The electronic band structure and optical properties of boron arsenide

TL;DR

This study uses advanced computational methods to accurately determine the electronic band structure and optical properties of boron arsenide, resolving previous discrepancies and confirming its covalent bonding nature.

Contribution

The paper applies a relativistic quasiparticle self-consistent GW approach with vertex corrections to improve band structure calculations of boron arsenide, providing more accurate results than previous methods.

Findings

Boron arsenide has an indirect band gap of 1.674 eV.

The direct band gap is 3.990 eV.

The material exhibits covalent bonding characteristics.

Abstract

We compute the electronic band structure and optical properties of boron arsenide using the relativistic quasiparticle self-consistent $GW$ approach, including electron-hole interactions through solution of the Bethe-Salpeter equation. We also calculate its electronic and optical properties using standard and hybrid density functional theory. We demonstrate that the inclusion of self-consistency and vertex corrections provides substantial improvement in the calculated band features, in particular when comparing our results to previous calculations using the single-shot $GW$ approach and various DFT methods, from which a considerable scatter in the calculated indirect and direct band gaps has been observed. We find that BAs has an indirect gap of 1.674 eV and a direct gap of 3.990 eV, consistent with experiment and other comparable computational studies. Hybrid DFT reproduces the indirect gap well, but provides less accurate values for other band features, including spin-orbit splittings. Our computed Born effective charges and dielectric constants confirm the unusually covalent bonding characteristics of this III-V system.

Tables

Table 1. Table 1: Calculated indirect band gap E g ind subscript superscript 𝐸 ind 𝑔 E^{\mathrm{ind}}_{g} , direct band gap E g dir subscript superscript 𝐸 dir 𝑔 E^{\mathrm{dir}}_{g} , valence band (conduction band) spin-orbit split off energy at the Γ Γ \Gamma point Δ SO Γ , V subscript superscript Δ Γ V SO \Delta^{\Gamma,\mathrm{V}}_{\mathrm{SO}} ( Δ SO Γ , C subscript superscript Δ Γ C SO \Delta^{\Gamma,\mathrm{C}}_{\mathrm{SO}} ), at the X point Δ SO X , V subscript superscript Δ X V SO \Delta^{\mathrm{X,V}}_{\mathrm{SO}} and at the L point Δ SO L , V subscript superscript Δ L V SO \Delta^{\mathrm{L,V}}_{\mathrm{SO}} , high-frequency dielectric constant ϵ ∞ superscript italic-ϵ \epsilon^{\infty} , longitudinal and transverse electron effective masses ( m e , l ∗ subscript superscript 𝑚 e 𝑙 m^{*}_{\mathrm{e},l} and m e , t ∗ subscript superscript 𝑚 e 𝑡 m^{*}_{\mathrm{e},t} , respectively), heavy hole ( m hh ∗ subscript superscript 𝑚 hh m^{*}_{\mathrm{hh}} ), light hole ( m lh ∗ subscript superscript 𝑚 lh m^{*}_{\mathrm{lh}} ) and split-off hole ( m so ∗ subscript superscript 𝑚 so m^{*}_{\mathrm{so}} ) effective masses of BAs. We compare our results obtained using DFT with the PBEsol functional, hybrid DFT with the HSE06 functional, the full QS G ​ W 𝐺 𝑊 GW approach and the scaled approach (sQS G ​ W 𝐺 𝑊 GW ) and with the inclusion of ladder diagrams via the Bethe-Salpeter equation (QS G ​ W 𝐺 𝑊 GW +BSE), with experiment, where available, and previous calculations employing wave function (WF) techniques and DFT, non-local DFT (including hybrid DFT, meta-GGA and the modified Becke Johnson exchange potential) and the G ​ W 𝐺 𝑊 GW approach. All energies are given in eV, while the effective masses are given in units of the electronic rest mass.

