Giant bowing of the band gap and spin-orbit splitting energy in GaP1−xBix dilute bismide alloys

Using spectroscopic ellipsometry measurements on GaP1−xBix/GaP epitaxial layers up to x = 3.7% we observe a giant bowing of the direct band gap (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{g}^{{\rm{\Gamma }}}$$\end{document}EgΓ) and valence band spin-orbit splitting energy (ΔSO). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{g}^{{\rm{\Gamma }}}$$\end{document}EgΓ (ΔSO) is measured to decrease (increase) by approximately 200 meV (240 meV) with the incorporation of 1% Bi, corresponding to a greater than fourfold increase in ΔSO in going from GaP to GaP0.99Bi0.01. The evolution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{g}^{{\rm{\Gamma }}}$$\end{document}EgΓ and ΔSO with x is characterised by strong, composition-dependent bowing. We demonstrate that a simple valence band-anticrossing model, parametrised directly from atomistic supercell calculations, quantitatively describes the measured evolution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{g}^{{\rm{\Gamma }}}$$\end{document}EgΓ and ΔSO with x. In contrast to the well-studied GaAs1−xBix alloy, in GaP1−xBix substitutional Bi creates localised impurity states lying energetically within the GaP host matrix band gap. This leads to the emergence of an optically active band of Bi-hybridised states, accounting for the overall large bowing of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{g}^{{\rm{\Gamma }}}$$\end{document}EgΓ and ΔSO and in particular for the giant bowing observed for x ≲ 1%. Our analysis provides insight into the action of Bi as an isovalent impurity, and constitutes the first detailed experimental and theoretical analysis of the GaP1−xBix alloy band structure.

with the GaP 1−x Bi x alloy VB edge then being a primarily Bi-derived band possessing an admixture of GaP VB edge Bloch character 30 , enabling optical coupling to the comparatively unperturbed Γ 6c CB states. Comparison between theory and experiment highlights the emergence of this Bi-derived impurity band lying energetically within the GaP band gap, close in energy to the unperturbed GaP VB edge. Our analysis reveals the giant bowing of Γ E g and Δ SO due to this VBAC interaction: Γ E g (Δ SO ) decreases (increases) by ≈200 meV (240 meV) when 1% Bi is incorporated substitutionally in GaP.
We begin here by describing our measurement procedure and the general features of our experimental results. Next, we describe our theoretical model, which is then applied to analyse the trends revealed by the experimental measurements. Finally, we use the results of this combined theoretical and experimental analysis to describe general features of the GaP 1−x Bi x band structure.

experimental Measurements
Spectroscopic ellipsometry (SE) was used to study bulk-like GaP 1−x Bi x epitaxial layers containing up to 3.7% Bi. The samples were grown on (001)-oriented GaP by MOVPE, with x-ray diffraction measurements confirming that the layers are in a state of compressive pseudomorphic strain. Full details of the sample growth and characterisation-including data from high-resolution x-ray diffraction, atomic force microscopy, secondary ion mass spectrometry, and scanning transmission electronic microscopy measurements-can be found in ref. 43 . The SE measurements were performed at room temperature using a J. A. Woollam Co. variable angle spectroscopic ellipsometer. Three incident beam angles were used to generate sufficient data to provide confidence in modelling fits to the measured spectra. Angles of 73.5°, 74.0° and 74.5° were chosen since they are close to the pseudo-Brewster angles of the samples under investigation, thereby ensuring that the phase change on reflection Δ (measured relative to the sample normal) remained close to 90°. When Δ is close to 0 or 180°, the level of noise in the measured SE spectra is increased, and the sensitivity of the ellipsometic parameters to small changes in the optical properties is decreased. Keeping Δ close to 90° increases both the precision and accuracy of the measurements 45 . A carefully defined modelling and fitting procedure was used to extract the energies corresponding to critical points in the band structure from the measured SE data, allowing the energies corresponding to the Γ E g and + Δ Γ E g SO inter-band transitions to be extracted 46 . Full details of this fitting procedure are presented as Supplementary Material.
