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 devices1. Thus NiO is used as a transparent hole transport layer in oxide p-n heterojunction devices such as electrical current rectifiers2,3, ultraviolet (UV) photodetectors4,5 and light-emitting diodes6,7. 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 NixMg1−xO thin films over the entire composition range can be grown with high crystalline quality on MgO single crystals8, having a structural mismatch of only 0.8% at maximum9. As compared to many other semiconductor alloys, the NixMg1−xO system is unique in demonstrating a great flexibility for band gap tuning from 3.7 eV to 7.8 eV10 in the deep UV region without the drawbacks of a significant change in lattice parameter or a phase transition. Therefore, NixMg1−xO is increasingly receiving interest for application in deep UV photodetectors8,11,12,13,14. To tune the photosensitivity of these devices, it is essential to understand the NixMg1−xO 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 AxB1−xC as a function of composition x follows the parabolic function15
where and 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 NixMg1−xO band gap dependence has been determined by optical absorption spectra of epitaxial thin films on MgO substrates prepared by molecular beam epitaxy17, textured thin films on quartz substrates prepared by magnetron sputtering14,18 and sol-gel spin coated thin films on quartz substrates19. Despite the large 7.8 eV MgO band gap, it has been observed that the NixMg1−xO band gap increases only marginally to 4.8 eV, even at a small NiO fraction of only 8 at.%. A density functional theory (DFT) calculation of the NixMg1−xO band gap dependence on composition has investigated superstructures with NiO contents of 25, 50 and 75 at.%20. However, since the study excludes the dilute NiO concentration regime for x < 0.25, no band gap discontinuity has been identified and the underlying NixMg1−xO electronic band structure has not been investigated.
The present work combines experiment and theory to resolve the controversy over the NixMg1−xO 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 NixMg1−xO system11,12,14,18,20,21. Through a detailed investigation of the NixMg1−xO microstructure and the underlying NixMg1−xO 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 NixMg1−xO band gap dependence shall not be attributed to apparent structural or compositional inhomogeneities in NixMg1−xO films and II) the standard bowing theory is inapplicable to describe the non-parabolic composition dependence of the NixMg1−xO band gap because deep Ni 3d gap states evoke a highly anomalous bowing trend and significantly reduce the NixMg1−xO band gap even at small NiO fractions of only 3.7 at.%.
Structural characterization of NixMg1−xO thin films
In the ω − 2θ high resolution X-ray diffraction (HRXRD) patterns of the NixMg1−xO thin films (x = 1, 0.83, 0.64, 0.41, 0.23, 0.12) only the 200 and 400 diffraction peaks are observed, indicating that films are single-phase and grown in 100-orientation on the MgO(100) substrates (Fig. 1). Due to the marginal difference in lattice parameters between NiO (4.177 Å) and MgO (4.212 Å)9, the NixMg1−xO 200 and 400 diffraction peaks overlap with the corresponding peaks of the MgO substrate, especially for the NixMg1−xO specimens of high MgO content. The insets show a clear shift of the NixMg1−xO 200 and 400 diffraction peaks with increasing MgO fraction towards the MgO substrate peaks, indicating an increase in the out-of-plane lattice parameter.
The projection of the 3D reciprocal space maps of the NixMg1−xO 200 reflection along the reciprocal lattice vector Qx 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 Kα1 X-ray source of 1.5406 Å allows for a significantly improved spatial resolution of the high intensity 200 reflections of the NixMg1−xO 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 NixMg1−xO 200 diffraction peak in the in-plane Qy 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 NixMg1−xO 200 diffraction peaks along Qz (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 NixMg1−xO 200 diffraction peak shifts to smaller Qz 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 ax is larger than the unstrained reference (see Supplementary Fig. S1), indicating that the NixMg1−xO thin films are not completely relaxed, but strained to match the in-plane MgO substrate lattice parameter.