	This Work					Previous Studies
	PBEsol	HSE06	QS$GW$	sQS$GW$	QS$GW$+BSE	WF/DFT	non-local DFT	$GW$	Experiment
$E_g^{\mathrm{ind}}$	1.010	1.693	1.895	1.705	1.674	$0.7-1.27$ bas_opw_calc_bands_prb_stukel1970 ; bas_apw_bands_bp_pssb_prasad1989 ; bas_calc_bands_prb_hart2000 ; bas_dft_props_lda_jphysc_wentzcovitch1986 ; iii-v_comparison_calcs_properties_semicondscitech_anua2013	$1.58-1.933$ bas_dft_opticalabsorb_bsb_arxiv_ge2019 ; bas_expt_dft_defects_thermtrans_prl_zheng2018 ; bas_expt_dft_hse_defects_bandgap_apl_lyons2018 ; bas_dft_defects_hse_apl_chae2018 ; bas_dft_elecprops_gaussianorbs_jap_nwigboji2016 ; iii-v_comparison_calcs_properties_semicondscitech_anua2013	1.6 bas_gw_bn_bp_bands_prb_surh1991	1.46 bas_expt_bandgap_powder_optictransmiss_jelecsoc_ku1966 ; bas_expt_ptype_electrode_bandgap_jacs_wang2012
								1.87 bas_dft_bgaas_binas_props_physicab_chimot2005	1.4 bas_expt_ptype_hall_jap_chu1972
								2.049 bas_dft_bands_gw_oneshot_apl_bushick2019	1.77 bas_expt_dft_hse_defects_bandgap_apl_lyons2018
$E_g^{\mathrm{dir}}$	3.032	3.846	4.216	3.966	3.990	3.56, bas_opw_calc_bands_prb_stukel1970 4.23 bas_apw_bands_bp_pssb_prasad1989	3.05, bas_calc_bands_prb_hart2000 3.7, bas_dft_opticalabsorb_bsb_arxiv_ge2019 3.301 bas_dft_elecprops_gaussianorbs_jap_nwigboji2016	4.2, bas_gw_bn_bp_bands_prb_surh1991 4.135 bas_dft_bands_gw_oneshot_apl_bushick2019
${\Delta }_{\mathrm{SO}}^{\Gamma ,C}$	0.210	0.202	0.196	0.197	0.197	0.21 bas_calc_bands_prb_hart2000		0.2, bas_gw_bn_bp_bands_prb_surh1991 0.206 bas_dft_bands_gw_oneshot_apl_bushick2019
${\Delta }_{\mathrm{SO}}^{\Gamma ,V}$	0.207	0.316	0.211	0.210	0.207	0.33, bas_opw_calc_bands_prb_stukel1970 0.21 bas_calc_bands_prb_hart2000 ; bas_dft_bgaas_binas_props_physicab_chimot2005	0.23 bas_expt_dft_hse_defects_bandgap_apl_lyons2018	0.22, bas_gw_bn_bp_bands_prb_surh1991 0.206 bas_dft_bands_gw_oneshot_apl_bushick2019
${\Delta }_{\mathrm{SO}}^{X,V}$	0.129	0.213	0.128	0.128	0.125	0.14 bas_calc_bands_prb_hart2000		0.1, bas_gw_bn_bp_bands_prb_surh1991 0.140 bas_dft_bands_gw_oneshot_apl_bushick2019
${\Delta }_{\mathrm{SO}}^{L,V}$	0.144	0.145	0.144	0.144	0.141	0.17 bas_calc_bands_prb_hart2000		0.1, bas_gw_bn_bp_bands_prb_surh1991 0.148 bas_dft_bands_gw_oneshot_apl_bushick2019
${ϵ}^{\infty }$	9.909	7.995	7.384	7.690	9.881	9.8, bas_dft_bgaas_binas_props_physicab_chimot2005 8.1779 iii-v_comparison_calcs_properties_semicondscitech_anua2013		9.02 bas_dft_bands_gw_oneshot_apl_bushick2019
$m_{e,l}^{\ast }$	1.128	1.016	1.028	1.036	1.004	1.2, bas_opw_calc_bands_prb_stukel1970 1.099 bas_dft_bp_bsb_elec-phon_prb_liu2018	1.16, bas_expt_dft_hse_defects_bandgap_apl_lyons2018 1.015 bas_dft_elecprops_gaussianorbs_jap_nwigboji2016	1.093 bas_dft_bands_gw_oneshot_apl_bushick2019
$m_{e,t}^{\ast }$	0.198	0.198	0.199	0.201	0.200		0.32, bas_expt_dft_hse_defects_bandgap_apl_lyons2018 0.232 bas_dft_elecprops_gaussianorbs_jap_nwigboji2016	0.239 bas_dft_bands_gw_oneshot_apl_bushick2019
$m_{\mathrm{hh}}^{\ast }$	0.420	0.374	0.605	0.610	0.624	0.51, bas_opw_calc_bands_prb_stukel1970 0.402, bas_dft_bp_bsb_elec-phon_prb_liu2018 0.403 bas_dft_bgaas_binas_props_physicab_chimot2005	0.64, bas_expt_dft_hse_defects_bandgap_apl_lyons2018 0.355 bas_dft_elecprops_gaussianorbs_jap_nwigboji2016	0.635 bas_dft_bands_gw_oneshot_apl_bushick2019
$m_{\mathrm{lh}}^{\ast }$	0.172	0.188	0.129	0.130	0.129	0.20, bas_opw_calc_bands_prb_stukel1970 0.182, bas_dft_bp_bsb_elec-phon_prb_liu2018 0.193 bas_dft_bgaas_binas_props_physicab_chimot2005	0.25, bas_expt_dft_hse_defects_bandgap_apl_lyons2018 0.107 bas_dft_elecprops_gaussianorbs_jap_nwigboji2016	0.192 bas_dft_bands_gw_oneshot_apl_bushick2019
$m_{\mathrm{so}}^{\ast }$	0.221	0.225	0.264	0.263	0.263		0.26 bas_expt_dft_hse_defects_bandgap_apl_lyons2018	0.192 bas_dft_bands_gw_oneshot_apl_bushick2019

Table 2. Table 2: Calculated bulk modulus B 0 subscript 𝐵 0 B_{0} , its derivative B 0 ′ subscript superscript 𝐵 ′ 0 B^{\prime}_{0} , elastic constants C 11 , C 12 and C 44 , zone-centre longitudinal (transverse) optical phonon frequency ω LO subscript 𝜔 LO \omega_{\mathrm{LO}} ( ω TO subscript 𝜔 TO \omega_{\mathrm{TO}} ), Born effective charges Z ∗ superscript 𝑍 Z^{*} and static dielectric constant ϵ 0 superscript italic-ϵ 0 \epsilon^{0} of BAs, as computed using DFT with the PBEsol functional.

