Band gap bowing in NixMg1−xO

Epitaxial transparent oxide NixMg1−xO (0 ≤ x ≤ 1) thin films were grown on MgO(100) substrates by pulsed laser deposition. High-resolution synchrotron X-ray diffraction and high-resolution transmission electron microscopy analysis indicate that the thin films are compositionally and structurally homogeneous, forming a completely miscible solid solution. Nevertheless, the composition dependence of the NixMg1−xO optical band gap shows a strong non-parabolic bowing with a discontinuity at dilute NiO concentrations of x < 0.037. Density functional calculations of the NixMg1−xO band structure and the density of states demonstrate that deep Ni 3d levels are introduced into the MgO band gap, which significantly reduce the fundamental gap as confirmed by optical absorption spectra. These states broaden into a Ni 3d-derived conduction band for x > 0.074 and account for the anomalously large band gap narrowing in the NixMg1−xO solid solution system.

The wide-band gap semiconductor NiO (3.7 eV) is one of the few p-type transparent conductive oxides demonstrating good electrical properties for application in optoelectronic devices 1 . Thus NiO is used as a transparent hole transport layer in oxide p-n heterojunction devices such as electrical current rectifiers 2,3 , ultraviolet (UV) photodetectors 4,5 and light-emitting diodes 6,7 . The epitaxial growth of high-crystal quality NiO thin films is desired, since polycrystalline films exhibit inferior electrical transport properties and may lead to higher leakage currents in p-n heterojunction devices.
The epitaxial growth of NiO on MgO single crystal substrates offers a promising platform for the preparation of complete UV-transparent oxide devices. Both NiO and MgO crystallize in the cubic rock-salt structure and Ni x Mg 1−x O thin films over the entire composition range can be grown with high crystalline quality on MgO single crystals 8 , having a structural mismatch of only 0.8% at maximum 9 . As compared to many other semiconductor alloys, the Ni x Mg 1−x O system is unique in demonstrating a great flexibility for band gap tuning from 3.7 eV to 7.8 eV 10 in the deep UV region without the drawbacks of a significant change in lattice parameter or a phase transition. Therefore, Ni x Mg 1−x O is increasingly receiving interest for application in deep UV photodetectors 8,11-14 . To tune the photosensitivity of these devices, it is essential to understand the Ni x Mg 1−x O band gap evolution as a function of composition and to be able to accurately describe it using an analytical equation.
Experimentally it has been shown that the band gap dependence in most semiconductor alloys or solid solutions A x B 1−x C as a function of composition x follows the parabolic function 15 where E g AC and E g BC are the band gaps of the pure compounds AC and BC, respectively, and b is the bowing parameter. Since the band gap of the semiconductor alloy is generally smaller than indicated by the linear interpolation between the band gaps of its pure end members, the bowing parameter b is a positive constant. It has been proposed that the band gap bowing results from the aperiodic variation of the crystal potential in substitutional alloys, arising from random variations in occupation of the metal sites in the alloy by elements A and B, and its magnitude is a symmetric function of composition proportional to x(1 − x) 16 .