A cross-sectional scanning transmission electron microscopy high angle angular dark-field (STEM-HAADF) image of the Ni0.23Mg0.77O film on MgO substrate acquired along the <001> zone axis is shown in Fig. 3(a). To locate the Ni0.23Mg0.77/MgO interface, a STEM energy dispersive X-ray spectroscopy (EDX) elemental map employing the Ni Lα,β (green) and Mg Kα (red) emission lines was recorded (Fig. 3(b)). The cross-sectional high resolution transmission electron microscopy (HRTEM) image of the Ni0.23Mg0.77O/MgO interface acquired along the <001> zone axis using multi-beam conditions shows coherency and a defect-free Ni0.23Mg0.77O 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 Ni0.23Mg0.77O 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).
Optical transmission and band gap dependence
The optical transmission spectra of NixMg1−xO thin films (x = 1, 0.83, 0.64, 0.41, 0.23, 0.17, 0.12) indicate about 80% transparency in the visible and UV region until the onset of their fundamental absorption (Fig. 4(a)). The optical absorption blue-shifts with increasing MgO content in the NixMg1−xO thin films, from 340 nm for pure NiO to 200 nm for Ni0.12Mg0.88O. 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 NixMg1−xO 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 NixMg1−xO thin films prepared by electron beam evaporation12, molecular beam epitaxy17, magnetron sputtering14,18, pulsed laser deposition23 and sol-gel spin coating19 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 Ni0.12Mg0.88O. 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.
Density functional theory calculation of the NixMg1−xO electronic structure
The calculated electronic band gaps obtained from DFT calculations of both ordered and special quasi-random structures (SQS) of NixMg1−xO are included in the band gap plot (Fig. 5). To account for the underestimation of the calculated band gaps for NiO (1.9 eV) and MgO (4.7 eV) applying the local density approximation (LDA + U) method, these values are shifted by +2 eV for comparison with experiment. The hybrid density functional calculations using the Heyd, Scuseria and Ernzerhof (HSE) approximation yields band gaps for NiO (4.2 eV) and MgO (6.5 eV) in good agreement with experiment. In consistence with the experimentally measured optical band gap trend, the calculations confirm that upon alloying with MgO, the NixMg1−xO band gap increases linearly over the entire investigated composition range (1 ≤ x ≤ 0.037). Both LDA + U and HSE calculation methods demonstrate that the variation in the band gap between NiO and Ni0.125Mg0.875O is only about 0.7 eV. However, the difference between the calculated band gaps of Ni0.037Mg0.963O and pure MgO of 1.8 eV is surprisingly large. More than 65% of the total band gap variation in the NixMg1−xO is observed after alloying just a few percent of NiO to MgO.
A superposition of the calculated band structure of Ni0.074Mg0.926O and that of pure MgO obtained within the LDA + U method is shown in Fig. 6(a). It is striking to observe that the Ni0.074Mg0.926O band structure shows deep localized levels almost entirely derived from Ni 3d eg states at about 3 eV inside the MgO band gap. The Ni 3d eg impurity-like level remains localized as there is only a negligible contribution from hybridization with O 2p states of the surrounding atoms. Both the Ni0.074Mg0.926O valence and conduction band show a nearly unperturbed electronic structure derived from pure MgO. Since the extended states of the MgO conduction band are not affected by the localized Ni 3d eg states, the approximation of employing a composition-independent +2 eV energy shift for the calculated band gaps obtained within the LDA + U method is justified to compare these results with experiment.
The calculated density of states (DOS) of NixMg1−xO (x = 1, 0.5, 0.037 and 0) is shown in 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 eg and t2g states just above the MgO valence band, which is occupied, and II) the energy level almost entirely derived from Ni 3d eg 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 eg and t2g levels merge with the O 2p states at the top of the valence band while the density of the higher energy Ni 3d eg impurity level increases and broadens to form the Ni0.5Mg0.5O conduction band.
