Band gap analysis and carrier localization in cation-disordered ZnGeN$_2$

Cation site disorder provides a degree of freedom in the growth of ternary nitrides for tuning the technologically relevant properties of a material system. For example, the band gap of ZnGeN$_2$ changes when the ordering of the structure deviates from that of its ground state. By combining the perspectives of carrier localization and defect states, we analyze the impact of different degrees of disordering on electronic properties in ZnGeN$_2$, addressing a gap in current studies which focus on dilute or fully disordered systems. The present study demonstrates changes in the density of states and localization of carriers in ZnGeN$_2$ calculated using band gap-corrected density functional theory and hybrid calculations on partially disordered supercells generated using the Monte Carlo method. We use localization and density of states to discuss the ill-defined nature of a band gap in a disordered material, comparing multiple definitions of the energy gap in the context of theory and experiment. Decreasing the order parameter results in a large reduction of the band gap in disordered cases. The reduction in band gap is due in part to isolated, localized states that form above the valence band continuum and are associated with nitrogen coordinated by more zinc than germanium. The prevalence of defect states in all but the perfectly ordered structure creates challenges for incorporating disordered ZnGeN$_2$ into optical devices, but the localization associated with these defects provides insight into mechanisms of electron/hole recombination in the material.

The band gap of ZnGeN 2 changes with the degree of cation site disorder and is sought in light emitting diodes for emission at green to amber wavelengths. By combining the perspectives of carrier localization and defect states, we analyze the impact of different degrees of disorder on electronic properties in ZnGeN 2 , addressing a gap in current studies which largely focus on dilute or fully disordered systems. The present study demonstrates changes in the density of states and localization of carriers in ZnGeN 2 calculated using band gap-corrected density functional theory and hybrid calculations on partially disordered supercells generated using the Monte Carlo method. We use localization and density of states to discuss the ill-defined nature of a band gap in a disordered material and identify site disorder and its impact on structure as a mechanism controlling electronic properties and potential device performance. Decreasing the order parameter results in a large reduction of the band gap. The reduction in band gap is due in part to isolated, localized states that form above the valence band continuum associated with nitrogen coordinated by more zinc than germanium. The prevalence of defect states in all but the perfectly ordered structure creates challenges for incorporating disordered ZnGeN 2 into optical devices, but the localization associated with these defects provides insight into mechanisms of electron/hole recombination in the material.

I. INTRODUCTION
Site disorder, the replacement of chemical species on a fixed crystallographic lattice, has recently grown in interest across semiconductor research areas as a means to control optoelectronic properties. While site disorder-referred to from here on simply as disorder or degree of order-has notably been studied as a mechanism for managing properties in chalcogenide transistor and solar cell materials for some time, [1][2][3][4][5] its application to such a vast array of ternary and multinary nitrides and phosphides is a more recent development. 6,7 Insight from broader comparisons of II IV N 2 materials has identified relationships between cation species, structural distortion and electronic structure due to this disorder ? and in some systems site disorder has been investigated as a means of lowering band gap energies to ideal ranges for targeted applications. In ZnGeN 2 , cation disorder is sought to reduce the band gap from the calculated 9? ,10 and measured [11][12][13][14] 3.0-3.6 eV to the 2.1-2.5 eV range desired for amber to green wavelengths in a light emitting diode (LED), often referred to as the green gap. 15 Disordered ZnGeN 2 , which is lattice matched to GaN, may be desirable as a replacement for high In content In x Ga 1−x N, which suffers from a miscibility gap and large lattice mismatch with GaN in heterostructure devices. 11,[16][17][18] Because disorder adds nuance to how a band gap is measured and calculated, when the term 'band gap' is used in this work, we refer to the energy difference between the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO) unless specified otherwise. However, this energy difference is not the only viable definition as will be discussed throughout this Article.
