Impact of disorder on the optoelectronic properties of GaN y As 1−x−y Bi x alloys and heterostructures
IImpact of disorder on the optoelectronic properties of GaN y As − x − y Bi x alloys andheterostructures Muhammad Usman, ∗ Christopher A. Broderick,
2, 3, † and Eoin P. O’Reilly
2, 4 School of Physics, University of Melbourne, Parkville, Melbourne, Victoria 3010, Australia Tyndall National Institute, Lee Maltings, Dyke Parade, Cork T12 R5CP, Ireland Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1UB, U.K. Department of Physics, University College Cork, Cork T12 YN60, Ireland
We perform a systematic theoretical analysis of the nature and importance of alloy disorder effectson the electronic and optical properties of GaN y As − x − y Bi x alloys and quantum wells (QWs), usinglarge-scale atomistic supercell electronic structure calculations based on the tight-binding method.Using ordered alloy supercell calculations we also derive and parametrise an extended basis 14-band k · p Hamiltonian for GaN y As − x − y Bi x . Comparison of the results of these models highlights therole played by short-range alloy disorder – associated with substitutional nitrogen (N) and bismuth(Bi) incorporation – in determining the details of the electronic and optical properties. Systematicanalysis of large alloy supercells reveals that the respective impact of N and Bi on the band structureremain largely independent, a robust conclusion we find to be valid even in the presence of significantalloy disorder where N and Bi atoms share common Ga nearest neighbours. Our calculations revealthat N- (Bi-) related alloy disorder strongly influences the conduction (valence) band edge states,leading in QWs to strong carrier localisation, as well as inhomogeneous broadening and modificationof the conventional selection rules for optical transitions. Our analysis provides detailed insight intokey properties and trends in this unusual material system, and enables quantitative evaluation ofthe potential of GaN y As − x − y Bi x alloys for applications in photonic and photovoltaic devices. I. INTRODUCTION
Over the past several decades significant research ef-fort has been dedicated to the development of III-Vsemiconductor alloys and quantum-confined heterostruc-tures such as quantum wells and quantum dots, as platforms for the development of a range of pho-tonic, photovoltaic, and spintronic devices. Despite thewidespread of use InP-based quantum well (QW) baseddevices in optical communications, an important factorlimiting overall device performance is the prevalence oftemperature-dependent loss mechanisms, including car-rier leakage, as well as non-radiative Auger recombina-tion and inter-valence band absorption (IVBA) processesinvolving transitions between the highest energy valencebands (VBs) and the spin-split-off (SO) band.
Asthese loss mechanisms are governed by the fundamentalparameters of material band structure, in particular theband gap ( E g ) and VB spin-orbit splitting energy (∆ SO ),any attempt to mitigate them must focus on developingmaterials and heterostructures whose electronic proper-ties can be flexibly engineered. Recently, dilute bismide alloys have emerged as apromising new material system whose band structurecan be engineered in order to suppress IVBA and Augerrecombination.
Dilute bismides are formed when adilute fraction of Bi atoms replace As in (In)GaAs, form-ing the (In)GaAs − x Bi x alloy. The incorporation of Bileads to an extremely strong reduction and composition-dependent bowing of E g , of ≈
90 meV between x = 0and 1% in GaAs − x Bi x . Contrary to N incorporationin GaAs, which strongly perturbs the conduction band(CB) structure leading to a reduction of E g of ≈ y = 0 and 1% in GaN y As − y , Bi – being significantly larger and more electropositive thanAs or N – primarily impacts the VB.
As a re-sult, the strong reduction of E g in GaAs − x Bi x is ac-companied by a similarly large increase and bowingof ∆ SO . This brings about the potential to engi-neer the band structure to achieve ∆ SO > E g , and tohence facilitate suppression of the dominant Auger re-combination and IVBA mechanisms. Given thestrong and complementary nature of the impact of Nand Bi incorporation on the GaAs band structure, co-alloying to form the quaternary dilute nitride-bismide al-loy GaN y As − x − y Bi x provides significant opportunitiesfor band structure engineering. N (Bi) incorpora-tion primarily impacts the CB (VB) structure and intro-duces tensile (compressive) strain with respect to a GaAssubstrate, suggesting that the band gap, VB structure,band offsets, and strain can all be readily engineered.Initial experimental studies have revealed the ex-pected giant reduction and bowing of E g , confirmingthat GaN y As − x − y Bi x alloys offer an interesting plat-form from the perspective of photonic and photovoltaicdevice development.To fully exploit the novel characteristics ofGaN y As − x − y Bi x , a comprehensive theoretical under-standing of the properties of this emerging semiconductoralloy must be developed. Given the highly-mismatchednature of N- and Bi-containing semicondutor alloys– in which substitutional N and Bi atoms act asisovalent impurities, generating localised states whichhybridise with the extended (Bloch) states of the hostmatrix semiconductor leading to a breakdown in Blochcharacter – conventional theoretical approaches tomodel alloy band structure, such as the virtual crystal(VC) approximation, break down (as is the case, to a a r X i v : . [ c ond - m a t . m t r l - s c i ] J un lesser extent, even in more conventional semicondutoralloys ). These unusual material properties then man-date that a direct, atomistic approach, free of limitingapproximations, is necessary to understand the stronglyperturbed electronic structure. In particular,based on previous investigations of GaN y As − y andGaAs − x Bi x alloys, it is expected that short-range alloydisorder – associated with the formation of pairs andlarger clusters of N and/or Bi atoms sharing commonGa nearest neighbours – will have a pronounced effecton GaN y As − x − y Bi x alloy properties. Previous studieshave been primarily based on continuum approaches,which typically ignore disorder effects and thereforeoffer limited scope to understand the full details of thecomplicated alloy electronic structure, or to analysethe results of experimental measurements . Here,we present a multi-scale framework based on atom-istic tight-binding (TB) and continuum k · p modelsto describe the electronic and optical properties ofGaN y As − x − y Bi x alloys and heterostructures. Throughsystematic large-scale atomistic TB calculations, wedevelop a detailed understanding of GaN y As − x − y Bi x alloys and QW structures, in particular highlightingthe crucial significance of alloy disorder effects and theassociated implications for the interpretation of futureexperimental measurements, and for development ofheterostructures for device applications.We begin our analysis by undertaking large-scale elec-tronic structure calculations on ordered alloy supercells,and quantify the impact of co-alloying N and Bi on theGaAs electronic structure in the ultra-dilute (dilute dop-ing) limit. Our results show that N and Bi perturb theelectronic properties effectively independently of one an-other. For large, disordered alloy supercells we com-pute the evolution of the electronic structure with alloycomposition, revealing general trends and demonstrating,somewhat surprisingly, that the respective impact of Nand Bi incorporation on the electronic properties remainsindependent, even in the presence of significant short-range alloy disorder. Our analysis demonstrates thatan extended basis set 14-band k · p Hamiltonian – whichexplicitly treats the localised impurity states associatedwith substitutional N and Bi atoms, and is parametriseddirectly via atomistic supercell calculations – describesthe main features of the band structure evolution withreasonable accuracy compared both to full atomistic cal-culations and experimental measurements. We performatomistic and continuum calculations of the electronicand optical properties of GaN y As − x − y Bi x /GaAs QWs,the results of which highlight the role played by alloydisorder in determining the properties of technologicallyrelevant heterostructures. On the basis of our analysiswe then evaluate the potential to develop devices forpractical applications, and (i) suggest, contrary to recentstudies, that GaN y As − x − y Bi x heterostructures arenot suitable for applications at 1.55 µ m, and (ii) con-firm that the most promising potential application ofGaN y As − x − y Bi x alloys is as an ≈ Overall, our resultselucidate the unusual properties of GaN y As − x − y Bi x ,highlight the importance of short-range alloy disorder indetermining the details of the material properties, andprovide guidelines for the development of optimised pho-tonic and photovoltaic devices based on this emergingsemiconductor alloy.The remainder of this paper is organised as follows.In Sec. II we describe our atomistic TB and continuum k · p models of the GaN y As − x − y Bi x electronic structure.Next, in Sec. III we present our results, beginning inSec.III A with an analysis of the impact of co-alloying Nand Bi on the electronic properties in the dilute dopinglimit, before turning in Sec. III B to analyse the evolu-tion of the electronic structure in disordered alloys, andthen the respective electronic and optical properties ofGaN y As − x − y Bi x /GaAs QWs in Secs. III C and III D.In Sec. 4 we describe the implications of the calculatedtrends in the electronic and optical properties for practi-cal applications. Finally, we summarise and conclude inSec. V. II. THEORETICAL MODELS
The unusual electronic properties of dilute nitride andbismide alloys derive from the fact that, when incorpo-rated in dilute concentrations, N and Bi act as isovalentimpurities which strongly perturb the band structure ofthe host matrix semiconductor. Due to the prominenceof N- and Bi-related impurity effects, conventional ap-proaches to analyse alloy band structures – e.g. the VCapproximation – break down, meaning that direct atom-istic calculations are generally required to provide quan-titative insight.
Furthermore, since the effects ofBi and N incorporation are prominent at dilute composi-tions, quantitative understanding of the properties of realmaterials must be built on analysis of systems contain-ing upwards of thousands of atoms, in order to mitigatefinite size effects and so that there is sufficient scope toanalyse important alloy disorder effects.
Here, we provide an overview of the atomistic TB andcontinuum k · p models we have developed to study theGaN y As − x − y Bi x electronic structure. Full details ofthese models, including the parameters used in our cal-culations, can be found in Sec. S1 of the SupplementaryMaterial. A. Atomistic: sp s ∗ tight-binding model Since the TB method employs a basis of localisedatomic orbitals, it is ideally suited to probe the elec-tronic structure of localised impurities. This, combinedwith its low computational cost compared to first prin-ciples methods, means that appropriately parametrisedTB models provide a physically transparent and highlyeffective means by which to systematically analyse theproperties of large, disordered systems and realistically-sized heterostructures. We have previously demonstratedthat the TB method provides a detailed understandingof the electronic and optical properties of GaN y As − y and GaAs − x Bi x alloys, and that calculations based onthis approach are in quantitative agreement with a widerange of experimental data. Here, we extendthis approach to GaN y As − x − y Bi x alloys.Our nearest-neighbour sp s ∗ TB model forGaN y As − x − y Bi x is closely based upon that devel-oped in Ref. 24 for dilute bismide alloys, which wehave previously employed to provide quantitative un-derstanding of the electronic, optical and spin properties of GaAs − x Bi x . In this model, which ex-plicitly includes the effects of spin-orbit coupling, theorbital energies at a given atomic site are computeddepending explicitly on the local neighbour environment,and the inter-atomic interactions are taken to vary withthe relaxed nearest-neighbour (i) bond lengths via ageneralisation of Harrison’s scaling rule, and (ii)bond angles via the two-centre expressions of Slaterand Koster. To treat GaN y As − x − y Bi x we have madeone significant modification to this model, by includingan on-site renormalisation that corrects the orbitalenergies at a given atomic site depending on the localdisplacement of the atomic positions due to latticerelaxation. This simple modification – which dependsonly on the differences in atomic orbital energies andbond lengths between the constituent GaN, GaAsand GaBi compounds – is motivated by our previousanalysis of GaN y As − y alloys, where we found that itsuitably describes the charge transfer associated withthe non-local character of the change in the supercellHamiltonian due to substitutional N incorporation. As in Ref. 24, the relaxed atomic positions in the alloysupercells are computed using a valence force field modelbased on the Keating potential.