By alloying with MgO, the NiO band gap increases about linearly from 3.7 eV to 4.9 eV in Ni0.037Mg0.963O. 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 NixMg1−xO 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 NixMg1−xO 200 diffraction spots obtained by HRXRD shown in Fig. 2 do not show any peak broadening in the out of plane Qz 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 NixMg1−xO solid solution24,25. 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 clustering26,27. The microstructural homogeneity and high-quality epitaxial growth of the NixMg1−xO is evident from the HRTEM images showing a defect-free NixMg1−xO/MgO interface and suggesting single-domain epitaxial growth without evidence of either ordering or clustering (cf. Fig. 3).
The NixMg1−xO 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 gap28. 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 levels29.
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 eg and t2g states30. Absorption spectra, cathodoluminescence and photoluminescence measurements of Ni-doped MgO crystals further indicate electronic transitions attributed to the energy levels of Ni2+ impurities31,32,33. 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 eg states broaden to form the NixMg1−xO conduction band. The anomalous NixMg1−xO 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 NixMg1−xO band gap bowing is fundamentally different from that of most substitutional semiconductor alloys, for which the standard bowing equation holds. Instead, the non-parabolic NixMg1−xO 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 GaNxAs1−x34 and II-VI semiconductor alloys such as ZnSxTe1−x35. 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 gap36,37. However, the NixMg1−xO solid solution system is different in that the localized Ni 3d eg 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 NixMg1−xO solid solution presents a unique system in which cation substitution of a small percentage of Ni2+ for Mg2+ leads to a fundamental change in the electronic structure, while maintaining complete miscibility as well as structural stability over the entire composition range.
The NixMg1−xO band gap trend is best described by a linear interpolation of the NiO band gap (3.7 eV) and that of MgO doped with an infinitesimal Ni impurity concentration (4.9 eV):
The results presented in this work show a critical concentration of about 7.4 at.% NiO in NixMg1−xO beyond which the defect-like impurity states become delocalized through the interaction of neighbouring Ni2+ cations distributed on the cation sublattice of the cubic rock-salt structure. For a random distribution of Ni2+ and Mg2+ cations on the face-centered cubic sublattice of NixMg1−xO, the calculated site percolation threshold of nearest-neighbour Ni2+ cations is 19.9 at.% NiO38. The observed threshold at about 7.4 at.% NiO defines a “localized-to-delocalized” transition (cf. ref. 29) at which the NixMg1−xO electronic structure changes from the characteristics of a concentrated NixMg1−xO solid solution with interacting Ni2+ cations to that of a dilute Ni:MgO solid solution with isolated Ni2+ impurity centres creating localized states in the MgO band gap. This indicates that the length scale over which Ni2+-Ni2+ interactions can appear is greater than predicted assuming only nearest-neighbour interactions, or the non-random ordering of Ni2+ atoms on the cation sublattice sites decreases the probability of Ni2+ nearest-neighbour site occupation.
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 NixMg1−xO 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 NixMg1−xO 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 NixMg1−xO 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.
Single-phase polycrystalline NixMg1−xO 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. NixMg1−xO thin films were grown on single-crystal MgO (100) substrates by pulsed laser ablation of the NixMg1−xO ceramic targets using a 248 nm KrF excimer laser. The growth temperature was 600 °C and the deposition pressure was 0.7 Pa O2. The NixMg1−xO ceramic targets were ablated with a laser beam fluency of 0.8 J/cm2 per pulse at a frequency of 10 Hz, resulting in a film thickness of about 300 nm.
The composition of the NixMg1−xO 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 Lα (852 eV), Ni Lβ (869 eV) and Mg Kα (1254 eV) X-ray emission lines were recorded. The background-corrected, integrated peaks of the Ni Lα, Ni Lβ and the Mg Kα emission for all NixMg1−xO thin films were evaluated based on a calibration curve obtained using data from the polycrystalline NixMg1−xO targets of known composition. It was confirmed that the 5 kV primary electron beam is entirely probing the NixMg1−xO thin films and does not reach the MgO substrate by investigating NixMg1−xO thin films of equal layer thickness grown onto Si substrates for which the Si Kα emission (1740 eV) was not observed.