The solid red, green and blue lines in Fig. 1(a-c) respectively show the measured SE data-where tan(Ψ) is related to the change in the ratio of the amplitudes of the p-and s-polarisations upon reflection 45 -in the GaP, GaP 0.987 Bi 0.013 and GaP 0.963 Bi 0.037 samples. Solid (dashed) black lines show the corresponding fits to Δ (Ψ). The GaP sample consists of an epitaxial GaP buffer layer grown on a GaP substrate, and was analysed first in order to obtain accurate input parameters for the SE fits. These parameters were then used to describe the substrate and buffer layer in the subsequent models of the Bi-containing samples. Following this procedure it was possible to achieve good fits to the key features observed in the measured GaP 1−x Bi x SE spectra (cf. Fig. 1(b,c)). In Fig. 1(a) a clear feature associated with Γ E g is visible in the measured Δ and Ψ spectra, which is well described by a modelling fit corresponding to a Γ-point GaP room temperature band gap = .
Γ E 2 76 g eV. The slight deviation from the accepted value of 2.78 eV is attributable to the sample growth taking place on n-doped GaP substrates 47 .
Turning to Fig. 1(b,c) we note from the measured Δ spectra that Bi incorporation gives rise to an additional feature at lower energy than Γ E g in GaP, which shifts to even lower energies with increasing x. This indicates a large reduction of Γ E g , in agreement with theoretical predictions 30,48,49 . The spectral features associated with Γ E g are significantly broader in the Bi-containing samples than in GaP. This is likely associated with the presence of Bi composition fluctuations across the samples, as well as short-range alloy disorder, associated with the formation of pairs and larger clusters of Bi atoms sharing common Ga nearest neighbours in a substitutional alloy 50,51 . Using the fitting procedure outlined in the Supplementary Material it was also possible to extract the energies associated with the + Δ Γ E g SO transitions in each sample. The values of Γ E g and Δ SO extracted in this manner are shown respectively in Fig. 2(b,c), using closed red circles and blue squares. We note that the uncertainties in these data are associated with the broadening of the corresponding features in the measured spectra 46 . Overall, the SE  43 . Solid red, green and blue lines respectively denote data measured for incident beam angles of 73.5°, 74.0° and 74.5°. Solid (dashed) black lines show the fits to the measured Δ (Ψ) spectra; the SE model and associated fitting procedure are outlined in the text.
www.nature.com/scientificreports www.nature.com/scientificreports/ measurements indicate that incorporation of dilute concentrations of Bi is sufficient to cause a giant reduction (increase) and bowing of Γ E g (Δ SO ).

theoretical Calculations
To understand this unusual behaviour we have used alloy supercell electronic structure calculations to analyse the contributions to the Bi-induced changes in the band edge energies, and to parametrise a suitable VBAC model for GaP 1−x Bi x . This approach does not rely on post hoc fitting to alloy band structure data, thereby providing a predictive capability commonly lacking in models of this type 32 . In ref. 30 we employed an atomistic tight-binding (TB) model to analyse the electronic structure of ordered and disordered GaP 1−x Bi x alloys. By directly constructing the T 2 -symmetric localised states |ψ Bi 〉 associated with an isolated, substitutional Bi impurity we predicted the presence of a VBAC interaction having a composition dependence β x . In the dilute doping (large supercell) limit we determined that the Bi-related localised states in GaP:Bi lie approximately 120 meV above the unperturbed GaP VB edge, in good agreement with experiment 44 . Analysis of the electronic structure of ordered Ga(P,As) 1−x Bi x alloys eludicates the differences in the impact of Bi incorporation on the band structure: the natural VB offsets between GaP, GaAs and GaBi lead to the 6p valence orbitals of Bi lying below the 4p valence orbitals of As in energy, but higher in energy than the 3p valence orbitals of P. As such, a substitutional Bi impurity forms a resonant localised state lying energetically below the VB edge in GaAs, but a bound localised state lying above the VB edge in energy in GaP 30 . Building on our initial analysis of GaP 1−x Bi x we have derived an extended basis set 12-band (VBAC) k ⋅ p Hamiltonian to describe the dilute bismide band structure 32 . Using the TB model of ref. 30 we have directly evaluated the Bi-related parameters of this model, including the distinct VBAC, virtual crystal (VC) and strain-related contributions to the Bi-induced shifts in the band edge energies 52 . To analyse the SE measurements we focus on the band edge energies at the zone centre: at Γ the 12-band Hamiltonian diagonalises into decoupled blocks describing the CB, heavy-hole (HH), light-hole (LH) and spin-split-off (SO) band edges 52 . As in GaAs 1−x Bi x , the energy of the GaP 1−x Bi x Γ-point CB state Γ 6c is well described as , where the zero of energy has been chosen at the unperturbed GaP VB edge, = .