We note that thisTB-based approach reproduces the detailed featuresof the electronic structure of N- and Bi-containingalloys compared to first principles calculations basedon density functional theory. A detailed description ofour theoretical framework can be found in our recentlypublished review of the theory and simulation of dilutebismide alloys, Ref. 36, where the validity of thisapproach is evaluated in the context of first principlescalculations and experimental measurements.To study the properties of bulk GaN y As − x − y Bi x al-loys we employ simple cubic supercells containing 4096atoms. We have previously demonstrated that this super-cell size is sufficiently large to (i) mitigate finite size ef-fects on the calculated electronic properties, by providinga suitably large basis of folded bulk states with which todescribe N- and Bi-related localised states, and (ii) pro-vide sufficient scope for the formation of a large variety ofdistinct local atomic environments to accurately reflectthe short-range alloy disorder inherent in real materials.Our heterostructure calculations are performed for re- alistically sized, [001]-oriented GaN y As − x − y Bi x /GaAsQWs. The 24576-atom supercells used to study thesestructures have a total length of 24 nm along the [001]direction, and 4 nm along each of the [100] and [010] in-plane directions. The thickness of the GaN y As − x − y Bi x QW layer is taken to be 8 nm in each case, with sur-rounding 8 nm thick GaAs barrier layers. The lateralextent and thickness of these QWs were chosen based onour analysis of GaAs − x Bi x /GaAs QWs and MQWs ,where we noted that (i) the lateral dimensions were suf-ficient to mitigate in-plane finite size effects, and (ii) thecalculated properties were robust to QW thickness fluc-tuations ∼ ± B. Continuum: 14-band k · p Hamiltonian Previous analysis has demonstrated it is usefulto derive continuum models that describe perturbedband structure of GaN y As − y and GaAs − x Bi x al-loys. Phenomenological approaches, principally theband-anticrossing (BAC) model, have originated frominterpretation of spectroscopic data and atomisticelectronic structure calculations, and are widelyemployed as a straightforward means by which to de-scribe the evolution with alloy composition of the mainfeatures of the band structure of bulk materials and het-erostructures. For GaN y As − y it is well established thatthe the CB structure can be described by a simple 2-band BAC model, in which the extended states of theGaAs host matrix CB edge interact with a set of higherenergy N-related localised states. In Ga(In)N y As − y the composition dependence of the BAC interaction be-tween these two sets of states results in a strong reduc-tion of the alloy CB edge energy with increasing y . Si-miliar behaviour is present in GaAs − x Bi x : the strongreduction (increase) and composition-dependent bowingof E g (∆ SO ) can be described in terms of a valence band-anticrossing (VBAC) interaction, between the ex-tended states of the GaAs VB edge and localised im-purity states associated with substitutional Bi impuri-ties, which pushes the alloy VB edge upwards in energywith increasing x . While (V)BAC models generally omiteffects associated with alloy disorder, they nonethelessprovide reliable descriptions of the main features of thealloy band structure and have been used as a ba-sis to provide quantitative prediction of the properties ofreal dilute nitride and bismide photonic and photovoltaicdevices. We have previously demonstrated that an appropriateset of k · p basis states for GaN y As − x − y Bi x alloys mustrepresent a minimum of 14 bands: the spin-degenerateCB, light-hole (LH), heavy-hole (HH) and SO bands ofthe GaAs host matrix (8 bands), the A -symmetric Nlocalised states (of which there is one spin-degenerateset; 2 bands), and the T -symmetric Bi localised states(of which there are two spin-degenerate sets; 4 bands).Atomistic supercell calculations confirm that the respec-tive impact of N and Bi on the band structure are de-coupled in ordered GaN y As − x − y Bi x alloys, confirmingthat an appropriate k · p Hamiltonian can be constructedby directly superposing the separate VC, (V)BAC andstrain-dependent contributions associated with N andBi incorporation. As such, the GaN y As − x − y Bi x bandstructure then admits a simple interpretation in termsof the respective perturbation of the CB and VB sepa-rately by N- and Bi-related localised states, with minorVC changes to the CB (VB) structure due to Bi (N)incorporation. We parametrise the 14-band k · p Hamil-tonian directly from TB supercell calculations, obtainingthe N- and Bi-related band parameters without the usualrequirement to perform post hoc fitting to the results ofalloy experimental data.
III. RESULTS AND DISCUSSIONSA. Co-alloying N and Bi: dilute doping limit
Analysis of large, ordered 2 M -atom Ga M N As M − and Ga M As M − Bi supercells – undertaken us-ing the empirical pseudopotential and TBmethods – have revealed the mechanisms bywhich substitutional N and Bi impurities perturb theGaAs band structure. Specifically, it has been shownthat the local relaxation of the crystal lattice (arisingfrom differences in covalent radius) and charge transfer(arising from differences in atomic orbital energies) dueto substitutional incorporation of an isolated N (Bi) im-purity gives rise to highly localised states. These statesare resonant with and couple to the GaAs CB (VB)in GaN y As − y (GaBi x As − x ), leading to hybridisedalloy CB (VB) edge states containing an admixture ofextended GaAs Bloch and localised N (Bi) character.These localised states can be described as linear com-binations of the supercell eigenstates that fold back toΓ. Since more eigenstates fold back to Γ with increasingsupercell size, systematic analysis has demonstrated theneed to use large supercells – i.e. ultra-dilute Bi and Ncompositions – to accurately quantify the nature andimpact of these localised impurity states in the dilutedoping limit. Here, we perform a similar analysis for a series of 4096-atom Ga N As Bi supercells. We vary the rela-tive positions of the N and Bi atoms to ascertain (i) anychanges to their respective impact on the electronic struc-ture, and (ii) the scale of any interactions between N- andBi-related localised states. In each case we compute thefractional GaAs Γ (Bloch) character of the supercell zone-centre eigenstates in order to quantify the changesin the electronic structure associated with the formationof different N and Bi local atomic environments. Full de-tails of the calculated Γ character spectra are presentedin Sec. S2 of the Supplementary Material.Table I summarises the results of our calculations fora range of Ga N As Bi supercells. We use sub-scripts B, C and D to describe the positions of Bi atomsrelative to a N atom at position A. Prior to relaxation ofthe atomic positions, the vectors describing the separa-tion between these atomic sites are: r AB = a ( (cid:98) x + (cid:98) y + (cid:98) z ), r AC = a ( (cid:98) x + (cid:98) y ) and r AD = a ( (cid:98) x + (cid:98) y ), where a is the GaAslattice constant. For example, the notation GaAs:N A Bi B describes a supercell in which the Bi atom is orientedalong [111] relative to the N atom, with the N and Biatoms occupying the same position in a given pair ofneighbouring 8-atom simple cubic unit cells, while thenotation GaAs:N A Bi D describes a [110] oriented N-Ga-Bicomplex in which the N and Bi atoms share a commonGa nearest neighbour.We begin with a Ga N As Bi supercell in whichthe N and Bi atoms are separated by √ a (where theydo not form a pair or cluster) and note that the asso-ciated reduction in symmetry, arising from relaxation ofthe crystal lattice about the impurity sites, lifts the de-generacy of the predominantly GaAs LH- and HH-likeVB edge states. As the N and Bi atoms are brought closertogether the splitting between these states increases dueto the resultant larger local relaxation of the lattice,with the largest calculated splitting of ≈
10 meV in aGa N As Bi supercell occurring for an N-Ga-Bicomplex oriented along the [110] direction (in which theN and Bi atoms are second-nearest neighbours and sharea Ga nearest neighbour). We note that this lifting ofthe VB edge degeneracy due to a reduction in symme-try is consistent with that calculated previously for dis-ordered GaAs − x Bi x . In the Ga N As andGa As Bi supercells we note the presence of asmall ( < From Table I, we see that as the N and Bi atomsare brought closer together the splitting between thetwo highest energy VB states ( E v and E v ) increasesdue to the resultant larger local relaxation of the lat-tice. The largest calculated splitting of ≈
10 meV in aGa N As Bi supercell occurs when the N and Biatoms are second-nearest neighbours sharing a Ga near-est neighbour (in the GaAs:N A Bi D supercell). We notethat this lifting of the VB edge degeneracy due to a re-duction in symmetry is consistent with that calculatedpreviously for disordered GaAs − x Bi x . We notealso that the calculated shift in the VB edge energy inthe supercells containing both N and Bi is significantlylarger than that in the equivalent N-free supercells. Our
TABLE I. Energies of the lowest energy CB ( E CB ), two highest energy VBs ( E VB,1 and E VB,1 ), and SO band edge ( E SO )calculated using the TB method for selected 4096-atom simple cubic GaN y As − x − y Bi x supercells. The notation describingdistinct (non-equivalent) local atomic configurations in the second column are defined in the text; accompanying schematicillustrations can be found in Fig. S1 of the Supplementary Material. The spatial distance between the N and Bi atoms for each(unrelaxed) configuration is denoted by r N,Bi and given in units of GaAs lattice constant a .Supercell Configuration r N,Bi ( a ) E CB (eV) E V1 (eV) E V2 (eV) E SO (eV)Ga As GaAs —– 1 .
519 0 . . − . N As GaAs:N A —– 1 .
500 0 . . − . As Bi GaAs:Bi B —– 1 .
518 0 . . − . N As Bi GaAs:N A Bi B √ .
500 0 . . − . N As Bi GaAs:N A Bi C √ .
506 0 . . − . N As Bi GaAs:N A Bi D √ .
495 0 . . − . analysis suggests that this is a result of the large localrelaxation of the lattice about the N atomic site which,within the framework of our TB model, generates sig-nificant shifts in the energies of the p orbitals localisedon the N atom. However, we calculate that this trenddoes not persist beyond the ultra-dilute regime, withthe calculated overall shift in the VB edge energy withcomposition in GaN y As − x − y Bi x closely tracking that inGaAs − x Bi x (cf. Sec. III B).The calculated trends in the impact of interaction be-tween N- and Bi-related localised states is consistent withprevious calculations, and can be understood generallyon the basis of their respective independent impact inGa N As and Ga As Bi supercells. For theCB edge state, having energy E CB , the impact of the in-teraction between the N and Bi atoms is minimal: theoverall character of the CB edge is predominantly deter-mined by the impact of the localised resonant state asso-ciated with the N atom. Even in the GaAs:N A Bi B super-cell – in which the N and Bi atoms are closest, and hencethe interaction of their associated localised states max-imised – the largest calculated Bi-induced shift in E CB isminimal when compared to the overall shift in E CB calcu-lated in an equivalent Bi-free supercell. In this case thecharacter of the calculated GaN y As − x − y Bi x CB edgestate results from a combination of an N-induced hybridi-sation and reduction in energy (described by the con-ventional 2-band BAC model), in addition to small fur-ther changes associated with (i) the VC-like Bi-inducedreduction in E CB , and (ii) the local compressive strainassociated with lattice relaxation about the Bi atomicsite. Contrary to the trends observed for the CB and VBedges, we note that the SO band edge energy is relativelyunaffected by N incorporation: the calculated trends inthe energy of Γ character of the SO band edge states areessentially identical to those in GaAs − x Bi x , with the im-pact of N manifesting primarily via small energy shiftsassociated with local lattice relaxation.Next, we turn our attention to the localised states as-sociated with N and Bi, which we construct explicitlyfor each supercell. Our analysis reveals a some-what surprising feature: the overall nature and charac-ter of the N- (Bi-) related localised states are found to be effectively identical to those in an equivalent Bi-freeGa N As (N-free Ga As Bi ) supercell. Inall cases we find that the character of the band edgeeigenstates is largely retained: the CB (VB) edge eigen-states are a linear combination of the unperturbed GaAsCB (VB) edge and N (Bi) localised states, describable viathe same 2- (4-) band (V)BAC model as in GaN y As − y (GaAs − x Bi x ). This confirms that the impact of N(Bi) on the CB (VB) structure is effectively independentof co-alloying with Bi (N) and, as we will see below, thatgeneral trends in the evolution of the GaN y As − x − y Bi x band structure can be described to a reasonable degreeof accuracy by superposing the established description ofimpact of both N and Bi incorporation. B. Band edge energies in disordered alloys
Having analysed the impact of co-alloying N and Biin the dilute doping limit, we now turn our attention tothe evolution of the electronic structure with alloy com-position in disordered alloys. We restrict our attentionto lattice-matched, 4096-atom supercells in which the Nand Bi compositions have been chosen to produce netzero macroscopic strain with respect to GaAs – i.e. werequire y = 0 . x so that the GaN y As − x − y Bi x latticeconstant computed using V´egard’s law is equal to thatof GaAs. At each alloy composition we compute theelectronic structure of five distinct supercells in whichthe N and Bi atoms are substituted at randomly chosensites on the anion sublattice. To determine the compo-sition dependence of the band edge energies we averageover the results of these five distinct calculations at eachcomposition. As a reference for the GaN y As − x − y Bi x calculations, we have performed the same analysis forequivalent Bi- (N-) free GaN y As − y (GaAs − x Bi x ) su-percells. The results of these calculations are summarisedin Figs. 1(a), 1(b) and 1(c) for GaN y As − y , GaAs − x Bi x and GaN y As − x − y Bi x , respectively.Beginning with GaN y As − y , the closed red circles inFig. 1(a) show the calculated dependence of the CB andVB edge energies E CB and E VB on y . Our calculations re-produce the well-known trends for GaN y As − y . Firstly, E CB ( y ) E VB ( y ) E g ( y ) (a) E CB ( x ) E VB ( x ) E g ( x ) (b) E CB ( x, y ) E VB ( x, y ) E g ( x, y ) (c) . . (d) x = 1 . % T = 300 K E g ∆ SO E n e r g y ( e V ) N composition, y (%)GaN y As − y TB k · p Bi composition, x (%)GaAs − x Bi x TB k · p Bi composition, x (%)GaN y As − x − y Bi x ( y = 0 . x ) TB k · p N composition, y (%)Theory vs. experiment FIG. 1. (a), (b) and (c) Calculated variation of the CB and VB edge energies E CB and E VB , and band gap E g = E CB − E VB ,with alloy composition in GaN y As − y , GaAs − x Bi x and GaN y As − x − y Bi x (lattice-matched to GaAs). Closed red circles (greensquares) show the values of E CB and E VB ( E g ) averaged at each composition over those calculated for a series of free-standing,disordered 4096-atom supercells using the sp s ∗ tight-binding model. Error bars denote the standard deviation about theseaverage values, computed from the corresponding energies for the distinct supercells considered at each composition. Solid(dashed) black lines show E CB and E VB ( E g ) calculated using the 14-band k · p Hamiltonian. (d) Variation of E g and ∆ SO withN composition y in GaN y As . − y Bi . ( x = 1 . k · p Hamiltonian (dashed and solid blacklines, respectively), compared to the results of room temperature spectroscopic ellipsometry (SE; closed green squares andcircles) and photo-modulated reflectance (PR; closed red squares) measurements. The experimental data are from Ref. 67.