$B_0$ (GPa)	$B_0^{\prime }$	C11 (GPa)	C12 (GPa)	C44 (GPa)	${\omega }_{\mathrm{LO}}$ (cm-1)	${\omega }_{\mathrm{TO}}$ (cm-1)	$Z_B^{\ast }$	$Z_{\mathrm{As}}^{\ast }$	${ϵ}^0$
139.0	3.99	275.8	73.3	168.7	692.4	696.0	-0.509	0.503	10.010

Full text

Introduction: As microelectronic devices become smaller and more powerful, efficient cooling of their components becomes a critical issue. The transfer of heat from the active component requires materials with very high thermal conductivities. Diamond and graphite, both of which have thermal conductivities $\sim 2000$ Wm*-1K-1*, have significant disadvantages: diamond is expensive to produce and graphite is highly anistropic. bas_review_properties_angchem_tian2018 A recent first-principles study predicted that BAs has a thermal conductivity comparable to that of diamond and a more favourable thermal expansion coefficient for incorporation in devices. bas_dft_highthermal_cond_prl_lindsay2013 ; bas_dft_gan_thermal_transport_prb_broido2013 Initially, experimental studies failed to confirm this prediction, bas_expt_thermal_cond_low_apl_kim2016 as high quality single crystals for measurement are difficult to produce. bas_review_properties_angchem_tian2018 Scattering from defects in the single crystals was shown to be a significant factor in the low measured conductivities, bas_dft_vacancies_thermal_cond_prb_protik2016 ; bas_expt_dft_defects_thermtrans_prl_zheng2018 but, after considerable effort in improving crystal growth using the chemical vapour transport method, thermal conductivities of $>1000$ Wm*-1K-1* have been observed, bas_expt_high_thermal_cond_science_kang2018 ; bas_expt_high_thermal_cond_single_crystal_science_li2018 ; bas_expt_high_thermal_cond_single_crystal_science_tian2018 which are in very close agreement with more recent calculations. bas_others_dft_4ph_scattering_lowthermcond_prb_feng2017 The material is therefore of great interest for potential device applications; many of its fundamental properties, however, remain poorly characterised.

BAs stabilises in the zinc blende phase, isostructural with other III-V semiconductors such as GaAs, and has been shown to be $p$-type as-grown. bas_expt_ptype_hall_jap_chu1972 ; bas_dft_bp_bsb_elec-phon_prb_liu2018 Difficulties in growing the material have meant that it has remained relatively obscure until recent years. Early experimental studies indicated a band gap of $\sim 1.5$ eV, bas_expt_bandgap_powder_optictransmiss_jelecsoc_ku1966 ; bas_expt_thinfilm_optical_jelecsoc_chu1974 but whether the gap was direct or indirect was inconclusive. A subsequent measurement on a thin film of BAs indicated an indirect gap of 1.46 eV. bas_expt_ptype_electrode_bandgap_jacs_wang2012 Computational studies using wave function methods or density functional theory (DFT) showed that the band structure was similar to that of Si; bas_opw_calc_bands_prb_stukel1970 ; bas_apw_bands_bp_pssb_prasad1989 ; bas_calc_bands_prb_hart2000 the conduction band minimum (CBM) occurs along the $\Gamma$ to X direction and has p character, in contrast to AlAs, GaAs or InAs, where the s-like CBM is at $\Gamma$. The difference has been attributed to the anomously low energy of the boron p orbitals and strong s-s repulsion, bas_calc_bands_prb_hart2000 indicating a high degree of covalency in the bonding between B and As. Moreover, calculations of Born effective charges suggest a reversal of the rôles of anions and cations in the material, bas_dft_gan_thermal_transport_prb_broido2013 ; zincblende_calc_lattice-dyn_elastic_physicab_pletl1999 in contrast to that expected from formal oxidation states. oxidation_states_letter_jphyschemlett_walsh2017 ; oxidation_states_perspective_nmater_walsh2018 The magnitude of the indirect band gap $E_{g}^{\mathrm{ind}}$, as well as that of the lowest direct gap (at $\Gamma$) $E_{g}^{\mathrm{dir}}$ and the band curvatures vary substantially depending on the level of theory applied, with values of $E_{g}^{\mathrm{ind}}$ ranging from $0.7-1.6$ eV. bas_expt_dft_defects_thermtrans_prl_zheng2018 ; bas_opw_calc_bands_prb_stukel1970 ; bas_apw_bands_bp_pssb_prasad1989 ; bas_calc_bands_prb_hart2000 ; bas_dft_bp_bn_pressure_prb_wentzcovitch1987 ; bas_gw_bn_bp_bands_prb_surh1991 ; bas_dft_bgaas_binas_props_physicab_chimot2005 ; bas_dft_alas_gaas_inas_elecprops_compmatersci_ahmed2007 ; iii-v_comparison_calcs_properties_semicondscitech_anua2013 ; bas_dft_bp_bsb_elec-phon_prb_liu2018 ; bas_dft_bands_gw_oneshot_apl_bushick2019 ; bas_dft_opticalabsorb_bsb_arxiv_ge2019 In a recent study combining theory and experiment, an indirect gap of 1.78 eV was calculated using hybrid DFT, bas_expt_dft_hse_defects_bandgap_apl_lyons2018 which agreed well with the value derived from a combination of photoluminescence measurements and computed defect states (1.77 eV). bas_expt_dft_hse_defects_bandgap_apl_lyons2018 Other hybrid DFT studies using the same functional, however, report indirect gaps of 1.58, bas_dft_opticalabsorb_bsb_arxiv_ge2019 1.62 bas_expt_dft_defects_thermtrans_prl_zheng2018 and 1.90 eV. bas_dft_defects_hse_apl_chae2018 Further studies using techniques beyond standard DFT, including $GW$, report values from 1.48$-$2.049 eV. iii-v_comparison_calcs_properties_semicondscitech_anua2013 ; bas_dft_elecprops_gaussianorbs_jap_nwigboji2016 ; bas_gw_bn_bp_bands_prb_surh1991 ; bas_dft_bgaas_binas_props_physicab_chimot2005 ; bas_dft_bands_gw_oneshot_apl_bushick2019