The Ni x Mg 1−x O band gap dependence has been determined by optical absorption spectra of epitaxial thin films on MgO substrates prepared by molecular beam epitaxy 17 , textured thin films on quartz substrates prepared by magnetron sputtering 14,18 and sol-gel spin coated thin films on quartz substrates 19  The present work combines experiment and theory to resolve the controversy over the Ni x Mg 1−x O band gap bowing. Regardless of the strongly non-parabolic band gap dependence on composition, the standard bowing theory has been rigidly applied in previous studies to describe the band gap evolution in the Ni x Mg 1−x O system 11,12,14,18,20,21 . Through a detailed investigation of the Ni x Mg 1−x O microstructure and the underlying Ni x Mg 1−x O electronic structure with a particular focus on the dilute NiO concentration regime (x = 0.125, 0.074 and 0.037), the present work demonstrates that I) the origin of the irregular Ni x Mg 1−x O band gap dependence shall not be attributed to apparent structural or compositional inhomogeneities in Ni x Mg 1−x O films and II) the standard bowing theory is inapplicable to describe the non-parabolic composition dependence of the Ni x Mg 1−x O band gap because deep Ni 3d gap states evoke a highly anomalous bowing trend and significantly reduce the Ni x Mg 1−x O band gap even at small NiO fractions of only 3.7 at.%. The projection of the 3D reciprocal space maps of the Ni x Mg 1−x O 200 reflection along the reciprocal lattice vector Q x recorded with 6 keV synchrotron radiation is given by an iso-intensity contour map on a logarithmic scale (Fig. 2). The increased X-ray wavelength of 2.067 Å as compared to the Cu K α1 X-ray source of 1.5406 Å allows for a significantly improved spatial resolution of the high intensity 200 reflections of the Ni x Mg 1−x O thin film and MgO substrate. Note that the observation of multiple peaks for the MgO 200 substrate indicates the presence of additional macroscopic crystallographic domains (twins). The relatively narrow Ni x Mg 1−x O 200 diffraction peak in the in-plane Q y direction in reciprocal space as compared to the MgO substrate indicates the high crystalline quality and low degree of mosaicity of the thin films. In particular, the out-of-plane width of the Ni x Mg 1−x O 200 diffraction peaks along Q z (x = 0.83, 0.64 in Fig. 2(b,c)) does not indicate any compositional broadening as compared to the pure NiO thin film ( Fig. 2(a)). The Ni x Mg 1−x O 200 diffraction peak shifts to smaller Q z values with increasing MgO content in accordance with the slight increase in lattice parameter from NiO (4.177 Å) to MgO (4.212 Å). Reciprocal space map analysis of the asymmetric NiO 204 reflection shows that the in-plane lattice parameter a x is larger than the unstrained reference (see Supplementary Fig. S1), indicating that the Ni x Mg 1−x O thin films are not completely relaxed, but strained to match the in-plane MgO substrate lattice parameter.

Structural characterization of Ni
A cross-sectional scanning transmission electron microscopy high angle angular dark-field (STEM-HAADF) image of the Ni 0.23 Mg 0.77 O film on MgO substrate acquired along the < 001> zone axis is shown in Fig. 3(a). To locate the Ni 0.23 Mg 0.77 /MgO interface, a STEM energy dispersive X-ray spectroscopy (EDX) elemental map employing the Ni L α,β (green) and Mg K α (red) emission lines was recorded ( Fig. 3(b)). The cross-sectional high resolution transmission electron microscopy (HRTEM) image of the Ni 0.23 Mg 0.77 O/MgO interface acquired along the < 001> zone axis using multi-beam conditions shows coherency and a defect-free Ni 0.23 Mg 0.77 O bulk, demonstrating the single-domain epitaxy of the film on the MgO substrate ( Fig. 3(c)). The very close lattice matching and coherent crystal interface between the Ni 0.23 Mg 0.77 O film and MgO substrate is observed in the high magnification average background subtraction filtered (ABSF) HRTEM image ( Fig. 3(d)) of the region indicated in Fig. 3(c).  Fig. 4(a)). The optical absorption blue-shifts with increasing MgO content in the Ni x Mg 1−x O thin films, from 340 nm for pure NiO to 200 nm for Ni 0.12 Mg 0.88 O. The calculated optical absorption spectra confirm the linear relationship between the photon energy hν and (αhν) 2 , where α denotes the absorption coefficient, which is valid for direct transitions in semiconductors at the absorption edge ( Fig. 4(b)) 22 .
