexes exist in as-grown and doped InN, at what concentrations, and with which carrier-producing or compensating levels, under growth conditions that are stated and anchored to physically realized InN growth environments (e.g. MBE/MOVPE temperature and flux ranges and chemical-potential limits set by the relevant second phases, consistent with observed defect populations), completes the problem.

References

  1. X. M. Duan and C. Stampfl, 'Defect complexes and cluster doping of InN: First-principles investigations', Physical Review B 79, 035207 (2009). DOI: 10.1103/PhysRevB.79.035207, https://doi.org/10.1103/PhysRevB.79.035207
  2. X. M. Duan and C. Stampfl, 'Nitrogen vacancies in InN: Vacancy clustering and metallic bonding from first principles', Physical Review B 77, 115207 (2008). DOI: 10.1103/PhysRevB.77.115207, https://doi.org/10.1103/PhysRevB.77.115207
  3. A. Janotti and C. G. Van de Walle, 'Sources of unintentional conductivity in InN', Applied Physics Letters 92, 032104 (2008). DOI: 10.1063/1.2832369, https://doi.org/10.1063/1.2832369
  4. A. Janotti, J. L. Lyons and C. G. Van de Walle, 'Hybrid functional calculations of native point defects in InN', physica status solidi (a) 209, 65 (2011). DOI: 10.1002/pssa.201100216, https://doi.org/10.1002/pssa.201100216
  5. J. L. Lyons, A. Janotti and C. G. Van de Walle, 'Effects of carbon on the electrical and optical properties of InN, GaN, and AlN', Physical Review B 89, 035204 (2014). DOI: 10.1103/PhysRevB.89.035204, https://doi.org/10.1103/PhysRevB.89.035204
  6. J. L. Lyons, A. Alkauskas, A. Janotti and C. G. Van de Walle, 'First-principles theory of acceptors in nitride semiconductors', physica status solidi (b) 252, 900 (2015). DOI: 10.1002/pssb.201552062, https://doi.org/10.1002/pssb.201552062
  7. C. Rauch, I. Makkonen and F. Tuomisto, 'Towards experimental identification of vacancy complexes in InN', physica status solidi (a) 208, 1548 (2011). DOI: 10.1002/pssa.201001160, https://doi.org/10.1002/pssa.201001160
  8. C. Rauch, I. Makkonen and F. Tuomisto, 'Identifying vacancy complexes in compound semiconductors with positron annihilation spectroscopy: A case study of InN', Physical Review B 84, 125201 (2011). DOI: 10.1103/PhysRevB.84.125201, https://doi.org/10.1103/PhysRevB.84.125201
  9. C. Rauch, F. Tuomisto, P. D. C. King, T. D. Veal, H. Lu and W. J. Schaff, 'Self-compensation in highly n-type InN', Applied Physics Letters 101, 011903 (2012). DOI: 10.1063/1.4732508, https://doi.org/10.1063/1.4732508 (preprint arXiv:1204.3299, https://arxiv.org/abs/1204.3299)
  10. S. Zhao, B. H. Le, D. P. Liu, X. D. Liu, M. G. Kibria, T. Szkopek, H. Guo and Z. Mi, 'p-Type InN Nanowires', Nano Letters 13, 5509 (2013). DOI: 10.1021/nl4030819, https://doi.org/10.1021/nl4030819
  11. S. Khromov, P. O. Å. Persson, X. Wang, A. Yoshikawa, B. Monemar, J. Rosen, E. Janzén and V. Darakchieva, 'Correlation between switching to n-type conductivity and structural defects in highly Mg-doped InN', Applied Physics Letters 106, 232102 (2015). DOI: 10.1063/1.4922301, https://doi.org/10.1063/1.4922301
  12. M. O. Eriksson, S. Khromov, P. P. Paskov, X. Wang, A. Yoshikawa, P. O. Holtz, B. Monemar and V. Darakchieva, 'Recombination processes in Mg doped wurtzite InN films with p- and n-type conductivity', AIP Advances 9, 015114 (2019). DOI: 10.1063/1.5052432, https://doi.org/10.1063/1.5052432
  13. F.-Q. Zhao, J.-J. Shi and M. Yang, 'First-Principles Study on Native Defect Complexes in InN', Communications in Theoretical Physics 53, 145 (2010). DOI: 10.1088/0253-6102/53/1/30, https://doi.org/10.1088/0253-6102/53/1/30