N incorporation causes a rapid decrease and large com-position dependent bowing of E CB , with the calculatedCB edge states consisting of an admixture of GaAs CBedge Bloch and N localised state character. Secondly,the VB and SO edge energies are relatively unperturbedfrom those calculated in ordered alloy supercells, andare well described via conventional VC energy shifts as E VB ( y ) = E VB (0) + κ N y and E SO ( y ) = E SO (0) − γ N y ,where E VB (0) and E SO (0) are the corresponding unper-turbed GaAs band edge energies. Thirdly, the bandgap E g = E CB − E VB (depicted by closed green squares inFig. 1(a)) is calculated to decrease by ≈
180 meV when1% of the As atoms are replaced by N, in good agreementwith a range of experimental measurements. The closed red circles in Fig. 1(b) show the calcu-lated dependence of E CB and E VB on x in GaAs − x Bi x .Again we recover the experimentally observed trends. Biincorporation causes a rapid decrease of the band gap(closed green squares), which is characterised by strongcomposition dependent bowing and is qualitatively sim-ilar to that in GaN y As − y , but instead originates fromstrong upward bowing of E VB . The calculated VB edgeeigenstates consist of an admixture of GaAs VB Blochand Bi localised state character, and the calculated ≈
100 meV decrease in E g in going from GaAs toGaAs . Bi . is in good agreement with experimentalmeasurements. We note that our calculations cor-rectly describe that the decrease of E g due to N incor-poration is larger than that associated with Bi incorpo-ration at fixed composition, reflecting the larger differ-ences in covalent radius and electronegativity between Nand As than between As and Bi. We further find that E CB and the SO band edge energy E SO in GaAs − x Bi x are well described via conventional VC energy shifts as E CB ( x ) = E VB (0) − α Bi x and E SO ( x ) = E SO (0) − γ Bi x . The calculated VC parameters κ N,Bi , α N,Bi and γ N,Bi aregiven, along with the remainder of the parameters of the14-band k · p model, in Sec. S1.B of the SupplementaryMaterial.The error bars in Figs. 1(a) and 1(b) denote the stan-dard deviations of the corresponding energies, computedfrom the values of E CB , E VB and E g calculated for each ofthe distinct supercells considered at fixed alloy composi-tion. For GaN y As − y we note that the calculated stan-dard deviations for E CB (and hence E g ) increase stronglywith increasing y , from 6 meV at an ultra-dilute N com-position y = 0 .
15% to >
80 meV for 2% (cid:46) y (cid:46) y As − y CB structure, with the different spa-tially random distributions of N atoms in the supercellsconsidered leading to large differences in the computedCB edge energy at fixed y . Conversely, we compute neg-ligible standard deviations (cid:46) E VB across therange of y considered, reflecting the weak impact of Nincorporation on the VB structure. We find that thesetrends are reversed in GaAs − x Bi x alloys, where we com-pute negligible standard deviations (cid:46) E CB ateach Bi composition x , reflecting that VC contributionsdominate the CB and SO band edge character. How-ever, we compute larger standard deviations for E VB , re-flecting the important role played by Bi-related localisedstates and alloy disorder in determining the nature of theGaAs − x Bi x VB edge states. However, the maximumcalculated value of 43 meV for the standard deviation of E VB at x = 1 .
95% is approximately one-half of that calcu-lated for GaN y As − y at similar composition indicating,as expected, that N-related alloy disorder more stronglyperturbs the electronic structure.Figure 1(c) summarises the results of the disorderedalloy supercell calculations for GaN y As − x − y Bi x lattice-matched to GaAs. The variation of E CB , E VB and E g areshown here as a function of the Bi composition x , for Ncompositions y = 0 . x . The calculated reduction of E g – approximately 220 meV (400 meV) in a lattice-matchedalloy having x = 1% ( y = 1%) – is significantly largerthan that in either GaN y As − y or GaAs − x Bi x . Thatthis giant band gap bowing can be achieved in alloyswhich are lattice-matched to GaAs suggests significantpotential for applications at infrared wavelengths. Wenote (i) the decrease in E CB is comparable to, but slightlylarger than, that in GaN y As − y , and (ii) the increase in E VB is approximately equal to that in GaAs − x Bi x . ∆ SO is calculated to increase by approximately 70 meV per %Bi in lattice-matched GaN y As − x − y Bi x – i.e. by approxi-mately the same amount as in GaAs − x Bi x , reflecting theweak impact of N incorporation on the VB structure.As in Figs. 1(a) and 1(b), the error bars in Fig. 1(c)denote the standard deviations computed for E CB , E VB and E g using the corresponding calculated values for thedistinct supercells considered at each fixed alloy com-position. As expected on the basis of the trends dis-cussed above for ternary GaN y As − y and GaAs − x Bi x alloys, (i) the impact of N and Bi incorporation leadsto large respective computed standard deviations in E CB and E VB , and (ii) the magnitude of the computed stan-dard deviation for E CB at fixed composition is largerthan that associated with E VB . We note then that thestandard deviation associated with E g , the maximumcalculated value of which was 98 meV at x = 3 . y As − x − y Bi x than that in ei-ther GaN y As − y or GaAs − x Bi x having the same Nor Bi composition, but is in general broadly compara-ble in magnitude to that associated with GaN y As − y .Overall, these results reaffirm that N- and Bi-related al-loy disorder leads to significant inhomogeneous spectralbroadening of the band edge transitions in GaN y As − y and GaAs − x Bi x alloys, and demonstrates that while al-loy disorder effects can be expected to have an impor-tant impact on the GaN y As − x − y Bi x electronic struc-ture, the magnitude of such effects should somewhat ex-ceed but nonetheless be broadly comparable to those inGaN y As − y .To confirm that these trends can be described as adirect combination of the separate impact of N (Bi) in-corporation primarily on the CB (VB) structure, we havealso used the 14-band k · p model to calculate the compo-sition dependent band edge energies. The results of the k · p calculations of E CB and E VB (solid black lines) and E g (dashed black lines) for GaN y As − y , GaAs − x Bi x andGaN y As − x − y Bi x are shown respectively in Figs. 1(a),1(b) and 1(c). In GaN y As − y (GaAs − x Bi x ) this corre-sponds to a 10- (12-) band k · p model of the band struc- ture. In all cases we note that the variation of E CB and E VB with composition are in good overall agreement withthe results of the disordered supercell calculations. Wenote some minor deviation between the calculated vari-ation of E g in GaN y As − y and GaN y As − x − y Bi x usingthe TB and k · p models, which we identify as being as-sociated with the relatively stronger impact of N-relatedcluster states – neglected in the k · p model – on the CBedge. These results suggest overall that the evolu-tion of the main features of the GaN y As − x − y Bi x bandstructure (i) is primarily determined by the influence oflocalised states associated with independent N and Biimpurities, and (ii) can be reliably described using an ex-tended k · p Hamiltonian whose basis explicitly includesthese localised states and their coupling to the GaAs hostmatrix band edge states.Finally, Fig. 1(d) compares the variation of E g (solidblack lines) and ∆ SO (dashed black lines) with N composi-tion y , calculated using the 14-band k · p model, to photo-modulated reflectance (PR) and spectroscopic ellipsome-try (SE) measurements undertaken on a series of pseudo-morphically strained GaN y As . − y Bi . ( x = 1 . The calculated variation of E g and ∆ SO with y are in good overall agreement with experiment. Firstly,the 14-band model accurately describes the measuredlarge reduction in E g compared to that in GaAs, andcaptures the evolution of E g with increasing N compo-sition. Secondly, the 14-band model describes well theoverall magnitude of ∆ SO – which is larger than that inGaAs due to the presence of Bi – and that incorporatingN tends to have little impact on ∆ SO , with the measuredand calculated values remaining approximately constantacross the range of N compositions considered, confirm-ing the predicted weak impact of N incorporation on theVB structure. C. Electronic properties of GaN y As − x − y Bi x /GaAsquantum wells In order to realise photonic devices based onGaN y As − x − y Bi x alloys, in practice it will likelybe required to develop quantum-confined heterostruc-tures. We therefore elucidate and analyse generalfeatures of the electronic and optical properties ofGaN y As − x − y Bi x /GaAs QWs, focusing in particular onthe bound lowest energy electron and highest energy holestates e h
1. In order to account for alloy disordereffects, for each QW structure having fixed N and Bicomposition we consider ten distinct supercells havingdifferent statistically random spatial distributions (RDs)of substitutional N and Bi atoms at anion lattice sitesin the GaN y As − x − y Bi x QW layer. We compare the re-sults of atomistic TB calculations to those obtained forthe same structures using the continuum 14-band k · p model in the envelope function approximation (EFA), inwhich the QWs are treated as idealised one-dimensional TABLE II. Details of the GaN y As − x − y Bi x /GaAs structures analysed in Secs. 3.3 and 3.4. Structures 1 and 2 contain ternary(N-free) GaAs − x Bi x /GaAs QWs, while structures 3 and 4 contain quaternary GaN y As − x − y Bi x QWs. In addition to theBi and N compositions x and y we summarise the results of the 14-band k · p calculations, which are independent of alloydisorder at fixed composition. We provide the computed in-plane component (cid:15) xx (= (cid:15) yy , compressive in all structures) of themacroscopic strain, as well as the CB and VB band offsets (∆ E CB and ∆ E HH ), ground state transition energy e h T = 300K, and the corresponding ground state emission/absorption wavelength for the e h x (%) y (%) (cid:15) xx (%) ∆ E CB (meV) ∆ E HH (meV) e h e h − .
74 106 359 1.097 11302 9.00 —– − .
06 153 456 0.960 12923 6.25 2.50 − .
23 419 366 0.791 15684 9.00 1.00 − .
86 306 459 0.808 1535 structures. Since the 14-band model does not explicitlyaccount for the presence of alloy disorder, the results ofthe k · p calculations provide a reference against whichto highlight the role played by alloy disorder in the fullatomistic calculations.Four compressively strained QW structures are consid-ered: structures 1 and 2 are N-free GaAs − x Bi x /GaAsQWs having respective Bi compositions x = 6 .
25 and9%, while structures 3 and 4 are GaN y As − x − y Bi x /GaAsQWs having respective Bi and N compositions x =6 . y = 2 .