Γ E (0) 2 78 g eV is the host matrix band gap, α describes the VC shift of the CB edge energy, and δE CB hy is the energy shift associated with the hydrostatic component of the compressive pseudomorphic strain in a GaP 1−x Bi x /GaP epitaxial layer 52 .
The energies of the HH-like alloy VB states are given in the 12-band VBAC model as the eigenvalues of the 2 × 2 matrix 30,52 ax describes the VC, hydrostatic and axial strain-induced shifts to the GaP HH band edge energy, and ax is the energy of the HH-like Bi-related localised states relative to the zero of energy at the unperturbed GaP VB edge 52 . The energies of the LH-and SO-like VB states are given as the eigenvalues of a 3 × 3 matrix which can be found, along with full details of the model, in ref. 52 .
The Bi-related band structure parameters computed for GaP 1−x Bi x are summarised in Table 1 where, for comparative purposes, the corresponding parameters computed for GaAs 1−x Bi x are provided. Comparing these two sets of VBAC parameters we firstly note that while a substitutional Bi impurity in GaAs leads to the formation of a resonant localised state lying 183 meV below the VB edge (ΔE Bi < 0), in GaP it leads to the formation of a bound localised state lying 122 meV above the VB edge (ΔE Bi > 0). In both cases the alloy VB edge consists of an admixture of extended (Bloch) and localised (Bi-related) character. However, in GaAs 1−x Bi x the alloy VB edge states are primarily GaAs-derived, retaining significant Bloch character, while in GaP 1−x Bi x the alloy VB edge states are primarily Bi-derived, having low overall Bloch character 30 . As we will describe in further detail below, www.nature.com/scientificreports www.nature.com/scientificreports/ this key difference describes the fundamentally distinct nature of the perturbed VB structure in GaAs 1−x Bi x and GaP 1−x Bi x alloys, and has significant consequences for the electronic and optical properties. We note that this behaviour is qualitatively similar to the stark differences in the CB structure of the dilute nitride alloys GaN x As 1−x and GaN x P 1−x : a substitutional N impurity in GaAs creates a N-related localised state which is resonant with the CB, while in GaP it creates a bound impurity state lying energetically within the band gap 35,38,53,54 .
Considering the VBAC coupling parameter β, we note that a larger value is calculated for GaP 1−x Bi x . This describes that the VBAC interaction is more pronounced in GaP 1−x Bi x than in GaAs 1−x Bi x -i.e. that a substitutional Bi impurity more strongly perturbs the VB structure in GaP than in GaAs-reflecting the larger differences in size and electronegativity between P and Bi than between As and Bi. We note that β in GaP 1−x Bi x is comparable to that calculated previously for the GaN x P 1−x CB (β = 1.74 eV) 38,54 . Turning our attention to the VC contributions to the band offsets-described via the parameters α, κ and γ for the CB, VB and SO offsets respectively -we again note that the calculated differences in these parameters for GaP 1−x Bi x and GaAs 1−x Bi x alloys reflect the associated trends in the lattice mismatch and natural band offsets between GaP, GaAs and the fictitious semimetallic zinc blende compound GaBi 30,32,33,55 . The calculated values α = 4.39 and 2.82 eV for GaP 1−x Bi x and GaAs 1−x Bi x respectively describe increases of approximately 44 and 28 meV per % Bi of the type-I zone-centre CB offset in free-standing GaP 1−x Bi x /GaP and GaAs 1−x Bi x /GaAs, reflecting the larger difference in energy between the Γ 6c CB edge states in GaP and GaBi compared to that between GaAs and GaBi. For the VB offset, the larger calculated value of κ for GaP 1−x Bi x describes the larger VB offset between GaP and GaBi than between GaAs and GaBi. Similarly, the smaller calculated value of γ for GaP 1−x Bi x describes the smaller SO band offset between GaP and GaBi than between GaAs and GaBi, and results from a combination of (i) a natural VB offset between GaP and GaAs which exceeds the difference in Δ SO between the two compounds 56 , and (ii) the extremely large predicted VB spin-orbit splitting in GaBi 39 .