To investigate the impact of disorder on the band gap of ZnGeN 2 , we utilize disordered structures in large supercells of 1,024 atoms 19 . These structure models incorporate site disorder consisting of cation antisite pairs, which numerous defect studies have highlighted as the dominant native defects in ZnGeN 2 . 10,20? -25 In contrast to a dilute defect model, site disorder accounts for the interaction of Zn Ge and Ge Zn present in high concentrations representative of materials grown under non-equilibrium conditions. This study separates the impact of site disorder explicitly from stoichiometry, non-native defects and crystalline quality, all known to further influence optical and electronic properties of interest. To illustrate the ordered system, Figure 1 provides the crystal structure, reciprocal space map and band structure of ZnGeN 2 .
In a dilute defect picture, defects do not interact and additional occupied or unoccupied states are viewed as defect states within an otherwise unchanged band gap. Historically, the theoretical discussion of differentiating band gaps and defect states has been held in this context of dilute point defects [26][27][28] or in fully random systems [29][30][31] , but misses systems with intermediate degrees of order with a few notable exceptions 32,33 .
In materials with both dilute and non-dilute defects, Urbach energy 34 describes how the optical absorption of a semiconductor tails off exponentially [35][36][37] at energies below the band gap due to transitions from within bands to defect states in the energy gap and at even lower energies directly between defect states in the gap. [38][39][40] Urbach tails are evident in Tauc 41 analyses of thin films as well as in bulk systems, where variations of the Kubelka-Munk method 42,43 are often used to interpret band gaps. These bulk and film methods frequently vary in interpretation of an optical band gap based on differences in their assumptions. [44][45][46] The difficulty in properly defining a band gap stems to a large extent from the fact that the band gap is used as a scalar metric to address a multitude of related but distinct phenomena and questions, either in experimental measurements or theoretical computation, and in various fields of research. Fundamentally, the band gap is the difference between ionization potential (electron removal energy) and electron affinity (electron addition energy). As such, it is not an optical or even excited-state property. However, most experimental approaches for band gap measurements are 3 Band gap analysis and carrier localization in cation-disordered ZnGeN 2 based on optical spectroscopy, as mentioned above. In such approaches, it is difficult to account for nontrivial physical mechanisms that modify the shape of the spectra from which the band gap value is deduced. For example, calculations using the Bethe-Salpeter equation (e.g., Ref. 47) show that excitonic effects (electron-hole interaction) tend to redshift the dielectric response above the absorption threshold compared to the independent particle approximation, and enable sub-band gap excitations (exciton binding energy). Similarly, the variation of oscillator strength resulting from wavefunction symmetries (direct vs indirect, allowed vs forbidden transitions) is often not precisely known, but can affect the spectra in ways that are not fully captured by model parameters used, e.g. in Tauc analysis. Furthermore, there is a fundamental difference between optical transition energies in absorption and emission, i.e., the Stokes shift 48 , which is non-radiatively converted to heat. These effects add significant uncertainties to band gap determination in all but the most thoroughly characterized systems (e.g., GaAs 49 , Cu 2 O 50 , ZnO 51 , GaN 52 ). These uncertainties are further exacerbated in disordered materials, where one must make additional assumptions or define models to discriminate between defect and continuum states.
This work addresses these challenges from the perspective of large-scale supercell electronic structure calculations in disordered ZnGeN 2 . Here, we investigate the consequences of disorder in ZnGeN 2 due to non-equilibrium synthesis on electronic structure. We use non-self-consistent hybrid functional calculations to enable analysis of the density of states (DOS) and carrier localization as a function of long-range order (LRO) and short-range order (SRO). The band gap of 3.5 eV of ordered ZnGeN 2 decreases with increasing degree of disorder and eventually closes for strongly disordered configurations. Calculated inverse participation ratios (IPR) allow us to assess the localization of states in this range of disordered ZnGeN 2 and discuss how localization impacts our interpretation of a band gap as well as device characteristics. By comparing the DOS of Zn-GeN 2 structures from band gap corrected calculations in 1,024 atom cells, we analyze the effect of disorder on the band gap of the system.

