Existing defect models for ${\mathrm{In}}_{2}{\mathrm{O}}_{3}$ and ZnO are inconclusive about the origin of conductivity, nonstoichiometry, and coloration. We apply systematic corrections to first-principles calculated formation energies $\ensuremath{\Delta}H$, and validate our theoretical defect model against measured defect and carrier densities. We find that (i) intrinsic acceptors (``electron killers'') have a high $\ensuremath{\Delta}H$ explaining high $n$-dopability, (ii) intrinsic donors (``electron producers'') have either a high $\ensuremath{\Delta}H$ or deep levels, and do not cause equilibrium-stable conductivity, (iii) the O vacancy ${V}_{\mathrm{O}}$ has a low $\ensuremath{\Delta}H$ leading to O deficiency, and (iv) ${V}_{\mathrm{O}}$ has a metastable shallow state, explaining the paradoxical coexistence of coloration and conductivity.

Dopability, intrinsic conductivity, and nonstoichiometry of transparent conducting oxides.

The paper critically reviews the state of the art in the field of experimental techniques possible to be applied to the study of conductometric gas sensors based on semiconducting metal oxides. The used assessment criteria are subordinated to the proposed R&D approach, which focuses on the study, and subsequent modelling, of sensors’ performance in realistic operation conditions by means of a combination of phenomenological and spectroscopic techniques. With this viewpoint, the paper presents both the to-date achievements and shortcomings of different experimental techniques, describes – by using selected examples – how the proposed approach can be used and proposes a set of objectives for the near future.

Metal oxide-based gas sensor research: How to?

Contemporary theories of defects and impurities in semiconductors rely to a large extent on supercell calculations within density-functional theory using the approximate local-density approximation (LDA) or generalized gradient approximation (GGA) functionals. Such calculations are, however, affected by considerable uncertainties associated with: (i) the ``band-gap problem,'' which occurs not only in the Kohn-Sham single-particle energies but also in the quasiparticle gap (LDA or GGA) calculated from total-energy differences, and (ii) supercell finite-size effects. In the case of the oxygen vacancy in ZnO, uncertainties (i) and (ii) have led to a large spread in the theoretical predictions, with some calculations suggesting negligible vacancy concentrations, even under Zn-rich conditions, and others predicting high concentrations. Here, we critically assess (i) the different methodologies to correct the band-gap problem. We discuss approaches based on the extrapolation of perturbations which open the band gap, and the self-consistent band-gap correction employing the $\text{LDA}+U$ method for $d$ and $s$ states simultaneously. From the comparison of the results of different gap-correction, including also recent results from other literature, we conclude that to date there is no universal scheme for band gap correction in general defect systems. Therefore, we turn instead to classification of different types of defect behavior to provide guidelines on how the physically correct situation in an LDA defect calculation can be recovered. (ii) Supercell finite-size effects: We performed test calculations in large supercells of up to 1728 atoms, resolving a long-standing debate pertaining to image charge corrections for charged defects. We show that once finite-size effects not related to electrostatic interactions are eliminated, the analytic form of the image charge correction as proposed by Makov and Payne leads to size-independent defect formation energies, thus allowing the calculation of well-converged energies in fairly small supercells. We find that the delocalized contribution to the defect charge (i.e., the defect-induced change of the charge distribution) is dominated by the dielectric screening response of the host, which leads to an unexpected effective $1/L$ scaling of the image charge energy, despite the nominal $1/{L}^{3}$ scaling of the third-order term. Based on this analysis, we suggest that a simple scaling of the first order term by a constant factor (approximately 2/3) yields a simple but accurate image-charge correction for common supercell geometries. Finally, we discuss the theoretical controversy pertaining to the formation energy of the O vacancy in ZnO in light of the assessment of different methodologies in the present work, and we review the present experimental situation on the topic.

Assessment of correction methods for the band-gap problem and for finite-size effects in supercell defect calculations: Case studies for ZnO and GaAs

Tin-doped In2O3 layers were prepared by the spray technique with doping concentrationsc
Sn between 1 and 20 at. % and annealed at 500 °C in gas atmospheres of varying oxygen partial pressures. The room-temperature electrical properties were measured. Maximum carrier concentrationsN=1.5×1021cm−3 and minimum resistivities ϱ=1.3×10−4 Ω cm are obtained if the layers are doped withc
Sn≈9 at. % and annealed in an atmosphere of oxygen partial pressurep
O2 ⋦10−20 bar. At fixed doping concentration, the carrier mobility increases with decreasing oxygen pressure. The maximum obtainable mobility can be described in terms of electron scattering by ionized impurities. From an analysis of the carrier concentration and additional precision measurements of the lattice constants and film thicknesses, a defect model for In2O3:Sn is developed. This comprises two kinds of interstitial oxygen, one of which is loosely bound to tin, the other forming a strongly bound Sn2O4 complex. At low doping concentrationc
Sn≲4 at. % the carrier concentration is governed by the loosely bound tin-oxygen defects which decompose if the oxygen partial pressure is low. The carrier concentration follows from a relationN=K
1 ·p
O2
−1/8 ·(3 ×1010 × cSn −N)1/4 with an equilibrium constantK
1=1.4×1015 cm−9/4bar1/8, determined from our measurements.

Electrical properties and defect model of tin-doped indium oxide layers

In this review, we discuss the merits of solution-processed metal oxide semiconductors and consider their application in thin-film transistors for large-area electronics.

G. Van Unen

Papers

Electron concentration and mobility in In2O3