In this Rapid Communication, we employ the relativistic quasiparticle self-consistent $GW$ (QS$GW$) approach questaal_qsgw_theory_prl_vanschilfgaarde2006 to compute the band structure and optical properties of BAs. The $GW$ approximation can be used to correct the one-electron eigenvalues obtained from DFT within a many-body quasiparticle framework, including the exchange and correlation effects in a self-energy term dependent on the one-particle Green’s function $G$ and the dynamically screened Coulomb interaction $W$. Substantial improvements on DFT energy eigenvalues can be obtained, depending on how the self energy is computed. questaal_qsgw_theory_prl_vanschilfgaarde2006 ; questaal_theory_paper_prb_kotani2007 Previous studies employing the $GW$ approach have been carried out to study the band dispersion of BAs: Surh et al., bas_gw_bn_bp_bands_prb_surh1991 using an approximation to self-consistent $GW$ that involves a linear interpolation of the DFT eigenvalues, calculated an indirect (direct) gap of 1.6 eV (4.2 eV); Bushick et al., bas_dft_bands_gw_oneshot_apl_bushick2019 using a single-shot $GW$ approach, where the DFT eigenvalues are used to compute $G$ within a single iteration, computed an indirect (direct) gap of 2.049 eV (4.135 eV); while Chimot et al., bas_dft_bgaas_binas_props_physicab_chimot2005 also using a single-shot $GW$ approach, reported $E_{g}^{\mathrm{ind}}=1.87$ eV. The substantial difference between these values indicates their dependency on the underlying DFT calculation (although different lattice parameters used in the calculations and the convergence criteria employed may also play a rôle). In contrast, the QS$GW$ method solves for the effective potential in a self-consistent manner, resulting in excellent agreement with experiment for a wide range of systems. questaal_theory_paper_prb_kotani2007 ; pbx_qsgw_bands_prb_svane2010 ; mapi_qsgw_bands_prb_brivio2014 ; sbchx_qsgw_bands_apl_butler2016 ; bichi_qsgw_dft_defects_chemmater_ganose2018 Errors introduced tend to be systematic, 111We note that convergence issues with such computationally expensive techniques may also result in some discrepancies in reported band gaps. and can be largely accounted for by scaling the QS$GW$ self energy in post-processing runs to compute band structures. questaal_calc_bandsplit_prl_chantis2006 Moreover, the errors originate from the lack of ladder diagrams in determining $W$; such effects can be included through solution of the Bethe-Salpeter equation (BSE). questaal_theory_paper_prb_kotani2007 By doing so, we expect the approach will give a very accurate energy dispersion, which will be useful for comparison with both experimental studies and computational studies using lower levels of theory.