II. DISORDERED ATOMIC STRUCTURE MODELS
This Article builds on results from previous work 19 in which disordered ZnGeN 2 structures were generated using Monte Carlo (MC) simulations, providing atomic structure models with systematic variation of the order parameters across the order-disorder transition. The degree of disorder is controlled by an effective temperature describing the site ordering of a cation configu-ration within a crystalline system. This model includes the configurational entropy contribution to the free energy of the system, but excludes factors such as decomposition reactions which dominate at higher actual synthesis or process temperatures. Thus, the effective temperature provides a link to map site disorder between MC simulations and non-equilibrium synthesis. 19 We focus in this work on four effective temperatures representing four separate regimes of ordering. 2,000 K and 2,500 K structures include the ground state, ordered configuration as well as mostly ordered structures with few antisites per cell. 3,000 K structures are disordered, but not random and 5,000 K structures are highly disordered but still not random. The level of disorder between 3,000 K and 5,000 K is best understood through differences in electronic properties as discussed later in this Article. Truly random configurations are not realized below effective temperatures of approximately 400,000 K 19 .
To relate DOS, IPR and ordering, we employ the fraction of nitrogen coordinated by exactly two Ge and two Zn (Zn 2 Ge 2 motif fraction) as a measure of SRO, as well as the Bragg-Williams LRO parameter, η: where r Zn (r Ge ) is the fraction of Zn (Ge) on Zn (Ge) ground state sites. 54 indicates that a given state at a given energy is localized on average on one out of IPR atoms. An IPR of 1 indicates perfect delocalization and a value of 1,024 indicates exclusive localization on a single atom within the supercell.  Up to T e f f = 2,000 K, the MC simulation largely retains the ordered ground state structure, but some Zn Ge and Ge Zn antisite configurations start to develop. Between 2,000 K and 2,500 K, the concentration of antisite defects increases with a concomitant decrease in the average band gap of 0.7 eV. Just above 2,500 K, the system undergoes an order-disorder phase transition 19 , assuming a state with both long-and short-range disorder. It is important to note, however, that the system retains a significant degree of non-random LRO and SRO up to much higher effective temperatures.
As seen in Figure 2, comparing T e f f = 2,500 K and 3,000 K, the phase transition is accompanied  structure as well as where those states cluster in energy. In Figure 3, the traditional, unlimited IPR definition shows how small fractions of defects drastically reduce the band gap, while ignoring highly localized states shows that this significant change is largely-but not exclusively-due to these isolated defects. In the band gap interpretations that do not consider highly localized states (e.g., IPR<10 and IPR<5), the band gap still decreases by roughly 2 eV, but this reduction occurs through continuous bands in energy in disordered configurations rather than through defect states in structures with near-perfect ordering.
Taking band gaps as the difference between highest occupied and lowest unoccupied states yields a change from 3.5 eV to 2.0 eV with only a drop in LRO parameter from η=1.00 to η=0.94.
For low order parameters, the difference between the unlimited gaps and gaps excluding states with IPR>5 is again significant. The IPR limitation places the gap in the amber/green region of the visible spectrum with some trend toward higher band gap with higher SRO parameter per Figure 3b). For largely disordered structures, this sizable transition creates very small band gaps less than 1.6 eV for η ≤ 0.20, a much larger change in band gap with ordering than predicted for the more researched Cu 2 ZnSnS (CZTS) system. 58,59 When these structures are fully random (i.e., at infinite effective temperature), the band gaps consistently drop to zero for the unlimited definition and are undefined for the cases with limited IPR.
Though the supercell size of structures used for the present analysis is large for typical DFT 9 Band gap analysis and carrier localization in cation-disordered ZnGeN 2 calculations (and especially so for band gap corrected electronic structure calculations), it is still limited in capturing the statistics of configurational disorder, particularly in the dilute defect limit (low effective temperature). The localization of these states in structures with small fractions of defects and the impact of the defects' spatial proximity were studied by Skachkov et al. 20 .