5% and x = 9%, y = 1%. These struc-tures are described in Table II, and the simulated ge-ometries are as described in Sec. II A above. The N andBi compositions for structures 3 and 4 were chosen toproduce ground state e h µ m wavelength range. Ta-ble II also summarises the results of the 14-band k · p calculations for each QW structure, including the in-plane compressive strain (cid:15) xx , CB and VB offsets ∆ E CB and ∆ E HH , and QW band gaps ( e h y As − x − y Bi x /GaAs heterostructures have largetype-I band offsets and can therefore be expected to pos-sess intrinsically high electron-hole spatial overlap, sug-gesting the possibility to achieve good optical efficiencyand indicating potential for the development of light-emitting/absorbing devices. We calculate respective CBand VB offsets ∆ E CB = 106 meV and ∆ E HH = 359 meVin structure 1 ( x = 6 . x = 9%). In these N-free struc-tures Bi incorporation brings about compressive strainand large VB offsets ∆ E HH , and while ∆ E CB is consider-ably smaller it is nonetheless sufficiently large to providegood confinement of electrons and holes to facilitate effi-cient photon emission/absorption. Beginning with structure 1 and incorporating 2.5%N in the QW layer to obtain structure 3 increases theground state transition wavelength by ≈
450 nm to1.55 µ m, while simultaneously reducing the net com-pressive strain by a factor of approximately three to (cid:15) xx = − . E CB , by a factor of approximatelyfour to 419 meV, corresponding to the strong N-inducedreduction of the QW CB edge energy. The VB offsetis effectively unchanged, again reflecting that N incorpo-ration has little effect on the VB. We observe a similartrend in incorporating 1% N to go from structure 2 tostructure 4. In all cases we find that the trends in thenet macroscopic strain, QW band offsets, and band gapcalculated using the 14-band model are consistent withthose obtained from the full atomistic calculations.Figures 2(a) and 2(b) show the probability density forthe e h y As − x − y Bi x /GaAs QWs (a)structure 3, and (b) structure 4, calculated using the TBmethod. For each structure we have plotted probabilitydensities for three of the ten different RDs of Bi and Natoms considered; plots for additional RDs can be foundin Fig. S6 of the Supplementary Material. Solid anddashed lines respectively denote the probability densityprojected to cations and anions, calculated using the TBmethod at each fixed position z along [001] by summingover the probability density associated with all atoms inthe plane perpendicular to [001]. Firstly, we note that thecalculated probability densities in the N-free QWs, struc-tures 1 and 2 – which, for brevity, are provided in Sec. S3of the Supplementary Material – demonstrate that short-range alloy disorder strongly perturbs the hole states inGaAs − x Bi x /GaAs QWs, while the electron states in therelatively unperturbed CB can be well described usingthe conventional EFA. The GaN y As − x − y Bi x /GaAs h h h e e h FIG. 2. Probability density associated with the lowest energy CB electron state ( e
1; upper row) and highest energy VB hole state( h
1; lower row) in 8 nm thick (a) GaN . As . Bi . /GaAs ( x = 6 . y = 2 . . As . Bi . /GaAs( x = 9%, y = 1%) QWs. Solid (dash-dotted) black lines and solid (dash-dotted) red lines respectively denote the e h z along the [001] growthdirection by summing over the probability densities associated with each atom in the plane. Dashed blue lines denote thewell/barrier interfaces. The TB calculations were performed for ten supercells containing different random spatial distributions(RDs) of substitutional N and Bi atoms on the anion sublattice: results are presented here for three representative structures ineach case, with plots for additional GaN y As − x − y Bi x /GaAs structures, as well as for GaAs − x Bi x /GaAs structures, providedin Figs. S5 and S6 of the Supplementary Material. ysis of the GaN y As − y (GaAs − x Bi x ) electronic struc-ture, the observed strong localisation of the CB (VB)edge eigenstates generally reflects strong hybridisationof the extended GaAs band edge states with a multi-plicity of N (Bi) localised states associated with pairsand larger clusters of substitutional N (Bi) impurities,resulting in significant degradation of the Bloch charac-ter of the corresponding alloy eigenstates. In theGaN y As − x − y Bi x QW structures considered here this be-haviour is reflected in the near complete breakdown of theEFA description of the carrier probability densities due tostrong electron and hole localisation, with the details ofthe QW electronic properties being strongly dependenton the precise nature of the short-range alloy disorderpresent in the GaN y As − x − y Bi x layer of the structure.Figures 3(a) – 3(d) provide a summary of the calcu-lated e h k · p (solid and dashed greenlines) models. The probability densities shown for theTB calculations are averaged over those computed forthe ten different supercells (RDs) used to represent eachstructure. As in our analysis of the bulk electronic struc-ture (cf. Sec. 3.2), we find that the k · p method reliablycaptures the general trends observed in the full atom-istic calculations. However, on average, alloy disordereffects cause a breakdown of the EFA for hole states inGaAs − x Bi x QWs, and for both electron and hole statesin GaN y As − x − y Bi x QWs. On this basis we concludethat the electronic properties of GaN y As − x − y Bi x /GaAsQWs are strongly influenced by the alloy microstructure,and hence that the bound states in GaN y As − x − y Bi x heterostructures are generally characterised by strong lo- calisation and a corresponding degradation in Bloch char-acter, compared to those in equivalent structures basedon conventional semiconductor alloys. D. Optical transitions in GaN y As − x − y Bi x /GaAsquantum wells We consider now the optical properties of these QWstructures and, again facilitated by comparison with theresults of the atomistic TB and continuum k · p calcula-tions, quantify the impact of alloy disorder on the QWband gap and ground state optical transition strengths.The results of our TB calculations of the optical prop-erties for structures 1 – 4 are summarised in Figs. 4 (a)– 4(d) respectively where, in each case, the results for allten different RDs are shown. Solid black (red) lines showthe computed TE- (TM-) polarised e h λ corresponding to the computed e h k · p calculations h k (cid:107) = 0). As such,the h − x Bi x /GaAsstructures 1 and 2: the calculated TM-polarised opticaltransition strengths are negligibly small, even in the pres-ence of significant alloy disorder at x = 9%. FIG. 3. (a), (b), (c) and (d) respectively show the probability density associated with the e h e hh
1) probability density calculated using the 14-band k · p Hamiltonian inthe EFA. Dashed blue lines denote the well/barrier interfaces.
The calculated TE-polarised optical transitionstrengths for structures 1 and 2 describe that the bandedge optical transitions in GaAs − x Bi x /GaAs QWs arecharacterised by inhomogeneous broadening associatedwith alloy disorder-induced fluctuations in the QWband gap, producing a spectral width ∆ λ of the e h e h x = 6 .
25% (9%).
The TB calculations indicate littledegradation in the optical transition strengths in goingfrom structure 1 to 2, reflecting that the slight reductionin the Bloch character of h ≈ x is offset by the increase in electron-holespatial overlap brought about by the accompanyingincrease in ∆ E CB . Indeed, the k · p calculationsfor structures 1 and 2 indicate a modest increase (cid:46) x = 6 .
25 to 9%: we attribute the deviationfrom this trend observed in the TB calculations to theimpact of alloy disorder on the h − x Bi x alloys and QWs.Turning our attention to the quaternary QWs – struc-tures 3 and 4 – in Figs. 4(c) and 4(d) we note that co-alloying N and Bi leads to significant modifications ofthe optical properties. Firstly, we note a breakdown ofthe conventional QW selection rules: the h e h h p -like orbital components po-larised along the growth direction – and hence to havenon-zero transition strengths at the zone centre for TM-polarised transitions involving e
1. While N incorporationleaves the overall character of the VB edge states largelyunchanged on average, this demonstrates that N cluster-ing can nonetheless bring about non-trivial modificationof the character of individual bound hole states. This fur-ther confirms that calculated breakdown of the conven-tional optical selection rules is associated with the impactof N-related alloy disorder on the VB structure. This ex-plicitly local effect is not accounted for in the 14-band k · p model considered here, nor in existing models of theGaN y As − x − y Bi x band structure. We note however thatthis calculated breakdown of the conventional QW opti-cal selection rules does not necessarily imply that mea-sured optical spectra for GaN y As − x − y Bi x -based QWswill have large TM-polarised components. Our analy-sis here focuses on individual QW eigenstates: more de-tailed analysis shows that this behaviour is associatedwith the formation of localised states about clusters ofN atoms, which occur relatively rarely and are hence un-likely to contribute significantly to the net optical emis-sion/absorption.Secondly, we note that the combination of N- and Bi-related alloy disorder leads to large variations of the QWband gap between the distinct supercells considered foreach structure, resulting in ∆ λ = 47 nm (99 nm) in struc-ture 3 (4). For some RDs we calculate anomalously low1 FIG. 4. TE-polarised (solid black lines) and TM-polarised(solid red lines) optical transition strengths between the e h − x Bi x /GaAs QWs structure (a) 1, and (b)2, and for the GaN y As − x − y Bi x /GaAs QWs structure (c) 3,and (b) 4 (cf. Table II). For each structure the computedtransition strengths are shown for all ten distinct supercellsconsidered. The spectral width of the ground state transitionwavelength (∆ λ ) is calculated as the standard deviation of thewavelengths computed for the ten distinct supercells (RDs)used to represent each structure. ground state optical transition strengths, which our anal-ysis associates with strong hybridisation of the e h x (i.e. in going from structure 1 to 3,or structure 2 to 4). The k · p calculations indicate thatthe TE-polarised optical transition strength in structure3 (structure 4) should be ≈
75% ( ≈ e h y As − x − y Bi x -based het-erostructures, and indicates that intrinsic alloy disordermay potentially limit the optical efficiency and perfor-mance of real GaN y As − x − y Bi x materials and devices.The calculated strong variation of the TE-polarisedoptical transition strengths for the distinct RDs con-sidered reaffirms that the large number of mixed an-ion local environments that occur in a randomly disor-dered GaN y As − x − y Bi x alloy has a significant impact onthe band edge eigenstates. In general, short-range al-loy disorder generates a multiplicity of band edge stateswhich are (i) spread over a relatively small range of en-ergies, and (ii) strongly localised, at different locationsin the different QW structures investigated. In a realGaN y As − x − y Bi x heterostructure the band-edge opticalemission/absorption will then consist of contributionsfrom a range of such states, having broadly similar char-acter to the e h This has allowed, e.g., quantitative prediction of theoptical gain in (In)GaN y As − y GaAs − x Bi x QW laserstructures.
As such, while full atomistic calcula-tions demonstrate a breakdown of the EFA for individ-ual eigenstates in GaN y As − x − y Bi x , and are requiredin general to understand the full details of the un-usual material properties, it is possible that EFA-basedmodels may provide an average description of the elec-tronic and optical properties which accurately describesthe measured trends in real heterostructures. However,the stronger localisation of electrons in the N-containingQWs (cf. Figs. 3(c) and 3(d)) suggest that the EFAmay overestimate the carrier spatial overlap in type-IIGaN y As − y /GaAs − x Bi x QW structures.Overall, our results indicate that short-range alloy dis-order has a marked impact on the GaN y As − x − y Bi x alloy properties, similar in nature to equivalent ef-fects in Ga(In)N y As − y and GaAs − x Bi x alloys andheterostructures. We expect that a simliar k · p -based approach to that developed for (In)GaN y As − y and (In)GaAs − x Bi x can be reliably applied to compute,analyse and optimise the properties of devices based ontype-I GaN y As − x − y Bi x heterostructures, but that de-tailed comparison to future experimental measurementswill be required to ascertain the validity of such ap-proaches when applied to type-II heterostructures basedon N- and Bi-containing alloys. IV. IMPLICATIONS FOR PRACTICALAPPLICATIONS
Having quantified key trends in the evolution of theGaN y As − x − y Bi x electronic and optical properties, weturn our attention to the consequences of the unusualmaterial band structure for device applications. As wehave demonstrated above that the 14-band k · p Hamil-tonian describes well the composition dependent bandedge energies, we employ this model to compute thedependence of the energy gaps on N and Bi composi-tion, and epitaxial strain. These calculations are sum-marised in Fig. 5, where solid blue (dashed red) linesdenote alloy compositions for which fixed band gap E g (strain (cid:15) xx ) can be achieved in pseudomorphicallystrained GaN y As − x − y Bi x /GaAs. The (cid:15) xx = 0 line de-scribes that lattice-matching to GaAs is achieved for y = 0 . x . The extremely strong reduction of E g allows long emission/absorption wavelengths – rangingfrom ∼ µ m through the near-infrared to mid-infraredwavelengths in excess of 4 µ m – to be achieved at Nand Bi compositions compatible with established epitax-ial growth. The dash-dotted green line in Fig. 5 denotes alloy com-positions for which E g = ∆ SO , so that alloys lying to theleft (right) of this contour have ∆ SO < E g (∆ SO > E g ).We recall that Auger recombination and IVBA processesinvolving the SO band play an important role in lim-iting the performance of near- and mid-infrared light-emitting devices. For processes involving the SO bandthere are three distinct cases to consider: (i) ∆ SO < E g ,(ii) ∆ SO = E g , and (iii) ∆ SO > E g . Case (i) is typi-cal of near-infrared GaAs- and InP-based light-emittingdevices operating at wavelengths between 0.9 and 1.7 µ m. In this regime CHSH Auger recombination andIVBA involving the SO band are present, with the mag-nitude of these effects increasing strongly as the differ-ence E g − ∆ SO decreases. In GaAs-based devices op-erating close to 1 µ m ( E g = 1 .
24 eV) one typically has E g − ∆ SO ≈ .
90 eV, with CHSH Auger recombinationand IVBA playing a minimal role. However, in InP-baseddevices operating at 1.55 µ m ( E g = 0 . E g − ∆ SO ≈ .
45 eV, with CHSH Auger recom-bination and IVBA assuming a dominant role in limit-ing device performance.
This trend has been clearlyobserved in temperature- and pressure-dependent experi-mental measurements performed on a range of InP-baseddevices, highlighting that the threshold current densityincreases superlinearly with emission wavelength as oneapproaches case (ii).