Calculations: To determine the ground state lattice parameter $a$ and electronic structure of BAs, we have used plane-wave DFT as implemented in the VASP code, vasp_prb_kresse1993 ; vasp_prb_kresse1994 ; vasp_compmatsci_Kresse1996 ; vasp_prb_kresse1996 utilizing the solids-corrected Perdew-Burke-Ernzerhof (PBEsol) generalised gradient approximation (GGA) exchange-correlation functional GGA_pbe_prl_perdew1996 ; GGA_pbesol_prl_perdew2008 with the projector augmented wave method paw_physrevb50_blochl1994 to model the interaction between core and valence electrons (with three valence electrons for B and five for As). The total energy of the BAs zinc blende primitive cell was calculated at a series of constant volumes, using a 500 eV plane wave cut off and a 12$\times$12$\times$12$\Gamma$-centred Monkhorst-Pack monkhorst_pack_prb_monkhorst1976 k-point mesh, which provided convergence in the total energy up to 10*-4* eV, fitting the resultant energy-volume data to the Murnaghan equation of state. The bulk modulus $B$ and its derivative $B^{\prime}$ were derived using this approach. Spin-orbit interactions were included in the electronic structure calculations, spin-orbit_vasp_prb_hobbs2000 while a finer 16$\times$16$\times$16 k-point grid was used when computing the density of states (DOS). Furthermore, the calculations were repeated using a hybrid density functional (HSE06 hse06_functional_jphyschem_heyd2006 ), which counters the well-known self-interaction error in DFT. To provide calculated values of physical properties, at the PBEsol level of theory we have computed the high frequency dielectric constant ($\epsilon^{\infty}$), static dielectric constant ($\epsilon^{0}$), Born effective charges ($Z^{*}$) and the zone-centre transverse and longitudinal phonon frequencies ($\omega_{\mathrm{TO}}$ and $\omega_{\mathrm{LO}}$, respectively) using density functional perturbation theory, as implemented in VASP. optical-prop_paw_rpa_prb_gajdos2006 We have also computed the elastic constants C11, C12 and C44, using the finite displacement approach available in VASP.

The QS$GW$ calculations were performed using the Questaal package, questaal_qsgw_theory_prl_vanschilfgaarde2006 which implements DFT and $GW$ within an all-electron linear muffin-tin orbital basis set. Using the relaxed lattice parameter as determined from the plane-wave DFT calculations, the initial wave functions were determined at the PBEsol level of theory, employing the automatically generated augmentation spheres and interstitial Hankel functions as the basis set. The self energy was then obtained using the QS$GW$ formalism employing a $12\times 12\times 12$ $\Gamma$-centred $k$-point grid for the PBEsol and QS$GW$ calculations which provided convergence in the energy eigenvalues of under 1 meV (a 16$\times$16$\times$16 grid was used to calculate the DOS). The spin-orbit interaction was included perturbatively when computing the eigenvalues using the self energy, as described in Ref. mapi_qsgw_bands_prb_brivio2014, . As noted above, QS$GW$ introduces some systematic errors, questaal_theory_paper_prb_kotani2007 the most significant of which results in band gap overestimation. This error, however, has been shown to be largely corrected for by scaling the QS$GW$ self energy by 0.8, questaal_calc_bandsplit_prl_chantis2006 which is then combined with 0.2 of the PBEsol effective potential when determining band structures. We have followed this approach in the current work, but we have also corrected for the error by including ladder diagrams in $W$ through solution of the BSE, questaal_theory_paper_prb_kotani2007 as implemented in Questaal. In this approach, the $G_{0}$, rather than $G$ is used, which avoids the introduction of unphysical contributions. questaal_theory_paper_prb_kotani2007 The procedure consists of a full QS$GW$ calculation, followed by a correction to the $W$ through solution of the BSE. A final calculation of the self energy using the non-interacting $G_{0}$ is performed in order to compute the band structure. questaal_bse_theory_prm_cunningham2018 We compare all three approaches here: the self-consistent method (QS$GW$), the scaled method (sQS$GW$) and the inclusion of vertex corrections (QS$GW$+BSE).

We have calculated the dielectric function $\epsilon(\omega)$ at several levels of theory for comparison. Firstly, we have the summation over empty bands approach implemented in VASP optics_sumbands_prb_adolph2001 to compute $\epsilon(\omega)$ at the DFT level of theory. 64 empty bands were found to be sufficient to achieve convergence within three significant figures in this approach. Following the QS$GW$ calculations, we have used the random phase approximation (RPA) to determine $\epsilon(\omega)$, including local field effects via a modified response function, as implemented in Questaal. Finally, going beyond the RPA, we have included excitonic effects through the BSE for the four-point polarisation to compute $\epsilon(\omega)$. We note that, as BAs is an indirect gap system, phonon-assisted absorption is expected to occur at frequencies below the direct optical gap. Such absorption processes are not included in our analysis here but have been calculated elsewhere and shown to be small. bas_dft_opticalabsorb_bsb_arxiv_ge2019

Results: Using DFT with the PBEsol functional we determine $a=4.779$ Å, while using hybrid DFT with the HSE06 functional yields $a=4.772$ Å, both of which are in excellent agreement with the experimental value $a=4.777$ Å. bas_expt_ptype_hall_jap_chu1972 For the subsequent QS$GW$ calculations, we have imposed the PBEsol-derived $a$ as we use PBEsol-generated wave functions determined with the LMTO basis as the starting point for the self-consistent calculation of the self energy. We discuss below the effect of changing the lattice parameter on the computed band gaps.