The crystal structure of the NixMg1−xO thin films on MgO substrates was studied by HRXRD employing a Philips PANalytical MRD diffractometer equipped with a Cu-Kα1 X-ray source (1.5406 Å) utilizing parallel beam geometry. The structure and crystalline quality of the NixMg1−xO 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 NixMg1−xO 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 NixMg1−xO 200 diffraction peak intensity as a function of the components of the scattering vector Qx, Qy and Qz in reciprocal space. In this notation, Qx and Qy represent two orthogonal in-plane components and Qz 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 μm, 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 NixMg1−xO/MgO interface. EDX spectral images were acquired recording Ni Lα, Ni Lβ and Mg Kα X-ray emission lines and were analysed using the INCA software (ETAS group). The optical transmission of the NixMg1−xO thin films were measured with a Cary 5000 UV-Vis-NIR spectrophotometer using a bare MgO substrate as 100% transmission reference.
To describe the NixMg1−xO band gap trend, DFT calculations were performed using the LDA + U approach of Dudarev et al.39 and the hybrid density functional HSE approximation40,41. 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 method42 as implemented in the Vienna ab initio simulation package (VASP) was applied43,44. A plane wave energy cut-off of 800 eV was used in the calculations and the minimum k-point density is 0.4 Å−1. For all NixMg1−xO calculations, the Coulomb and exchange energy parameters, U = 4.3 eV and J = 1.0 eV, were applied to the Ni 3d states according to previous studies45. It is noted that the Dudarev approach to LDA + U only depends on an effective energy parameter Ueff = U − J of 3.3 eV. The HSE approximation was chosen since it provides more accurate band gap values for semiconductors and insulators compared to traditional density functional approximations. In the HSE approximation the usual Hartree-Fock exchange mixing coefficient of 0.25 was used and the range separation parameter was set to 0.2 Å−1.
NiO is an antiferromagnet (AF), in which different Ni layers along the  direction have magnetic moments pointing in opposite directions. Thus magnetic moments on the Ni atoms in NixMg1−xO may order antiferromagnetically and therefore calculations on NixMg1−xO were performed based on the rock-salt derived structure of NiO with AF ordered layers of Ni atoms along the  direction (cf. refs 20 and 46). In the case of Ni0.074Mg0.926O, a 3 × 3 × 3 repetition of the primitive rock-salt structure was used in which two Ni atoms substitute for two Mg atoms, while maintaining the AF coupling of the Ni moments. To describe the unordered NixMg1−xO solid solution, special quasi-random structures (SQS) were used47,48 and set up using the Alloy Theoretical Automated Toolkit (ATAT)49 for x = 0.75, 0.5 and 0.25. For all concentrations, the pair correlation functions for the first four atomic shells matched exactly with the pair correlation functions of the ideal unordered NixMg1−xO solid solution. Using this method the obtained structures contained a total of 64, 32 and 64 atoms for x = 0.75, 0.5 and 0.25, 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 NixMg1−xO solid solution were determined from the experimentally measured lattice constants of NiO (4.177 Å) and MgO (4.212 Å) according to Vegard’s law9.
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 alloys50. The CPA is an effective method in describing the band structures of III-V alloys when disorder determines the band gap bowing51,52, and in particular for calculation of detailed band properties such as effective masses and absorption line broadening53. However, the single-site CPA does not account for local environment effects which can be important to describe semiconductors and require cluster expansion methods54. 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 bowing48,55, while benefiting from a limited system size and reasonable computational effort.
How to cite this article: Niedermeier, C. A. et al. Band gap bowing in NixMg1−xO. Sci. Rep. 6, 31230; doi: 10.1038/srep31230 (2016).
We acknowledge support from the Leverhulme Trust via M.A. Moram’s Research Leadership Award (RL-0072012) and from Diamond Light Source via proposal MT9149. M.A. Moram acknowledges further support from the Royal Society through a University Research Fellowship. We acknowledge use of the Imperial College high performance computing service, via our membership of the UK’s HEC Materials Chemistry Consortium funded by EPSRC (EP/L000202), and use of the ARCHER UK national supercomputing service (http://www.archer.ac.uk).