The composition dependence of the Ni x Mg 1−x O optical band gap as determined from the absorption spectra presented in this work is shown Fig. 5. The optical band gaps which have been determined in previous studies of Ni x Mg 1−x O thin films prepared by electron beam evaporation 12 , molecular beam epitaxy 17 , magnetron sputtering 14,18 , pulsed laser deposition 23 and sol-gel spin coating 19 are included for comparison. In this work, the measured optical band gap of pure NiO is 3.7 eV and by alloying with MgO it increases linearly to 4.8 eV for Ni 0.12 Mg 0.88 O. Despite the small NiO fraction of only 12 at.%, it is striking to observe that the measured optical band gap remains 3 eV below that of pure MgO.     Fig. 6(b). A small concentration of 3.7 at.% NiO in MgO creates localized states inside the MgO band gap: I) the energy level derived from Ni 3d e g and t 2g states just above the MgO valence band, which is occupied, and II) the energy level almost entirely derived from Ni 3d e g states at about 3.5 eV. Both of these features show localized levels similar to those introduced by impurities, even though a concentration of 3.7 at.% NiO is considered in the alloying regime. With increasing NiO concentration, the lower energy Ni 3d e g and t 2g levels merge with the O 2p states at the top of the valence band while the density of the higher energy Ni 3d e g impurity level increases and broadens to form the Ni 0.5 Mg 0.5 O conduction band.

Discussion
By alloying with MgO, the NiO band gap increases about linearly from 3.7 eV to 4.9 eV in Ni 0.037 Mg 0.963 O. For MgO doped with impurity concentrations below 3.7 at.% NiO a band gap discontinuity is expected. It is obvious that the standard bowing equation cannot be applied to describe the anomalous Ni x Mg 1−x O band gap trend, because the bowing term bx(1 − x) only accounts for a parabolic deviation of the interpolated band gaps as a function of composition x, as a result of the aperiodic variation of the crystal potential in the random alloy.
The very narrow Ni x Mg 1−x O 200 diffraction spots obtained by HRXRD shown in Fig. 2 do not show any peak broadening in the out of plane Q z component in reciprocal space indicating that compositional fluctuations on a macroscopic level are insignificant. Compositional inhomogeneities are further very unlikely to be observed as it has been shown experimentally that NiO and MgO are fully miscible, typically forming a homogeneous Ni x Mg 1−x O solid solution 24,25 . A calorimetric investigation as well as thermodynamic modelling of the NiO-MgO system report a negative enthalpy of mixing describing a tendency towards atomic ordering rather than clustering 26,27 . The microstructural homogeneity and high-quality epitaxial growth of the Ni x Mg 1−x O is evident from the HRTEM images showing a defect-free Ni x Mg 1−x O/MgO interface and suggesting single-domain epitaxial growth without evidence of either ordering or clustering (cf. Fig. 3).
The Ni x Mg 1−x O band gap bowing cannot be attributed to any transition in crystal structure, direct-to-indirect electronic transition or variation in compositional homogeneity. However, calculations of the electronic properties of semiconductor alloy systems have previously shown that the band gap bowing coefficient may nevertheless become largely composition dependent if the dilute alloy shows a localized deep impurity level in the band gap 28 . When there is a pronounced difference in properties between the alloy end members, the impurities in the dilute alloy may create new electronic states rather than altering the host energy levels 29 .
Electronic structure calculations of 3d transition metal (Fe, Co, Ni) impurities in MgO reveal that deep energy levels in the band gap are formed by combination of the metal 3d orbitals with O 2p orbitals to form e g and t 2g states 30 . Absorption spectra, cathodoluminescence and photoluminescence measurements of Ni-doped MgO crystals further indicate electronic transitions attributed to the energy levels of Ni 2+ impurities [31][32][33] . The DFT calculations presented in this work demonstrate that even at dilute NiO concentrations of 3.7 at.%, well beyond typical defect concentrations, there is only a small hybridization between the Ni 3d and MgO extended states, creating such localized impurity-like levels inside the band gap. With increasing NiO content, the localized Ni 3d e g states broaden to form the Ni x Mg 1−x O conduction band. The anomalous Ni x Mg 1−x O band gap narrowing is thus attributed to the fundamental difference in electronic structure between NiO and MgO, resulting in a remarkable modification of the MgO host band structure upon alloying with only a dilute amount of NiO (x ≤ 0.037).