Our calculated band structures determined using DFT with the PBEsol functional, hybrid DFT with the HSE06 functional and QS$GW$+BSE are shown in Fig. 1. All three approaches show energy dispersion that is similar to that found in other studies, bas_apw_bands_bp_pssb_prasad1989 ; bas_calc_bands_prb_hart2000 ; bas_opw_calc_bands_prb_stukel1970 ; bas_expt_dft_hse_defects_bandgap_apl_lyons2018 ; bas_dft_bands_gw_oneshot_apl_bushick2019 with an indirect energy gap between the valence band maximum (VBM) at the $\Gamma$ point and the conduction band minimum (CBM) along the $\Gamma$ to X direction, at about 80% of the distance to the X point and a minimum direct gap at $\Gamma$, but there are differences in particular features such as the band gaps, curvatures and spin-orbit splittings. Indeed, we exclude the band structures determined using QS$GW$ and sQS$GW$ from Fig. 1 as they are quite similar to the HSE06 and QS$GW$+BSE results. We instead highlight the differences between the methods in Table 1, where the gaps $E^{\mathrm{ind}}_{g}$ and $E^{\mathrm{dir}}_{g}$, spin orbit splittings in the conduction bands at the $\Gamma$ point ($\Delta^{\Gamma,\mathrm{C}}_{\mathrm{SO}}$) and the valence bands at the $\Gamma$, X and L points ($\Delta^{\Gamma,\mathrm{V}}_{\mathrm{SO}}$, $\Delta^{\mathrm{X,V}}_{\mathrm{SO}}$ and $\Delta^{\mathrm{L,V}}_{\mathrm{SO}}$, respectively), $\epsilon^{\infty}$, and the longitudinal (transverse) electron effective mass $m^{*}_{\mathrm{e},l}$ ($m^{*}_{\mathrm{e},t}$) at the CBM, the heavy, light and spin-orbit split off hole effective masses ($m^{*}_{\mathrm{hh}}$, $m^{*}_{\mathrm{lh}}$ and $m^{*}_{\mathrm{so}}$, respectively) at the VBM are presented and compared with available values from previous studies.

In all three cases presented in Fig. 1, the total DOS and partial DOS indicates that the upper valence band consists of mixing between B and As p orbitals, while the lower conduction bands are dominated by B p states, demonstrating the high degree of covalency in this system. This result is in contrast to other III-V arsenides, but consistent with the study of Hart and Zunger. bas_calc_bands_prb_hart2000 From our DFT calculation using the PBEsol functional we find that $E^{\mathrm{ind}}_{g}=1.010$ eV, which is lower than that found in other studies using similar GGA functionals bas_dft_alas_gaas_inas_elecprops_compmatersci_ahmed2007 ; bas_dft_bp_bsb_elastic_elec_props_pssb_meradji2004 ; iii-v_comparison_calcs_properties_semicondscitech_anua2013 ; bas_dft_opticalabsorb_bsb_arxiv_ge2019 ; bas_expt_dft_defects_thermtrans_prl_zheng2018 by about 0.2 eV. We note, however, that those studies do not include the spin-orbit interaction (SOI); we attribute the discrepancies to differences in the GGA functionals used and the inclusion of the SOI. Indeed, we find that, in all our calculations, including the SOI reduces $E^{\mathrm{ind}}_{g}$ by 0.07 eV and $E^{\mathrm{dir}}_{g}$ by 0.2 eV (apart from the HSE06 case, where $E^{\mathrm{ind}}_{g}$ is reduced by 0.14 eV and $E^{\mathrm{dir}}_{g}$ by 0.27 eV).

Using the HSE06 functional increases the band gaps, as expected. Our value of $E^{\mathrm{ind}}_{g}=1.693$ eV using HSE06 functional is larger than those of Ge et al. bas_dft_opticalabsorb_bsb_arxiv_ge2019 (1.58 eV) and Zheng et al. bas_expt_dft_defects_thermtrans_prl_zheng2018 (1.62 eV), who did not include the SOI. Zheng et al. employed lower convergence criteria and Ge et al. used a different approach, in both cases finding lattice parameters larger than our calculated value using HSE06 (4.80 and 4.818  Å  versus 4.772  Å), which may account for the lower computed band gaps. Chae et al. bas_dft_defects_hse_apl_chae2018 determined $E^{\mathrm{ind}}_{g}=1.90$ eV using the HSE06 functional without the SOI and with a lower plane-wave cut off (400 eV); their result is reasonably close to our value excluding the SOI (1.833 eV). We note that Lyons et al. bas_expt_dft_hse_defects_bandgap_apl_lyons2018 did include the SOI and calculated $E^{\mathrm{ind}}_{g}=1.78$ eV, using an approach very similar to ours (but also with the lower cut off of 400 eV). They included 3d states as valence states in As, which they determined to lie $\sim 40$ eV below the VBM. It is not clear, therefore, what advantage is gained by their inclusion; we note that the pseudopotential that excludes the 3d states from the core is in much less use than the standard, five valence electron pseudopotential, and may therefore be less robust.