Finally, since the III-V compound AlAs has approximately the same natural VB offset relative to GaAs as GaP 56 , we note that the VB structure of the AlAs 1−x Bi x alloy-growth of which has also recently been established 57is expected to be qualitatively the same as in GaP 1−x Bi x . That is, based on the known chemical trends for III-V compounds, it can be expected based on the analysis presented here that (i) a substitutional Bi impurity in AlAs leads to the formation of a bound Bi-related localised impurity state, having the same general character as that described here in GaP, and (ii) this leads in turn to a large decrease (increase) and composition-dependent bowing of Γ E g (Δ SO ) at dilute Bi compositions x 58 . Bi at x = 0, we calculate that − E HH ( ± E HH ) decreases (increases) by 79 meV (103 meV) between x = 0 and 1%. Similarly, the VBAC interaction produces a set of LH-and SO-like Bi-hybridised bands 52 , the energies of which are again strongly dependent on x and characterised by strong composition-dependent bowing. As ± E LH moves downwards in energy towards E SO with increasing x the coupling between the LH-and SO-like states-which is brought about by the axial component of the pseudomorphic strain 52 -leads to an anticrossing which is manifested in an abrupt increase in the rate at which E SO decreases for  x 1%. We note also that this axial strain-induced anticrossing between the LH-and SO-like states leads to a change of the VB ordering at Γ, with > − −

E E
LH HH for  x 1%. Since the SE measurements do not detect optically forbidden transitions, such as those between the Γ 6c CB edge states and the Bi-related localised states 32,61 , quantitative assessment of the evolution of Γ E g and Δ SO with Bi composition x depends critically on the character of the hybridised GaP 1−x Bi x /GaP alloy VB edge eigenstates. As described above, since the GaP 1−x Bi x /GaP epitaxial layers under investigation are in a state of compressive pseudomorphic strain, the highest energy alloy VB states are expected to be HH-like (cf. Fig. 2(a)). It is therefore sufficient to investigate the character of the HH-like alloy eigenstates ψ ψ ψ . These energies represent the distinct separations evolving from the GaP direct band gap Γ E g and VB spin-orbit splitting energy Δ SO , as a direct result of the VBAC-induced hybridisation producing states having GaP HH character at distinct energies ± E HH . The results of these calculations are shown, using shaded lines, in Fig. 2(b,c)-for Γ E g and Δ SO , respectively-where they are compared to the values of Γ E g and Δ SO extracted from the SE measurements of Fig. 1(a-c). To describe the optical activity of these transitions the lines denoting the calculated transition energies are shaded according to the GaP HH Γ character Γ ± f , HH of the corresponding HH-like alloy VB edge states ψ | 〉 The quantitative agreement between the calculated and measured data in Fig. 2(b,c) confirms that the extremely large observed reduction (increase) and bowing of Γ E g (Δ SO ) results from the emergence of an optically active band of primarily Bi-derived impurity states lying energetically within the GaP band gap. That this impurity band lies within the GaP band gap accounts quantitatively for the observed trends: the contribution of the strong, composition-dependent bowing of + E HH to the decrease (increase) of Γ E g (Δ SO ) is combined with the binding energy ΔE Bi of the Bi-related localised states. This behaviour is qualitatively distinct from that in GaAs 1−x Bi x , where substitutional Bi atoms generate localised states which are resonant with the GaAs VB 30,33,62,63 , but similar to that in dilute nitride GaN x P 1−x , where substitutional N atoms produce a band of primarily N-derived states lying deep within the GaP band gap 37,38,53,54,[64][65][66][67] .
From the SE measurements we extract = . Γ E (0) 2 78 g eV. This is in good agreement with the calculated reduction of 284 meV in Γ E g between x = 0 and 1.3% in a pseudmorphically strained, ordered GaP 1−x Bi x /GaP alloy. Given the calculated reduction of 38 meV in E CB between x = 0 and 1.3%, we conclude that the majority (87%) of the reduction in Γ E g is associated with the emergence of the + E HH impurity band. Similarly, we measure an extremely large (>fourfold) increase of Δ SO , from 80 meV in GaP to approximately 360 meV at x = 1.3%. This is again in excellent agreement with the calculated value Δ SO = 355 meV, with the majority (92%) of the increase in Δ SO associated with the emergence of the + E HH band. Increasing x from 1.3 to 3.7% we note that the change in Γ E g and Δ SO per % Bi is significantly reduced. The measured (calculated) value = . ) represents a further reduction of 240 meV (207 meV) from that at x = 1.3%, while the measured (calculated) value Δ SO = 1 eV (0.590 eV) at x = 3.7% Bi represents a further increase of 250 meV (235 meV) from that at x = 1.3%. For Γ E g this change is only 73% of that between x = 0 and 1.3%, despite occuring over a 2.4% increase in x. The measured and calculated changes of Δ SO between x = 1.3 and 3.7% are approximately equal to those between x = 0 and 1.3%, again representing a significantly reduced change per % Bi. These trends highlight the strong dependence of the bowing of Γ E g and Δ SO on x. Our measured and calculated variation of Γ E g and Δ SO with x differs from that predicted using first principles electronic structure calculations 48 , but is close to that calculated via a VBAC model using parameter estimates based on available data for related alloys 49 .