These mid-gap states isolated in energy are generally accepted as detrimental to optoelectronic properties by decreasing the quantity of carriers collected, reducing the lifetimes of those carriers or inhibiting radiation of a photon. 60,61 However, at higher defect concentrations, where defect bands widen in energy as in largely disordered supercells in Figure 2b), conflicting theories exist as to the impact of defect density on the relative rate of non-radiative recombination.
In one theory, Luque et al. directly connect non-radiative Shockley-Read-Hall recombination to the localization caused by a low density or irregularity of impurities within a lattice, but see a reduction in non-radiative recombination as defect density increases above a certain threshold 62 .
In this theory, lower densities of defects correspond to more spatially isolated and therefore localized defects and spatially connected states exhibit more benign electronic properties 63 . However, gap states in 2b) show a comparable maximum IPR for every structure other than the ground state, independent of the degree of disorder of those configurations. These comparable degrees of localization independent of defect density align better with prevalent studies in the InGaN 2 system. In InGaN 2 and similar III-V alloys, higher defect densities and deep gap states cause higher rates of non-radiative recombination [64][65][66] . Based on the high degree of localization in disordered configurations, this latter theory of higher defect densities negatively impacting radiative recombination applies to ZnGeN 2 as well.
In order to address the non-radiative energy loss in disordered ZnGeN 2 , we performed calcula- The soft pseudopotential, N_s, allows for a low energy cutoff of 380 eV that benefits the feasibility of calculations using large supercells. 69 Each supercell achieved convergence when the difference in energy between steps of the ionic relaxation dropped below 10 −5 eV and forces below 0.02 eV Å −1 on each atom. These calculations used a Coulomb potential, U − J = 6 eV, applied to the Zn d orbital following the Dudarev approach 70 .
The large size of the supercells precludes the possibility of applying the GW approach 71 for each structure. In place of GW methods, the DOS and IPR of relaxed structures were calculated using a parameterized single-shot hybrid functional with an additional Coulomb potential V (SSH+V) of -1.5 eV (comparable to a U parameter of +3 eV) applied to Zn d orbitals. 72 The singleshot functional avoids the computationally expensive iteration to self-consistency of the hybrid functional Hamiltonian by holding the initial wavefunctions of the DFT+U calculation fixed. 72 This non-self-consistent approach closely reproduces the GW electronic structure for Zn-IV-N 2 nitrides and nitride-oxide alloys ? . However, since the hybrid functional Hamiltonian depends on the band occupancies, ambiguities occur when the band gap incorrectly closes in the underlying DFT calculation. In this case, we perform a second SSH+V iteration with updated band occupancies. This extra step, which could introduce some additional uncertainty in the electronic structure, was needed for most 5,000 K configurations. The Hartree-Fock exchange mixing parameter of the SSH+V functional was set to 0.19 and screening parameter to 0 for all structures. These parameters were fitted to replicate the total DOS produced by GW calculations for the ground state structure with a band gap of 3.5 eV calculated in SSH+V. The same hybrid functional and V parameters were also used for the self-consistent calculation for non-radiative energy losses due to electron and hole trapping.
To be able to plot the DOS and IPR of various disordered configurations on a common energy axis (cf. Figure 2), a potential reference needs to be defined. The bare band energies (defined Band gap analysis and carrier localization in cation-disordered ZnGeN 2 relative to the average electrostatic potential), are rather sensitive to changes of the cell volume.
The volume of the disordered supercells increases by up to 0.9% compared to the ground state for strongly disordered cells (T e f f = 5,000 K) and by about 2% for fully random cation disorder.
To eliminate the shift of the band energies with cell volume, we performed a sequence of ordered ZnGeN 2 calculations with varying cell volumes. Using the potential alignment approach of Ref. ?
and linear regression, we obtained the potential shift ∆V pot = -85 meV × ∆V vol , where ∆V vol is the volume change in percent. This offset is subtracted before plotting the electronic structure in

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.