The origin of the Ni x Mg 1−x O band gap bowing is fundamentally different from that of most substitutional semiconductor alloys, for which the standard bowing equation holds. Instead, the non-parabolic Ni x Mg 1−x O bowing behaviour induced by the Ni 3d-derived localized states can be related to the band gap narrowing observed for highly mismatched III-V semiconductor alloys such as GaN x As 1−x 34 and II-VI semiconductor alloys such as ZnS x Te 1−x 35 . Alloying of GaAs (ZnTe) with a small amount of N (S) introduces localized states at an energy level close to the conduction band edge, resulting in splitting of the conduction band into subbands and an effective narrowing of the fundamental gap 36,37 . However, the Ni x Mg 1−x O solid solution system is different in that the localized Ni 3d e g states remain as localized impurity states well inside the MgO band gap while no perturbation of the host conduction band structure is observed (cf. Fig. 6(a)). The Ni x Mg 1−x O solid solution presents a unique system in which cation substitution of a small percentage of Ni 2+ for Mg 2+ leads to a fundamental change in the electronic structure, while maintaining complete miscibility as well as structural stability over the entire composition range.
The Ni x Mg 1−x O band gap trend is best described by a linear interpolation of the NiO band gap E g NiO (3.7 eV) and that of MgO doped with an infinitesimal Ni impurity concentration E g Ni:MgO (4.9 eV): indicates that the length scale over which Ni 2+ -Ni 2+ interactions can appear is greater than predicted assuming only nearest-neighbour interactions, or the non-random ordering of Ni 2+ atoms on the cation sublattice sites decreases the probability of Ni 2+ nearest-neighbour site occupation.

Conclusion
Despite forming a completely miscible and compositionally homogeneous solid solution, alloys of NiO and MgO exhibit a strikingly anomalous band gap bowing behaviour. The non-parabolic Ni x Mg 1−x O band gap dependence is attributed to the fundamental difference in electronic structure between NiO and MgO, resulting in a remarkable modification of the MgO host band structure upon alloying with only a dilute amount of NiO (x ≤ 0.037). DFT calculations of the Ni x Mg 1−x O band structure and density of states demonstrate that localized Ni 3d impurity levels are introduced at an energy well below the MgO conduction band and account for the pronounced band gap narrowing as confirmed by optical absorption spectra. The standard bowing theory is inapplicable to describe the Ni x Mg 1−x O band gap dependence and may in general not be applied to semiconductor systems, for which one of the alloy end members creates such deep levels in the band gap.

Methods
Single-phase polycrystalline Ni x Mg 1−x O ceramic targets were prepared by sintering high purity NiO (99.999%) and MgO (99.995%) powders at temperatures of 1200-1550 °C for 5 h, applying higher temperatures for the targets of high MgO content. Ni x Mg 1−x O thin films were grown on single-crystal MgO (100) substrates by pulsed laser ablation of the Ni x Mg 1−x O ceramic targets using a 248 nm KrF excimer laser. The growth temperature was 600 °C and the deposition pressure was 0.7 Pa O 2 . The Ni x Mg 1−x O ceramic targets were ablated with a laser beam fluency of 0.8 J/cm 2 per pulse at a frequency of 10 Hz, resulting in a film thickness of about 300 nm.
The composition of the Ni x Mg 1−x O thin films was accurately determined by EDX in a JEOL JSM-6010LA scanning electron microscope (SEM). Using a primary beam voltage of 5 kV, EDX spectra of the Ni L α (852 eV), Ni L β (869 eV) and Mg K α (1254 eV) X-ray emission lines were recorded. The background-corrected, integrated peaks of the Ni L α , Ni L β and the Mg K α emission for all Ni x Mg 1−x O thin films were evaluated based on a calibration curve obtained using data from the polycrystalline Ni x Mg 1−x O targets of known composition. It was confirmed that the 5 kV primary electron beam is entirely probing the Ni x Mg 1−x O thin films and does not reach the MgO substrate by investigating Ni x Mg 1−x O thin films of equal layer thickness grown onto Si substrates for which the Si K α emission (1740 eV) was not observed.