In case (ii), which is denoted by the closed black cir-cle in Fig. 5, the CHSH Auger recombination and IVBAprocesses are resonant and can be expected to place se-vere limitations on efficiency, thereby all but eliminatingthe potential for sustainable device operation. For case(iii) an electron-hole pair recombining across the bandgap provides insufficient energy to promote an electronfrom the SO band to a VB edge hole state, thereby sup- C o m p . s t r a i n T e n s . s t r a i n ǫ x x = ǫ x x = − . % ǫ x x = + . % µ m . µ m . µ m µ m µ m µ m E g = ∆ S O C o m p . s t r a i n T e n s . s t r a i n ǫ x x = ǫ x x = − . % ǫ x x = + . % µ m . µ m . µ m µ m µ m µ m E g = ∆ S O C o m p . s t r a i n T e n s . s t r a i n ǫ x x = ǫ x x = − . % ǫ x x = + . % µ m . µ m . µ m µ m µ m µ m E g = ∆ S O C o m p . s t r a i n T e n s . s t r a i n ǫ x x = ǫ x x = − . % ǫ x x = + . % µ m . µ m . µ m µ m µ m µ m E g = ∆ S O N c o m p o s i t i o n , y ( % ) Bi composition, x (%)GaN y As − x − y Bi x /GaAs, T = 300 K N c o m p o s i t i o n , y ( % ) Bi composition, x (%)GaN y As − x − y Bi x /GaAs, T = 300 K N c o m p o s i t i o n , y ( % ) Bi composition, x (%)GaN y As − x − y Bi x /GaAs, T = 300 K N c o m p o s i t i o n , y ( % ) Bi composition, x (%)GaN y As − x − y Bi x /GaAs, T = 300 K FIG. 5. Composition space map calculated using the 14-band k · p Hamiltonian, describing the variation of the in-plane strain ( (cid:15) xx = (cid:15) yy ) and room temperature band gap ( E g )in pseudomorphically strained GaN y As − x − y Bi x alloys grownon [001]-oriented GaAs. Solid blue and dashed red lines re-spectively denote paths in the composition space along which E g and (cid:15) xx are constant. The dashed-dotted green line de-notes alloy compositions for which E g = ∆ SO ; alloys lying tothe right of this contour have ∆ SO > E g and are predicted tohave suppressed CHSH Auger recombination and IVBA. pressing CHSH Auger recombination and IVBA by con-servation of energy. This is the case in GaSb-basedmid-infrared devices operating at wavelengths between 2and 3 µ m where, in line with the expected suppression ofCHSH Auger recombination and IVBA, reduced thresh-old current densities compared to those in equivalent InP-based near-infrared devices are observed in experiment. As such, there are two desirable scenarios from the per-spective of the relationship between E g and ∆ SO . Firstly,when ∆ SO < E g with the ratio ∆ SO E g significantly lessthan one, in which case CHSH Auger recombination andIVBA occur but at sufficiently low rates so as not to im-pede device performance. Secondly, the ideal scenarioin which ∆ SO > E g and CHSH Auger recombinationand IVBA can be expected to be suppressed. While theideal ∆ SO > E g scenario can in principle be achieved inGaAs − x Bi x ( y = 0) with E g = 1 . µ m, our calcula-tions show that incorporating N requires a reduction in x to maintain a fixed emission wavelength, and there-fore reduces ∆ SO relative to E g : an undesirable changewhich pushes the band structure from case (iii) towardscase (ii). N incorporation then quickly brings E g and∆ SO into resonance – i.e. case (ii), E g = ∆ SO – the worstpossible scenario in terms of the suitability of the bandstructure for the development of a light-emitting device.The results in Fig. 5 then indicate that N-freeGaAs − x Bi x alloys provide the ideal scenario for the real-isation of efficient light emission at 1.55 µ m, and that co-alloying N with Bi to achieve 1.55 µ m emission presentsa band structure which is not compatible with the de-3sign of light-emitting devices. As a result of the strongincrease in ∆ SO brought about by Bi incorporation, at 1.3 µ m alloys containing Bi will have ∆ SO < E g but with thedifference E g − ∆ SO being smaller than in a conventionalInP-based material. This suggests that increased lossesdue to CHSH Auger recombination and IVBA can be ex-pected in a 1.3 µ m GaAs-based GaN y As − x − y Bi x devicecompared to a conventional InP-based device operatingat the same wavelength.Additionally, we note that it is generally desirable toexploit strain in QW lasers to enhance performance viareduction of the density of states close in energy to theVB edge (in compressively strained structures), or via po-larisation selectivity (in tensile strained structures). At fixed emission wavelength one usually seeks to utilisean alloy having as large a strain | (cid:15) xx | – compressiveor tensile – as is compatible with epitaxial growth.The decrease in compressive strain compared to that inGaAs − x Bi x associated with incorporating N while main-taining a fixed emission wavelength can be expected to re-duce the performance of a QW laser structure comparedto that expected for an equivalent GaAs − x Bi x /GaAslaser structure. Therefore, the expected degradation inthe suitability of the band structure in utilising quater-nary GaN y As − x − y Bi x in favour of (i) a conventionalInP-based (Al)In − x Ga x As heterostructure to achieve1.3 µ m emission, or (ii) a ternary GaAs − x Bi x het-erostructure to achieve 1.55 µ m emission suggests, con-trary to recent analysis, that GaN y As − x − y Bi x al-loys and heterostructures containing N are not suitablefor applications at 1.3 or 1.55 µ m.Our analysis does however confirm the potentialof GaN y As − x − y Bi x alloys for applications in multi-junction solar cells, since E g ≈ y = 0 . x , x ≈ . E g − ∆ SO for such alloys would not be as favourableas in an InP-based material designed to have a sim-ilar absorption wavelength, it is expected to be suffi-ciently large so as not to impede performance. Fur-thermore, the carrier densities present in an illuminatedsolar cell are significantly lower than those in an elec-trically pumped semiconductor laser, making Auger re-combination – which scales roughly as the cube of theinjected carrier density – less of an impediment to de-vice operation. The modest N and Bi compositions re-quired to reach a band gap ∼ y As − x − y Bi x /GaAsepitaxial layers. Analysis of prototypicalGaAs − x Bi x /GaAs QW solar cells indicates that one po-tential factor limiting the photovoltaic performance oflattice-matched GaN y As − x − y Bi x junctions may be thelarge inhomogeneous spectral broadening of the bandedge optical absorption, associated with the presence ofshort-range alloy disorder and crystalline defects. How-ever, we expect that these issues could be mitigated tosome degree via a combination of refinement of the epi-taxial growth, sample preparation, and device design. This approach has been successfully employed todevelop multi-junction solar cells incorporating the dilutenitride alloy (In)GaN y As − y , which have demonstratedrecord-breaking efficiency. Finally, since our calculations indicate thatGaN y As − x − y Bi x alloys display effectively identicalenhancement of the spin-orbit coupling to that inGaAs − x Bi x , similarly large enhancement of the Rashbaspin-orbit interaction to that in GaAs − x Bi x can beexpected in GaN y As − x − y Bi x , potentially opening upapplications in spintronic devices. V. CONCLUSIONS
We have developed a multi-scale theoretical frame-work to calculate the properties of GaN y As − x − y Bi x highly-mismatched alloys and heterostructures, basedon carefully derived atomistic TB and continuum k · p Hamiltonians. We have performed a systematic in-vestigation revealing key trends in the electronic andoptical properties of bulk GaN y As − x − y Bi x alloys andGaN y As − x − y Bi x /GaAs QWs. Our analysis indicatesthat GaN y As − x − y Bi x alloys provide broad scope forband structure engineering: incorporating N (Bi) allowsto manipulate the CB (VB) structure close in energy toband edges, offering significant control over the bandgap, VB spin-orbit splitting energy, and band offsets.Since incorporating N (Bi) brings about tensile (com-pressive) strain with respect to a GaAs substrate, co-alloying N and Bi then further delivers significant controlover the strain in epitaxial layers and heterostructures,providing further opportunities to tailor the electronicand optical properties. The intrinsic flexibility of theGaN y As − x − y Bi x alloy band structure is therefore par-ticularly appealing for practical applications.Through systematic analysis of large-scale atomisticelectronic structure calculations we demonstrated thatthe respective impact of N and Bi incorporation onthe CB and VB structure remain effectively indepen-dent, even in the presence of significant short-range al-loy disorder. Comparison of atomistic and continuumcalculations highlights that a 14-band k · p Hamiltonianis sufficient to describe the evolution of the main fea-tures of the GaN y As − x − y Bi x band structure, with thepredicted evolution of the band gap and VB spin-orbitsplitting energy in good agreement with experimentalmeasurements. Applying the TB model to computethe electronic and optical properties of realistically sizedGaN y As − x − y Bi x /GaAs QWs demonstrates that short-range alloy disorder produces strong carrier localisation,ultimately leading to significant inhomogeneous spectralbroadening as well as a breakdown of the conventional se-lection rules governing QW optical transitions. On thisbasis we conclude that alloy disorder effects are likelyto play an important role in determining the proper-ties of real GaN y As − x − y Bi x alloys and heterostructures.While our analysis suggests that k · p -based models in-4corporating appropriately parametrised inhomogeneousspectral broadening are likely to be suitable when ap-plied to type-I heterostructures, further detailed anal-ysis of N- and Bi-containing type-II heterostructures –e.g. GaN y As − y /GaAs − x Bi x QWs – may require a the-oretical description based upon direct atomistic calcula-tions.Based on our analysis of the electronic structurewe discussed implications of the unusual properties ofGaN y As − x − y Bi x alloys for practical applications. Con-trary to the existing literature our analysis suggests thatGaN y As − x − y Bi x heterostructures are not suitable forthe development of GaAs-based semiconductor lasers op-erating at 1.3 or 1.55 µ m: the reduction in compres-sive strain and spin-orbit splitting energy compared toequivalent N-free GaAs − x Bi x structures is expected tocompromise the proposed benefits of Bi incorporationand hence place significant limitations on performance.Overall, we conclude that the features of the electronicstructure revealed through our analysis indicate thatGaN y As − x − y Bi x alloys are most promising as a suit-able ∼ ACKNOWLEDGEMENTS
M.U. and C.A.B. contributed equally to this work.This work was supported by the European Commission(project no. FP7-257974), by Science Foundation Ire-land (SFI; project no. 15/IA/3082), and by the Engineer-ing and Physical Sciences Research Council, U.K. (EP-SRC; project no. EP/K029665/1). M.U. acknowledgesthe use of computational resources from the NationalScience Foundation (NSF, U.S.A.) funded Network forComputational Nanotechnology (NCN) through http://nanohub.org . The authors thank Dr. Zoe L. Bushelland Prof. Stephen J. Sweeney of the University of Surrey,U.K., for useful discussions and for providing access tothe results of their experimental measurements prior topublication.5
Supplementary Material Section
Section S1: Theoretical Models
The unusual electronic properties of dilute nitrideand bismide alloys derive from the fact that, whenincorporated in dilute concentrations, N and Bi act asisovalent impurities which strongly perturb the bandstructure of the host matrix semiconductor.
Asa result, the details of electronic structure are stronglyinfluenced, even at dilute compositions, by short-rangealloy disorder – i.e. the formation of pairs andlarger clusters of N and Bi atoms sharing common cationnearest neighbours – meaning that highly-mismatchedalloys pose significant challenges for theory. Firstly,due to the prominence of impurity effects conventionalapproaches to analyse alloy band structures, such as thevirtual crystal approximation, break down. Secondly,since the effects of Bi and N incorporation are prominentat dilute compositions and in the presence of alloydisorder, quantitative understanding of the propertiesof real materials must be built on analysis of systemscontaining upwards of thousands of atoms: theminimum N or Bi composition that can be consideredusing a 2 M -atom Ga M As M − X (X = N, Bi) supercellis M − . For example, to consider an alloy compositionof 0 .
1% then requires a supercell containing a minimumof 2000 atoms ( M − = 10 − ) and, since a 2000-atomsupercell corresponds only to a single substitutionalimpurity, there is no scope to investigate disorder effectsunless the system size is significantly increased.These features generally place the study of highly-mismatched alloys beyond the reach of contemporaryfirst principles methods, since sufficiently advancedapproaches cannot be applied directly to such largesystems due to their associated computational expense.This then mandates the development of alternativetheoretical approaches. Indeed, while first principlesanalyses have provided valuable insights and informa-tion, detailed understanding of the properties of realdilute nitride and bismide alloys has to date primarilybeen built via the development and application ofappropriate semi-empirical pseudopotential and tight-binding (TB) models. Furtheranalysis has shown that simple continuum models –derived and parametrised on the basis of either (i)experimental measurements, or (ii) electronic struc-ture calculations – provide an effective means tounderstand and analyse the main features of the bandstructure of N- and Bi-containing alloys, by describinggeneral trends in important material parameters (despiteomitting the detailed features of the electronic struc-ture). Here, in the following two sections, we present twomodels of the electronic structure of GaN y As − x − y Bi x alloys: (i) an atomistic sp s ∗ TB model, and (ii) acontinuum, extended basis set 14-band k · p Hamiltonian.