Including quasiparticle energies with the QS$GW$ approach increases the band gap significantly. 222Using the LMTO basis, with the PBEsol functional we determine $E^{\mathrm{ind}}_{g}=1.015$ eV, in very good agreement with our plane-wave DFT-PBEsol calculation. We first note that, with a single-shot $GW$ calculation based on the PBEsol-derived wave functions, we determine $E^{\mathrm{ind}}_{g}=1.805$ eV and $E^{\mathrm{dir}}_{g}=4.118$ eV (including the SOI, as we do for all the QS$GW$ calculations reported below). As expected, these results differ from previous one-shot $GW$ calculations, bas_gw_bn_bp_bands_prb_surh1991 ; bas_dft_bgaas_binas_props_physicab_chimot2005 ; bas_dft_bands_gw_oneshot_apl_bushick2019 which were based on the local density approximation (LDA). The self-consistent prodecure requires three further iterations, resulting in wider gaps of $E^{\mathrm{ind}}_{g}=1.895$ eV and $E^{\mathrm{dir}}_{g}=4.216$ eV. These values are overestimates, due to systematic errors in the approach used, questaal_theory_paper_prb_kotani2007 but can be largely corrected for using the QS$GW$+BSE method, which reduces the band gaps by over 0.2 eV (see Table 1). In fact, the conduction bands are shifted down consistently, as can be seen in Fig. 2, where we also include the single-shot $GW$ bands for comparison (as the QS$GW$+BSE procedure includes the $G_{0}$, rather than $G$ determined in the QS$GW$ method). questaal_bse_theory_prm_cunningham2018 Including a simple scaling of the self energy (sQS$GW$) results in band gaps that are within a few tens of meV to the full QS$GW$+BSE calculation, but the derived optical properties have significant differences (see below). It is worth noting that our calculated fundamental gap agrees well with a previous calculation which used the modified Becke-Johnson exchange potential tb-mbj_exchange_test_prl_tran2009 , excluding the SOI (1.73 eV). iii-v_comparison_calcs_properties_semicondscitech_anua2013

Using the QS$GW$+BSE approach, the resulting $E^{\mathrm{ind}}_{g}=1.674$ eV is larger than that measured in early experimental studies (about 1.5 eV) bas_expt_bandgap_powder_optictransmiss_jelecsoc_ku1966 ; bas_expt_ptype_hall_jap_chu1972 and in a more recent study using thin films (1.46 eV), bas_expt_ptype_electrode_bandgap_jacs_wang2012 but crystal quality, strain in the case of the thin film, as well as temperature effects may account for the discrepancies. A value of 1.77 eV, derived from photoluminescence measurements, bas_expt_dft_hse_defects_bandgap_apl_lyons2018 depends strongly on computed defect levels, which are notoriously difficult to determine with accuracies better than $\sim 0.1$ eV in many cases. in2o3_sno2_zno_prm_buckeridge2018 Our result would be compatible with the experimental measurement if slightly shallower defect states exist in the material. We note that, from our results the HSE06 functional reproduces the fundamental gap quite well. Other features, however, such as $E^{\mathrm{dir}}_{g}$, $\Delta^{\Gamma,\mathrm{V}}_{\mathrm{SO}}$ and $\Delta^{\mathrm{X,V}}_{\mathrm{SO}}$ disagree substantially with those calculated using QS$GW$, sQS$GW$ and QS$GW$+BSE (and indeed DFT-PBEsol). The $a$ used in the HSE06 calculations differs to that in the PBEsol-derived QS$GW$ approaches; if we instead use the $a$ determined using DFT-PBEsol in the HSE06 calculation, we find only a very slight change in the band gaps: by 4 meV for $E^{\mathrm{ind}}_{g}$ and by -6 meV for $E^{\mathrm{dir}}_{g}$.

We have calculated the carrier effective masses at the band edges (see Table 1). For the case of the DFT and hybrid DFT calculations, we have applied quadratic fits to the bands within 0.001 eV of the appropriate band extremum, along and perpendicular to the $\Gamma$ to X direction for the case of electrons, and along the $\Gamma$ to X, L and K directions for holes. For the hole effective masses, where the bands are non-spherical, we took an average of the values obtained. Our results are in reasonable agreement with other calculations using similar approaches (DFT and hybrid DFT, see Table 1). bas_dft_bp_bsb_elec-phon_prb_liu2018 ; bas_opw_calc_bands_prb_stukel1970 ; bas_dft_bgaas_binas_props_physicab_chimot2005 ; bas_expt_dft_hse_defects_bandgap_apl_lyons2018 ; bas_dft_elecprops_gaussianorbs_jap_nwigboji2016 For the QS$GW$ calculations, however, we took advantage of the Questaal code, which allows one to fit a quadratic function to a set of points forming an icosohedron about the band edges and derive the effective mass tensor. We therefore expect this approach to provide more accurate results than simply fitting to the bands along high symmetry directions (although for electrons the two approaches should be equivalent). The resulting hole effective masses were taken to be the averages of the diagonal mass tensor components. In all cases, the agreement between the results obtained using the different approaches is reasonable. We find that, similar to other studies, bas_dft_bands_gw_oneshot_apl_bushick2019 holes are lighter than electrons and should therefore be more mobile.