The crystal structure of the Ni x Mg 1−x O thin films on MgO substrates was studied by HRXRD employing a Philips PANalytical MRD diffractometer equipped with a Cu-K α1 X-ray source (1.5406 Å) utilizing parallel beam geometry. The structure and crystalline quality of the Ni x Mg 1−x O thin films on MgO were further investigated by HRXRD carried out on beamline B16 at the Diamond Light Source, UK, using a 6 keV (2.067 Å) monochromated X-ray source. The area detector was fixed at the expected 2θ value of the Ni x Mg 1−x O 200 diffraction peak while scanning the incidence angle ω to record 2-dimensional δ-χ diffraction patterns, where δ is the angle between the detector arm and the horizontal plane and χ is the angle between the detector arm and the vertical plane. 3-dimensional (3D) reciprocal space maps were calculated from the obtained data set which show the Ni x Mg 1−x O 200 diffraction peak intensity as a function of the components of the scattering vector Q x , Q y and Q z in reciprocal space. In this notation, Q x and Q y represent two orthogonal in-plane components and Q z represents the out-of-plane component of the scattering vector in the specimen frame of reference.
The electron transparent cross-sectional TEM specimen was prepared by grinding, polishing and dimpling until the specimen thickness was below 10 μ m, followed by Ar ion milling using a PIPS Ion miller (Gatan USA). Conventional HRTEM and STEM using the HAADF detector and STEM-EDX was performed using a JEOL 2100 microscope equipped with a field emission gun operating at 200 keV. The high magnification HRTEM image was filtered using an ABSF. STEM-HAADF and STEM-EDX studies were used to identify the Ni x Mg 1−x O/MgO interface. EDX spectral images were acquired recording Ni L α , Ni L β and Mg K α X-ray emission lines and were analysed using the INCA software (ETAS group). The optical transmission of the Ni x Mg 1−x O thin films were measured with a Cary 5000 UV-Vis-NIR spectrophotometer using a bare MgO substrate as 100% transmission reference.
To describe the Ni x Mg 1−x O band gap trend, DFT calculations were performed using the LDA + U approach of Dudarev et al. 39 and the hybrid density functional HSE approximation 40,41 . The LDA + U method was chosen in order to accurately describe the Ni 3d states because it significantly improves the performance of DFT applied to systems containing localised d electrons. The projector augmented wave method 42 as implemented in the Vienna ab initio simulation package (VASP) was applied 43,44  respectively. In addition to the SQS model structures, calculations were performed using ordered supercells based on 2 × 2 × 2 and 3 × 3 × 3 repetitions of the rock-salt derived AF NiO structure. Both approaches were applied to investigate the influence of atomic ordering on the calculation results. The lattice constants of the Ni x Mg 1−x O solid solution were determined from the experimentally measured lattice constants of NiO (4.177 Å) and MgO (4.212 Å) according to Vegard's law 9 .
It shall be noted that other approaches such as the Green's function based methods employing the coherent potential approximation (CPA) can provide an accurate description of random compound alloys 50 . The CPA is an effective method in describing the band structures of III-V alloys when disorder determines the band gap bowing 51,52 , and in particular for calculation of detailed band properties such as effective masses and absorption line broadening 53 . However, the single-site CPA does not account for local environment effects which can be important to describe semiconductors and require cluster expansion methods 54 . The supercells used in this study are only idealized descriptions of the random system, since the structures are periodically repeated in space and therefore have a translational symmetry in contrast to the real random alloy. Therefore, certain effects found in real disordered systems, such as transition energy broadening due to disorder and corresponding lifetime effects are neglected in the supercell methods. Even so, for this study, SQS in combination with ordered supercells are employed which closely reproduce the physically most relevant correlation functions of the infinite, random alloy and thus provide an accurate description of the band gap bowing 48,55 , while benefiting from a limited system size and reasonable computational effort.