Section S1.A: Atomistic Tight-binding Model
Since the TB method employs a basis set of lo-calised atomic orbitals, it is ideally suited to probethe electronic structure of localised impurities. This,combined with its low computational cost, means thatappropriately parametrised TB models provide a physi-cally transparent and highly effective means by which tosystematically analyse the properties of large, disorderedsystems. We have previously demonstrated that theTB method provides a detailed understanding of theelectronic and optical properties of GaN y As − y andGaAs − x Bi x alloys, and that the results of calculationsbased on this approach are in quantitative agreementwith a wide range of experimental data. Here,we extend this approach to quaternary GaN y As − x − y Bi x alloys.To accurately describe states lying close in energy tothe VB and CB edges we employ an sp s ∗ basis set, thereby allowing the dispersion of the lowest CB to bedescribed throughout the entire Brillouin zone withoutexplicit introduction of d orbitals, which avoids signifi-cant parametric complexity. Our TB parameters for theband structures of the GaN, GaAs and GaBi compoundsare obtained on the basis of first principles band structurecalculations. Spin-orbit coupling between p orbitalsis included explicitly in the model, in order to accountfor the known, strong relativistic effects associated withBi incorporation in (In)GaAs. To construct alloy su-percells we replace As atoms by N or Bi atoms at selectedpositions on the anion sublattice. To account for the im-portant effects of lattice relaxation about N andBi lattice sites, the relaxed atomic positions in a givensupercell are determined by minimising the total elasticenergy using a valence force field (VFF) model, basedon the Keating potential and including anharmonic cor-rections to account accurately for the large local strainsarising due to the significant differences in the covalentradii of N, As and Bi.
In our TB model the atomic orbital energies for a givenatom are taken to depend on the local neighbour envi-ronment by (i) averaging over the orbital energies of thecompounds formed by the atom and each of its nearestneighbours, and (ii) renormalising the nearest-neighbouraveraged orbital energy obtained in this manner to ac-count firstly for the mixed-anion local environment (seenby a cation having nearest neighbour anions of differentatomic species) and, secondly, for the non-local natureof the lattice relaxation brought about by the aforemen-6tioned large differences in the anion covalent radii. Inthis approach the energy of orbital α (= s, p x , p y , p z , s ∗ ) in the atom located at lattice site n (= 1 , . . . , M ) isgiven by E nα = 14 (cid:88) j (cid:20) ∆ E VB ( nj ) + E α ( nj ) (cid:124) (cid:123)(cid:122) (cid:125) Nearest-neighbour bond + (cid:88) k (cid:54) = j (cid:18) ∆ E α ∆ d (cid:19) njk (cid:18) d ( jk ) − d ( jk ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Mixed anion local environment + (cid:80) X M X (cid:16) ∆ E α ∆ d (cid:17) GaAsX (cid:80) X M X (cid:18) d ( nj ) − d ( nj ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Non-local lattice relaxation (cid:21) , (S1)where X = N or Bi, and the sum runs over the fournearest neighbours j of atom n . E α ( nj ), ∆ E VB ( nj ) and d ( nj ) respectively denote the energy of orbital α , theVB offset, and the unstrained nearest-neighbour bondlength of the compound formed by atoms n and j . d ( nj )is the relaxed bond length between atoms n and j in thesupercell, M X is the total number of N or Bi atoms, and (cid:80) X M X is the total number of N and Bi atoms. Themagnitude of the second and third terms in Eq. (S1) isdetermined by (cid:18) ∆ E α ∆ d (cid:19) njk = E α ( nj ) − E α ( nk ) d ( nj ) − d ( nk ) , (S2)which is the ratio of the difference between the energy oforbital α to that between the equilibrium bond lengths,for the compounds formed by atom n and its nearestneighbours j and k (with E α assumed here to include thecorresponding VB offset ∆ E VB for a given compound).The first term in Eq. (S1) describes conventional aver-aging of the orbital energy over nearest-neighbours. Thesecond and third terms in Eq. (S1) describe a renormal-isation of the orbital energy, and respectively accountfor (i) the large, bond length dependent differences be-tween the orbital energies in the compounds formed bya given atom and nearest neighbours of differing atomicspecies, and (ii) the relaxation of the crystal lattice inresponse to substitutional incorporation of an impurityX atom. Previous analysis has shown that it is perti-nent to include an orbital energy renormalisation of thisform to account for the fact that substitutional N andBi atoms have significant differences in electronegativ-ity and covalent radii compared to the As atoms theyreplace, meaning that they perturb the structural andelectronic properties much more severely than in conven-tional III-V alloys.The nature of the orbital energy renormalisation is bestunderstood by considering its action in the presence of anisolated substitutional impurity. In a Ga M As M − X su-percell a cation (Ga atom) having the X atom as a nearestneighbour sits in a mixed-anion nearest-neighbour envi-ronment (formed by the X atom and three As atoms).Due to relaxation of the crystal lattice, the length of TABLE S1. Band structure parameters for the con-stituent compounds in the sp s ∗ tight-binding model ofGaN y As − x − y Bi x , including the valence band offsets ∆ E VB ,orbital energies E α ( α = s, p and s ∗ ), inter-atomic interactionmatrix elements V δ ( δ = ssσ , ppσ , ppπ , s c p a σ , s a p c σ , s ∗ c p a σ and s ∗ a p c σ ), and renormalised atomic spin-orbit splitting en-ergies ∆ c,a . All parameters are given in units of eV.Parameter GaN GaAs GaBi∆ E VB − .
28 0 .
00 1 . E s c − . − . − . E s a − . − . − . E p c . . . E p a − . . − . E s ∗ c . . . E s ∗ a . . . V ssσ − . − . − . V ppσ . . . V ppπ − . − . − . V s c p a σ . . . V s a p c σ . . . V s ∗ c p a σ . . . V s ∗ a p c σ . . . c . . . a . . . η δ ( δ = ssσ , ppσ , ppπ , s c p a σ , s a p c σ , s ∗ c p a σ and s ∗ a p c σ ) used to rescale theinter-atomic interaction matrix elements V δ for the con-stituent compounds in the sp s ∗ tight-binding model ofGaN y As − x − y Bi x (cf. Eq. (S3)).Parameter GaN GaAs GaBi η ssσ .
030 3 .
512 3 . η ppσ .
090 3 .
204 2 . η ppπ .
720 4 .
326 3 . η s c p a σ .
500 4 .
500 4 . η s a p c σ .
000 4 .
100 4 . η s ∗ c p a σ .
200 7 .
200 7 . η s ∗ a p c σ .
000 4 .
200 4 . the Ga-X bond is smaller (larger) than the three Ga-7 TABLE S3. Unstrained bond lengths ( d ), and bond stretch-ing ( α ) and bond-angle bending ( β ) force constants for theconstituent compounds in the valence force field potentialused to relax the atomic positions in GaN y As − x − y Bi x su-percells.Parameter GaN GaAs GaBi d (˚A) 1 .
944 2 .
448 2 . α (N m − ) 158 .
00 41 .
18 61 . β (N m − ) 14 .
66 8 .
95 6 . As nearest-neighbour bonds when X = N (Bi).
Thesecond term in Eq. (S1) proceeds in a pairwise manner,adjusting the contribution of a given nearest-neighbourbond to the Ga orbital energy by accounting for the largedifferences in Ga orbital energies between GaAs and GaX– i.e. between those in compounds formed by the Gaatom and its X or As nearest neighbours. For a givennearest-neighbour bond the magnitude of the renormal-isation term is determined by the multiplicative factor d ( jk ) − d ( jk ), which is non-zero only when the relaxedbond length differs from d ( jk ), thus describing a sim-ple linear variation of the Ga orbital energy with bondlength in going from GaAs to GaX.The relaxation of the crystal lattice about the Xatomic site generally propagates through the crystal ona length scale which exceeds typical nearest-neighbourbond lengths. However, by definition the action of thesecond term in Eq. (S1) is explicitly limited to nearestneighbours: the As second-nearest neighbours of the Xatom in a Ga M As M − X supercell ( M X = (cid:80) X M X = 1)have four Ga nearest neighbours, in which case the fac-tor defined by Eq. (S2) vanishes, despite that the relaxedsecond-nearest neighbour bond lengths will be differentfrom those in unstrained GaAs. This non-local pertur-bation of the crystal structure is taken into account bythe third term in Eq. (S1), which renormalises the or-bital energy via a weighted average of the ratio definedin Eq. (S2) over all substitutional impurities in the su-percell. The magnitude of this correction is again de-termined by the difference between the relaxed and un- strained nearest-neighbour bond lengths, meaning that(i) it applies at all atomic sites that have been affectedby the lattice relaxation, and (ii) its magnitude decreasesas the distance from the X atomic site increases, in linewith the magnitude of the lattice relaxation.This orbital energy renormalisation therefore providesa suitable manner by which to explicitly incorporatethe non-local nature of the perturbation to the supercellHamiltonian – associated with N and/or Bi incorporation– into what is implicitly a nearest-neighbour model. Also,since this approach relies only on the known atomic or-bital energies and nearest-neighbour bond lengths, it hasthe benefit of circumventing the need to introduce ad-ditional parameters to describe these important beyond-nearest-neighbour effects.The inter-atomic interaction matrix elements betweena given pair of nearest neighbour atoms are computed inthe model by (i) using the two-centre integrals of Slaterand Koster to take into account changes in bond angle,and (ii) incorporating a bond length dependent rescalingto account for the difference between the relaxed bondlength between a pair of neighbouring atoms in the su-percell, and the equilibrium bond length of the compoundformed by the same two atoms. For each distinct type ofinteraction, the bond length dependent rescaling of theinteraction matrix element between nearest neighbours n and j is given by V δ ( d ( nj )) = V δ ( nj ) (cid:18) d ( nj ) d ( nj ) (cid:19) η δ , (S3)where V δ ( nj ) is the corresponding interaction matrix el-ement in the unstrained compound formed by atoms n and j , and δ = ssσ , ppσ , ppπ , s c p a σ , s a p c σ , s ∗ c p a σ or s ∗ a p c σ describes the symmetry of the interaction (with c and a denoting cations and anions, respectively). Theexponent η δ takes a distinct value for each δ – the full setare determined for each compound by fitting to the hy-drostatic deformation potentials obtained from first prin-ciples calculations. In the TB method we compute the momentum ma-trix element between the supercell eigenstates | n k σ (cid:105) and | m k σ (cid:48) (cid:105) , at wave vector k and having spins σ and σ (cid:48) , inthe position basis as P ( (cid:98) e ) nm ( k ) = − im (cid:126) (cid:88) σ,σ (cid:48) (cid:104) n k σ | (cid:104) (cid:92) ( (cid:98) e · r ) , (cid:98) H (cid:105) | m k σ (cid:48) (cid:105) = − im (cid:126) (cid:88) σ,σ (cid:48) (cid:88) p,q (cid:88) α,β a ∗ npασ ( k ) a mqβσ (cid:48) ( k ) ( (cid:98) e · r p − (cid:98) e · r q ) (cid:104) pασ | (cid:98) H | qβσ (cid:48) (cid:105) , (S4)where p and q index the atomic sites in the supercell, α and β index the atomic orbitals localised at each atomicsite, and (cid:98) e is a unit vector defining the direction alongwhich the corresponding emitted/absorbed photon is po- larised. The final expression for P ( (cid:98) e ) nm ( k ) is obtainedusing the spectral resolution of the position operator, (cid:92) ( (cid:98) e · r ) = (cid:80) p,α,σ ( (cid:98) e · r p ) | pασ (cid:105)(cid:104) pασ | , in addition to assum-8ing that there is no overlap between basis states localisedon different lattice sites, (cid:104) pασ | qβσ (cid:48) (cid:105) = δ pq δ αβ δ σσ (cid:48) (i.e. anorthogonal TB model).The TB model was implemented within the frameworkof the NEMO 3-D NanoElectronic MOdeling software,which was used to carry out the supercell calculations. Section S1.B: B. 14-band k · p Hamiltonian Previous analysis has shown that it is possible and usefulto derive simple continuum models that describe the per-turbed band structure of GaN y As − y and GaAs − x Bi x alloys. Phenomenological approaches, principallythe band-anticrossing (BAC) model, have originatedfrom interpretation of spectroscopic data and atomisticelectronic structure calculations, and are widely em-ployed as a straightforward and efficient means by whichto describe the evolution of the main features of theband structure (principally the band gap and band edgeenergies) with alloy composition, both in bulk materialsand in heterostructures.For GaN y As − y alloys it is well established that theevolution of the CB structure can be described by a sim-ple 2-band BAC model, in which the extended states ofthe GaAs host matrix CB edge undergo a composition de-pendent repulsive interaction (of magnitude β N √ y , where β N is the BAC coupling strength and y is the N com-position) with a set of localised states associated withsubstitutional N impurities (having energy E N ). In thecase where the N-related localised states lie energeti-cally within the host matrix CB – as is the case in(In)GaN y As − y – the composition dependence of theBAC interaction between these two sets of states re-sults in a strong reduction of the alloy CB edge energywith increasing y . It