The calculated values of $\epsilon^{\infty}$ given in Table 1 indicate the importance of including excitonic effects. $\epsilon^{\infty}$ decreases approximately with increasing band gap, from 9.909 (PBEsol) to 7.384 (QS$GW$), but increases substantially to 9.881, very close to the PBEsol value, when the QS$GW$+BSE approach is used. We have calculated the dielectric functional $\epsilon(\omega)=\epsilon_{r}(\omega)+\epsilon_{i}(\omega)$ using the summation over empty bands approach with the HSE06 functional, the sQS$GW$+RPA and the QS$GW$+BSE methods, as explained above. The results are shown in Fig. 3, with the derived optical absorption $\alpha(\omega)$ shown in the bottom panel. The HSE06 and sQS$GW$+RPA results are very similar, apart from an additional peak at about $\hbar\omega=7.5$ eV in $\epsilon_{i}(\omega)$, which may be due to some small differences in the band splittings between the HSE06 and sQSGW band structures. Including excitonic effects, however, substantially changes the optical properties. An onset in absorption is observed just below the direct gap (3.990 eV), a large additional peak is observed in $\epsilon_{i}(\omega)$ at about $\hbar\omega=5$ eV and the next peak is shifted down by about 0.5 eV, as a result of the shifting downwards of the conduction bands (see Fig. 2). As BAs is an indirect gap system, phonon-assisted transitions will occur below the direct gap, which we do not inculde in our analysis. Their effect, however, is expected to be minor. bas_dft_opticalabsorb_bsb_arxiv_ge2019 We note that the result of including the excitonic effects is similar to that seen in Ref. bas_dft_bands_gw_oneshot_apl_bushick2019, .

Finally, we have computed the elastic, dielectric and lattice dynamical properties of BAs using DFT with the PBEsol functional. The elastic constants and bulk modulus are in good agreement with previous calculations, iii-v_comparison_calcs_properties_semicondscitech_anua2013 ; zincblende_calc_lattice-dyn_elastic_physicab_pletl1999 ; bas_dft_bgaas_binas_props_physicab_chimot2005 as are the zone centre optical phonon frequencies. bas_dft_gan_thermal_transport_prb_broido2013 ; bas_dft_opticalabsorb_bsb_arxiv_ge2019 ; bas_dft_vacancies_thermal_cond_prb_protik2016 ; bas_dft_bgaas_binas_props_physicab_chimot2005 ; zincblende_calc_lattice-dyn_elastic_physicab_pletl1999 $\epsilon^{0}$, at 10.010, is very close to $\epsilon^{\infty}$ (9.909, see Table 1); consequently we have a lattice contribution to the dielectric constant of 0.101, indicating the high degree of covalency in this system. This result is in good agreement with Bushick et al. bas_dft_bands_gw_oneshot_apl_bushick2019 The $Z^{*}$ indicate a surprising rôle reversal of the cation (B) and anion (As), with a negative value of $Z^{*}_{\mathrm{B}}=-0.509$. Combined with our calculated refractive index $n=3.15$, this result is in good agreement with previous studies that report the reduced charge, $Z^{*}/n$, bas_dft_gan_thermal_transport_prb_broido2013 ; zincblende_calc_lattice-dyn_elastic_physicab_pletl1999 and is consistent with the bonding description given by Hart and Zunger. bas_calc_bands_prb_hart2000

Summary: We report the band structure and optical properties of BAs derived using the QS$GW$+BSE approach, determing an indirect gap of 1.674 eV and direct gap of 3.990 eV. We have demonstrated the importance of including self consistency in the $GW$ calculation and the inclusion of excitonic effects when determining the quasiparticle spectra by comparing our results to those we have obtained using DFT and hybrid DFT, as well as other previous calculations and experiment. Our computed effective masses agree with previous studies, as do the calculated bulk elastic, dielectric and lattice dynamical properties obtained using the PBEsol functional within DFT. The results indicate the unusually strong covalent bonding properties in this III-V arsenide and a swapping of the anion and cation rôles, consistent with previous studies.

Acknowledgment

The authors acknowledge the use of the UCL Legion and Grace High Performance Computing Facilities (Legion@UCL and Grace@UCL) and associated support services, and the ARCHER supercomputer through membership of the UK’s HPC Materials Chemistry Consortium, which is funded by EPSRC grant EP/L000202, in the completion of this work. D. O. S. acknowledges membership of the Materials Design Network. We would like to thank A. J. Jackson, B. A. D. Williamson and C. N. Savory for useful discussions.