We now turn our attention to two key qualitative features of the GaP 1−x Bi x electronic structure. Firstly, GaP has an indirect band gap due to the X 6c CB states lying ≈0.5 eV below Γ 6c , while in semimetallic GaBi the X 6c states lie ≈2 eV above Γ 6c 30,55 . Applying the VC approximation in conjunction with the TB model we estimate that the X 6c states shift downwards in energy by ≈12 meV per % Bi in free-standing GaP 1−x Bi x . This is less than the 44 meV per % Bi reduction of the Γ 6c state energy described by the VC parameter α (cf. Table 1), suggesting that Bi incorporation may bring about a direct band gap for sufficiently high x. Based on the calculated evolution of the GaP 1−x Bi x CB structure with x (cf. Supplementary Material) we estimate that a direct band gap exists for high Bi compositions  x 30%, suggesting that a direct band gap cannot be achieved at Bi compositions which are compatible with epitaxial growth. This highlights an important qualitative difference between the GaP 1−x Bi x and GaN x P 1−x band structures, since substitutional N in GaP generates localised states lying below the host matrix X 6c states in energy, bringing about a quasi-direct band gap even at ultra-dilute N compositions 37,53,54,[64][65][66] .
Secondly, our analysis in ref. 30 demonstrated that the VBAC description of the GaP 1−x Bi x VB structure breaks down with increasing x in the presence of short-range alloy disorder. Bi clustering creates a distribution of Bi-related localised states with which the GaP VB edge states strongly hybridise. This leads to a distribution of GaP VB edge Γ character over a multiplicity of impurity levels, suggesting that there is no single band possessing sufficient Bloch character to allow for appreciable absorption or emission of light 30 . While our results above demonstrate that the VBAC model provides a useful approach to analyse the main features of the band structure, the details of the electronic structure are in practice determined primarly by the impact of short-range alloy disorder. The difficulty in obtaining photoluminescence from the samples studied here, despite their high crystalline quality 43 , supports this interpretation: the GaP 1−x Bi x optical properties are intrinsically limited not solely by growth-related defects commonly associated with Bi incorporation, but by a combination of the indirect band gap and breakdown in VB edge Bloch character.
Despite having lattice constants commensurate with growth on Si, our analysis suggests that refinement of the epitaxial growth of GaP 1−x Bi x alloys is unlikely to lead to efficient light emitters: the optical properties are expected to be intrinsically limited by the nature of the material band structure. However, just as quaternary GaN x As y P 1−x−y alloys have found applications in III-V semiconductor lasers monolithically integrated on Si 68 , it is possible that similar progress could be made using As-rich quaternary GaP 1−x−y As y Bi x alloys for, e.g., applications in multi-junction solar cells, due to the fact that these alloys can be grown lattice-matched to either GaAs or germanium (Ge) while having band gaps close to 1 eV [69][70][71] .

Conclusion
In conclusion, we have presented a combined experimental and theoretical investigation of the GaP 1−x Bi x band structure. Measurements performed on GaP 1−x Bi x /GaP epitaxial layers reveal giant bowing of Γ E g and Δ SO , whereby Γ E g (Δ SO ) decreases (increases) by approximately 200 meV (240 meV) between x = 0 and 1%. These changes are characterised by strong, composition-dependent bowing. Electronic structure calculations confirm that substitutional Bi in GaP generates localised impurity states lying energetically within the GaP band gap, and that the main features of the GaP 1−x Bi x band structure can be understood in terms of a VBAC interaction between the extended states of the GaP VB edge, and highly localised Bi-related impurity states. A VBAC model was derived and parametrised directly from atomistic supercell calculations, allowing quantitative prediction of the evolution of the main features of the band structure with x. Our analysis suggests that the highest energy VB in GaP 1−x Bi x is a hybridised impurity band: admixture of the GaP VB edge Γ character into this primarily Bi-derived band allows optical coupling to the comparatively unperturbed CB states. Aspects of the GaP 1−x Bi x band structure are broadly comparable to GaN x P 1−x , but key qualitative differences highlight the distinction between Bi and N as isovalent impurities in conventional III-V semiconductors.