generally established that simi-lar behaviour is present in GaAs − x Bi x : the strong re-duction (increase) and composition-dependent bowing ofthe band gap (VB spin-orbit-splitting energy) can be de-scribed in terms of a valence band-anticrossing (VBAC)interaction between the extended states of the GaAs VBedge, and localised impurity states associated with sub-stitutional Bi impurities, which pushes the alloy VB edgeupwards in energy with increasing x . Detailed analy-sis has demonstrated that (i) incorporation of an iso-lated, substitutional N (Bi) impurity in GaAs leads tothe formation of A ( T ) symmetric localised impuritystates, which lie energetically within – i.e. are resonantwith – the GaAs CB (VB), and (ii) an appropriate 10-band (12-band) k · p Hamiltonian can be derived usingan extended basis set which includes these s -like ( p -like)localised states, in addition to the conventional 8-bandbasis defined by the GaAs zone-centre Bloch states asso-ciated with the lowest energy CB, and light-hole (LH),heavy-hole (HH), and spin-split-off (SO) hole VBs. De-spite their simplicity, these extended k · p models haveproven to be highly effective tools for the analysis of realmaterials and devices. For example, theoretical mod-els based on the 10- and 12-band models have provided significant insight into the electronic and optical prop-erties of quantum well (QW) heterostructures, and havebeen used as a basis for quantitative prediction and de-sign of key properties of electrically pumped semiconduc-tor lasers, including the optical gain and carrier recom-bination rates. However, formally, these (V)BAC-based models suf-fer from two fundamental limitations. Firstly, they con-sider the impact on the band structure of isolated, non-interacting impurities only : they are hence formally ap-plicable only to ordered alloys, thereby limiting theirability to predict the properties of real materials (whichinevitably contain some degree of alloy disorder). Sec-ondly, uncertainty surrounding the (V)BAC parameterscan lead to parametric ambiguity which further reducesthe predictive capability. In practice this first issue isnot a significant limitation, since it is typically found that(V)BAC models provide reasonably accurate descriptionsof the evolution of band properties which are not partic-ularly sensitive to the presence of short-range alloy dis-order. Instead, one must be careful in the interpretationof the results of such calculations, in the knowledge thatthe model will describe only band properties which arenot strongly effected by N and/or Bi clustering – e.g. theband gap and band edge energies, but not the effectivemasses or Land´e g factors. This second issue can, incertain circumstances, represent an impediment to thedevelopment and application of BAC models due to aninability to unambiguously determine a consistent set ofband parameters (of which there can be many) from agiven set of experimental or theoretical band structuredata (which are typically few in number). We havecircumvented this problem in general by developing aTB-based approach that allows the matrix elementsof the alloy Hamiltonian to be directly calculated andparametrised in a chosen basis of crystal eigenstates andwhich – via construction of the localised impurity statesassociated with substitutional N and/or Bi incorporation– allows explicit evaluation of the various contributionsto each matrix element – including direct computation ofthe (V)BAC parameters as well as virtual crystal (con-ventional alloy) contributions to the band edge energies– without the usual requirement to undertake post hocfitting to the results of experimental measurements onGaN y As − y or GaAs − x Bi x alloys.Turning our attention to GaN y As − x − y Bi x alloys, theanalysis summarised above suggests that a suitable ba-sis set must contain a minimum of 14 bands: the spin-degenerate CB, LH, HH and SO bands of the GaAshost matrix (8 bands), the A -symmetric N-related lo-calised states (of which there is one spin-degenerate set; 2bands), and the T -symmetric Bi-related localised states(of which there are two spin-degenerate sets; 4 bands).TB calculations on ordered GaN y As − x − y Bi x supercellsindicate that the respective impact of N and Bi on theCB and VB structure are decoupled from one another.As such, the GaN y As − x − y Bi x band structure then ad-mits a simple interpretation in terms of perturbation of9the CB and VB separately by N- and Bi-related localisedstates, respectively. As we demonstrate in Sec. III B ofthe main paper, the 14-band model defined in this man-ner – and parametrised directly from TB calculations –provides a simple and predictive means by which to de-scribe the band edge energies GaN y As − x − y Bi x alloys,even in the presence of significant alloy disorder. Detailsof the derivation of the 14-band k · p model can be foundin Ref. 38.Here, we focus on the calculation of the band edge en- ergies in pseudomorphically strained GaN y As − x − y Bi x alloys and QWs using the 14-band k · p Hamiltonian. Atthe centre of the Brillouin zone ( k = 0), and in the pres-ence of pseudomorphic strain corresponding to growthalong the (001) direction, the 14-band Hamiltonian blockdiagonalises into decoupled sub-matrices describing theCB, HH, and LH and SO states. There are six suchmatrices in total, three of which are distinct as a resultof spin degeneracy. The band edge energies are thengiven by the eigenvalues of the following spin-degeneratematrices H CB = (cid:32) E (0) g + ∆ E N + δE hy N β N √ yβ N √ y E (0) g + δE VCCB + δE hy CB (cid:33) , (S5) H HH = (cid:18) δE VCVB + δE hy VB − δE ax VB β Bi √ xβ Bi √ x ∆ E Bi + δE hy Bi − δE ax Bi (cid:19) , (S6) H LH,SO = δE VCVB + δE hy VB + δE ax VB −√ δE ax VB β Bi √ x −√ δE ax VB − ∆ (0) SO + δE VCSO + δE hy VB β Bi √ x E Bi + δE hy Bi + δE ax Bi , (S7)where E (0) g and ∆ (0) SO are the band gap and VB spin-orbitsplitting energy of the GaAs host matrix, and the zero ofenergy has been set at the (unperturbed) GaAs VB edge.These matrices can be diagonalised to provide analyticalexpressions for the GaN y As − x − y Bi x band edge energies:the lower eigenvalue of Eq. (S5) is the alloy CB edgeenergy, the upper eigenvalue of Eq. (S6) is the alloy HHband edge energy, and the highest and lowest eigenvaluesof Eq. (S7) are, respectively, the alloy LH and SO bandedge energies.In Eqs. (S5) – (S7) ∆ E N (∆ E Bi ) denotes the energy ofthe N- (Bi-) related localised states relative to the un-perturbed GaAs CB (VB) edge: ∆ E N > <
0) cor-responds to a resonant (bound) N-related localised statelying energetically within the GaAs CB (band gap), while∆ E Bi < >
0) corresponds to a resonant (bound) Bi-related localised state lying energetically within the GaAsVB (band gap). The virtual crystal (conventional al-loy) shifts to the CB, VB and SO band edge energies –which, by definition, are linear in the N and Bi composi-tions – are respectively defined as δE VCCB = − α Bi x − α N y , δE VCVB = κ Bi x + κ N y and δE VCSO = − γ Bi x − γ N y . The energy shifts due to the hydrostatic and axial com-ponents of the strain are given respectively by δE hy i = a i ( (cid:15) xx + (cid:15) yy + (cid:15) zz ) and δE ax i = − b i ( (cid:15) xx + (cid:15) yy − (cid:15) zz ),where i = CB, VB, N or Bi denotes the hydrostaticand axial deformation potentials a i and b i associated re-spectively with the GaAs host matrix CB and VB edges,and with the N- and Bi-related localised states. Wenote that (i) for pseudomorphic strain, (cid:15) xx = (cid:15) yy and (cid:15) zz = − C C (cid:15) xx , and (ii) by symmetry, b N = 0 (since theaxial component of the strain has no effect on the purely TABLE S4. N- and Bi-related parameters for the 14-band Hamiltonian of GaN y As − x − y Bi x , obtained on the ba-sis of the tight-binding supercell calculations. In GaAs − x Bi x (GaN y As − y ) ∆ E is given relative to the unperturbed GaAsvalence (conduction) band edge. Due to their A symmetry,the energy of the localised states associated with isolated sub-stitutional N impurities is not affected by axial strain: thereis no associated axial deformation potential b N .Parameter GaAs − x Bi x GaN y As − y ∆ E (eV) − .
183 0 . α (eV) 2 .
63 1 . β (eV) 1 .
13 2 . γ (eV) 0 . − . κ (eV) 1 .
01 1 . a (eV) − .
11 0 . b (eV) − .
71 —– s -like N-related localised states). The components of thestrain tensor are determined via the mismatch betweenthe lattice constants a ( x, y ) and a S of GaN y As − x − y Bi x and the GaAs substrate: (cid:15) xx = a S − a ( x,y ) a S . The latticeand elastic constants, and CB and VB edge deforma-tion potentials, are determined for GaN y As − x − y Bi x byinterpolating linearly between those of GaN, GaAs andGaBi. The N- and Bi-related parameters of the14-band model, derived on the basis of TB supercellcalculations, are provided in Table S4.For the analysis of the electronic and optical proper-ties of QWs, the 14-band Hamiltonian is solved directlyin the envelope function approximation (EFA) for eachQW heterostructure using a numerically efficient recipro-0cal space plane wave approach. The QW band structurecalculations are carried out in the axial approximation.In the plane wave approach the momentum matrix el-ements P ( (cid:98) e ) nm ( k (cid:107) ) – between the QW eigenstates | n k (cid:107) σ (cid:105) and | m k (cid:107) σ (cid:105) at in-plane wave vector k (cid:107) – are computedin reciprocal space using the general formulation due toSzmulowicz P ( (cid:98) e ) nm ( k (cid:107) ) = m (cid:126) (cid:88) σ,σ (cid:48) (cid:104) n k (cid:107) σ | (cid:98) e · ∇ k (cid:98) H | m k (cid:107) σ (cid:48) (cid:105) , (S8)where (cid:98) e · ∇ k (cid:98) H is the operator obtained by (i) takingthe directional derivative of the matrix elements of thebulk k · p Hamiltonian with respect to k along (cid:98) e , and(ii) symmetrising with respect to position-dependentmaterial parameters and quantising k z in the usualmanner. Using Eqs. (S4) and (S8) we compute theoptical transition strength (in units of energy) directlyfor each structure in terms of the zone-centre momentummatrix element as m (cid:126) | P ( (cid:98) e ) nm (0) | , where (cid:98) e = (cid:98) x and (cid:98) z for transverse electric- (TE-) and transverse magnetic-(TM-) polarised transitions, respectively. We notethat this approach to calculating the optical transitionstrengths directly employs the supercell Hamiltonianand computed eigenstates for a given structure, meaningthat the full effects of N- and Bi-induced hybridisationare explicitly accounted for in the analysis of the opticalproperties. Section S2: Dilute doping limit: impact ofco-alloying N and Bi on the GaAs electronicstructure
The interactions of individual substitutional N andBi atoms with the host matrix states in the GaN y As − y and GaAs − x Bi x alloys, respectively, have been wellstudied in the literature. An isolated N atomintroduces a resonant impurity state above the CB edgein GaAs , while an isolated Bi atom introduces a im-purity state lying below the GaAs VB edge in energy. However, no quantitative analysis has been undertakento date to quantify the extent of any interaction betweenthe localised impurity states associated with N and Biwhen both are incorporated substitutionally into GaAs.Since their individual behaviours are characterized byBAC interactions in the CB and VB, it is pertinent toinvestigate the degree to which this behaviour remainsintact in the quaternary GaN y As − x − y Bi x alloy. Here,we systematically analyse the interaction between theselocalised states by placing single substitutionaly N andBi atoms inside a large 4096-atom Ga N As Bi supercell, and probe the interaction as a function ofthe spatial separation of these impurities. We beginby placing the N and Bi atoms sufficiently far apartthat the interaction between their associated localisedimpurity states is minimal, and then gradually bringthem closer together, finally studying the case wherethe N and Bi atoms are located at adjacent sites on the anion sublattice (i.e. so that they are second-nearestneighbours, sharing a common Ga nearest neighbour).This latter case is that of maximum spatial overlapbetween the N- and Bi-related localised impurity states,and hence represents the case in which the interactionbetween these states would be maximised in a substitu-tional GaN y As − x − y Bi x alloy. As described in the mainpaper, we analyse the character of this interaction bycomputing the fractional GaAs Γ character of the alloyCB and VB edge states in each supercell.Figure S1 contains schematic illustrations of the dif-ferent local neighbour environments considered in thisanalysis. The relative positions of the N and Bi atoms ineach case are defined in Sec. III A of the main text. Thedistinct local neighbour environments considered are (a)an unperturbed GaAs supercell, as well as a GaAs su-percell containing (b) a single N impurity, (c) a single Biimpurity, and GaAs supercells in which a single Bi im-purity is placed with respect to the N atom at the (d)third-closest, (e) second-closest, and (f) closest sites onthe anion sub-lattice. This final case is that referred toabove, whereby the N and Bi atoms are second-nearestneighbours sharing a common Ga nearest neighbour. Thelabelling scheme used to denote these distinct local neigh-bour environments is defined in Sec. III A of the mainpaper.Figure S2 shows the fractional GaAs CB and VBedge Γ character G Γ ( E ) spectra for the ternaryGa N As and Ga As Bi supercells. Thesespectra clearly indicate hybridisation between the GaAsCB (VB) edge states and a resonant state associatedwith the substitutional N (Bi) atoms, lying energeticallywithin the CB (VB) of the GaAs host matrix semiconduc-tor. This hybridisation is the signature of the (V)BACinteractions described in Sec. S1 above, and is consistentwith the established trends described in the referencesprovided therein. For the Ga N As in Fig. S2 (a)we calculate the the alloy CB edge state has 87.4% GaAsCB edge character and is shifted downwards in energy by19 meV with respect to the unperturbed GaAs CB edge.Almost the entirety of the remainder of the GaAs CBedge character is calculated to reside on a state lying atan energy of approximately 1.64 eV, and which is highlylocalised about the N atomic site. We compute that the Nlocalised resonant state if of energy 1.62 eV in this super-cell – i.e. lying approximately 190 meV above the roomtemperature GaAs CB edge. Figure S2 (b) demonstratesthat the VB edge states in Ga N As are virtuallyunchanged compared to those in a pure Ga As su-percell, with the alloy VB edge states having > . As Bi supercell we ob-serve complementary behaviour: we find that the ma-jority ( > ∼
100 meVbelow the unperturbed GaAs VB edge in energy, withfurther smaller amounts projected onto lower lying VBstates (off scale in Fig. S2 (d)). Again, this hybridisa-tion describes the presence of BAC behaviour, this timeresulting from coupling between the VB edge states ofthe GaAs host matrix, and impurity states that are reso-nant with the GaAs VB and strongly localised about theBi atomic site. Fig. S3 (c) highlights that the calculatedoverlap between the alloy CB edge state and the CB edgestate of unperturbed GaAs is > . N As Bi supercells containing both N andBi, and analyse the changes in the calculated fractionalGaAs CB and VB edge Γ character brought aboutby co-alloying N and Bi. Table I in the main textprovides the band edge energies calculated for each ofthese supercells. Examining the fractional GaAs CBedge Γ character Figs. S3 (a), (c) and (e), we note thatthe trends observed in the presence of Bi remain rela-tively unchanged from those calculated for the ternaryGa N As supercell. This indicates that theimpact of N on the CB structure in GaN y As − x − y Bi x remains close to that in the ternary GaN y As − y alloywhen Bi is incorporated, providing further confirmationof the trends identified in the main paper: Bi incorpo-ration produces only minor quantitative changes to thenature of the CB structure, with the overall character re-maining qualitatively unchanged, even in the case whensubstitutional N and Bi atoms share common Ga nearestneighbours. Turning our attention to the VB structure,we again note the same general trend: the overallcharacter of the alloy VB edge states is qualitativelysimilar to that in the ternary GaAs − x Bi x alloy, withrelatively minor quantitative changed brought aboutby co-alloying with N. The primary feature revealed bythe calculated fractional GaAs VB edge Γ character inFigs. S3 (b), (d) and (f) is that the presence of N tendsto lift degeneracy of the LH- and HH-like VB edge states.This effect is associated with the fact that the differentlocal neighbour environments containing both N andBi represent disorder alloy microstructure in which thetranslational symmetry of the crystal lattice is broken,leading to a breakdown of the T d point group symmetryof the GaAs lattice. This lack of isotropy in the localcrystal structure, which is compounded by the largelocal relaxation of the crystal lattice about complexes ofspatially proximate N and Bi atoms, is then manifestedin the lifting of the degeneracy of the p -like alloy VBedge states due to the associated non-equivalence of the[001], [010] and [001] crystal directions. This lifting ofthe VB edge energy is qualitatively identical to that we have calculated previously in disordered GaAs − x Bi x alloys, but in quantitative terms tends to be larger inmagnitude in GaN y As − x − y Bi x due to the aforemen-tioned large local relaxation of the crystal lattice in caseswhere N and Bi atoms are located close to one another. Section S3: Electron and hole probability densityplots for GaBiAs/GaAs QWs:
Here we consider the electronic structure of the N-free GaAs − x Bi x /GaAs QWs having x = 6 .
25 and 9%(structures 1 and 2 of Secs. III C and III D in the mainpaper). Figures S5 (a) and (b) respectively show theprobability density along the (001) direction, associatedwith the lowest energy electron state e h z along (001) by summing over the probabilitydensity associated with all atoms in the plane perpen-dicular to (001). The plots from k · p model are shownin the Fig. 3 of the main text. Comparison of the TBand k · p calculations suggest that the electron states inGaAs − x Bi x /GaAs QWs are broadly similar in natureto those in conventional QWs – e.g. In x Ga − x As/GaAs– since the electron eigenstates are (i) well describedin terms of envelope functions which vary slowly andsmoothly with position, (ii) effectively insensitive to thepresence of underlying short-range alloy disorder, and(iii) relatively insensitive to changes in Bi composition(in this case, from x = 6 .
25 to 9% between structures 1and 2). This is consistent with the expected behaviourfor GaAs − x Bi x alloys and heterostructures: Bi incor-poration strongly perturbs the VB while leaving the CBrelatively unaffected, with the evolution of the latterreadily captured by conventional alloy descriptions. Wenote also that in all supercells considered e − x Bi x layer, reflecting the appreciabletype-I GaAs − x Bi x /GaAs CB offset ∆ E CB . This is confirmed by the calculated increase in the e E CB ( ∼ − α Bi x + δE hyCB )brought about by increasing the Bi composition fromthat in structure 1.By contrast, the h k · p method in the EFA. Whilethe h e e
1, which is spread across the2extent of the QW, for h − x Bi x alloy. As described above, these pairs and clusters create afull distribution of localised states lying close in energyto the VB edge, which then hybridise strongly withextended GaAs VB states leading to a multiplicity ofVB edge states having significant localised character(low Bloch character). Further analysis reveals thatthis strong spatial localisation of the hole eigenstatestypically occurs about the largest Bi clusters present ina given alloy (reflected here in the sense that a largeTB calculated probability density reflects the presenceof Bi pairs and/or clusters in the plane at that valueof z ). The precise distribution of Bi-related localisedstates depends closely on the precise short-range alloydisorder present in the GaAs − x Bi x layers in these QWstructures, thereby accounting for the observed strongvariation of the localisation of the h h k · p model in the EFA (shown in Fig. 4 of the mainpaper), we conclude that (i) the strongly perturbativeimpact of Bi incorporation and alloy disorder on the VBstructure in GaAs − x Bi x leads to a strong breakdown ofthe envelope function description of the hole eigenstatesin GaAs − x Bi x /GaAs heterostructures, and (ii) thisbreakdown of the EFA for hole eigenstates is a genericfeature of the electronic properties GaAs − x Bi x /GaAsin the sense that it is intrinsic and effectively insensitiveto the Bi composition. Section S5: Probability densities inGaN y As − x − y Bi x /GaAs quantum wells In the main paper we present probabil-ity densities calculated for the same 8 nmthick GaN . As . Bi . /GaAs andGaN . As . Bi . /GaAs QWs (structures 3 and4, respectively) using the TB method, for selectedsupercells containing distinct RDs of N and Bi atomssubstituted on the anion sublattice in the well region(cf. Fig. S4). Figures S6 (a) and (b) respectively presentprobability densities calculated for these two QWs, forseveral additional RDs in each case. The line types areas in Fig. 2 of the main paper. The calculated trendsare qualitatively similar here to those discussed for theother RDs in the main paper and, as such, we refer thereader to Sec. III C of the main paper for a discussion ofthe associated trends.3 FIG. S1. Schematic illustration of a single, simple cubic 8-atom GaAs unit cell within the 4096 atom supercells considered. Nand Bi atoms are substituted at specific As sites A, B, C and D on the anion sublattice (Ga and As atoms depicted respectivelyin black and white). (a) A pure Ga As supercell, with the location of the anion lattice sites A – D indicated. (b) AGa N As supercell containing an isolated substitutional N impurity (depicted in red) at the anion lattice site A (denotedby GaAs:N A ). (c) A Ga As Bi supercell containing an isolated substitutional Bi impurity (depicted in green) at the anionlattice site A (denoted by GaAs:Bi A ). (d) A Ga N As Bi supercell containing single substitutional N and Bi impuritiesat the respective anion lattice sites A and B (denoted by GaAs:N A Bi B ), so that the N and Bi atoms are third-nearest neighbours.(e) A Ga N As Bi supercell containing single substitutional N and Bi impurities at the respective anion lattice sites Aand C (denoted by GaAs:N A Bi C ), so that the N and Bi atoms are second-nearest neighbours. (f) A Ga N As Bi supercellcontaining single substitutional N and Bi impurities at the respective anion lattice sites A and D (denoted by GaAs:N A Bi D ),so that the N and Bi atoms are second-nearest neighbours. FIG. S2. Calculated GaAs fractional Γ character spectra G Γ ( E ) for the CB and VB edge states in ordered 4096-atomGa N As and Ga As Bi supercells. (a) and (b) show G Γ ( E ) associated respectively with the GaAs CB andVB (i.e. combined LH and HH) edge states in Ga N As . (c) and (d) show G Γ ( E ) associated respectively with the GaAsCB and VB edge states in Ga As Bi . Band-anticrossing interactions associated with N- (Bi-) related localised resonantstates are clearly visible in G Γ ( E ) calculated for the Ga N As CB (Ga As Bi VB), consistent with our previouscalculations in Refs. and .FIG. S3. Calculated GaAs fractional Γ character spectra G Γ ( E ) for the CB and VB edge states in 4096-atom Ga N As Bi supercells containing one N and one Bi impurity arranged in different local configurations (cf. Fig. S1). (a) and (b) show G Γ ( E )associated respectively with the GaAs CB and VB (i.e. combined LH and HH) edge states in the GaAs:N A Bi B supercell. (a) and(b), (c) and (d), and (e) and (f) show, respectively, G Γ ( E ) associated respectively with the GaAs CB and VB (i.e. combinedHH and LH) edge states in the GaAs:N A Bi B , GaAs:N A Bi C , and GaAs:N A Bi D supercells. In moving the Bi atom from the I toE to B anion lattice site (cf. Fig. S1 the Bi atom is brought closer to the N atom at anion site A. The reduced symmetry inthese supercells lifts the degeneracy of the LH- and HH-like Bi-related and VB edge states, with the energy splitting betweenthese states increasing as the Bi atom is brought closer to the N site, due to larger local relaxation of the crystal lattice. FIG. S4. Schematic illustration of the structures considered in the analysis of the electronic and optical properties ofGaN y As − x − y Bi x /GaAs quantum wells. We consider an 8 nm thick quantum well surrounded by 8 nm thick GaAs barri-ers, giving a total length of 24 nm along the (001) growth direction and a separation of 16 nm between image quantum wellsassociated with the Born von Karman boundary conditions. The lateral dimensions are taken to be 4 nm along the (100) and(010) in-plane directions. In the tight-binding calculations this corresponds to a supercell of volume 384 nm containing atotal of 24,576 atoms. The same geometry along (001) is employed in the 14-band k · p calculations with the exception that thecalculation proceeds in one dimension only, so that the lateral dimensions are inconsequential. FIG. S5. Probability density associated with the lowest energy conduction electron state (upper row) and highest energy valencehole state (lower row) in N-free (a) GaAs . Bi . /GaAs ( x = 6 . . Bi . /GaAs ( x = 9%) quantumwells (cf. Fig. S4). Solid (dash-dotted) black lines and solid (dash-dotted) red lines show, respectively, the electron and holeprobability density at cation (anion) sites calculated using the sp s ∗ tight-binding model, obtained at each position z alongthe growth direction by summing over the probability densities associated with each atom in the plane. Vertical dashed blacklines denote the well/barrier interfaces. The tight-binding calculations were performed for five different random distributions(RDs), corresponding to structures with fixed x and y in which the N and Bi atoms were substituted at randomly selectedsites on the anion sublattice. It can be seen clearly that Bi-related alloy disorder strongly perturbs the VB leading to stronghole localisation, but that the CB is relatively unaffected with the electron probability densities retaining the smooth envelopefunction-like behaviour. FIG. S6. Probability density associated with the lowest energy CB electron state ( e
1; upper row) and highest en-ergy VB hole state ( h
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