Intrinsic point defects and the $n$- and $p$-type dopability of the narrow gap semiconductors GaSb and InSb

TL;DR

This study uses hybrid density functional theory to analyze intrinsic point defects in GaSb and InSb, revealing how defect types influence their natural doping behavior and optoelectronic properties.

Contribution

It provides detailed defect formation energies and dopability insights of GaSb and InSb using advanced hybrid DFT calculations including spin orbit coupling effects.

Findings

Antisite disorder dominates defect landscape.

Cation vacancies are significant in GaSb under certain conditions.

Intrinsic n- and p-type behaviors are confirmed by defect calculations.

Abstract

The presence of defects in the narrow-gap semiconductors GaSb and InSb affects their dopability and hence applicability for a range of optoelectronic applications. Here, we report hybrid density functional theory based calculations of the properties of intrinsic point defects in the two systems, including spin orbit coupling effects, which influence strongly their band structures. With the hybrid DFT approach we adopt, we obtain excellent agreement between our calculated band dispersions, structural, elastic and vibrational properties and available measurements. We compute point defect formation energies in both systems, finding that antisite disorder tends to dominate, apart from in GaSb under certain conditions, where cation vacancies can form in significant concentrations. Calculated self-consistent Fermi energies and equilibrium carrier and defect concentrations confirm the intrinsic $n$- and $p$-type behaviour of both materials under anion-rich and anion-poor conditions. Moreover, by computing the compensating defect concentrations due to the presence of ionised donors and acceptors, we explain the observed dopability of GaSb and InSb.

Tables

Table 1. Table 1: Calculated lattice parameter a 𝑎 a , bulk modulus B 0 subscript 𝐵 0 B_{0} , elastic constants C 11 , C 12 and C 44 , band gap E g subscript 𝐸 𝑔 E_{g} , spin-orbit split off energy Δ SO subscript Δ SO \Delta_{\mathrm{SO}} , electron ( m e ∗ subscript superscript 𝑚 e m^{*}_{\mathrm{e}} ), light hole ( m lh ∗ subscript superscript 𝑚 lh m^{*}_{\mathrm{lh}} ) and heavy hole ( m hh ∗ subscript superscript 𝑚 hh m^{*}_{\mathrm{hh}} ) effective masses and zone-centre longitudinal optical phonon frequency ω LO subscript 𝜔 LO \omega_{\mathrm{LO}} of GaSb and InSb, compared with experimental results.  gasb_expt_lowT_lattconst_doklakadsssr_siroto1962 ; gasb_expt_elastic_bulk_mod_pressure_jap_mcskimin1968 ; gasb_gap_expt_elastic_consts_prb_boyle1975 ; gasb_expt_lowt_bandgap_jap_wu1992 ; gasb_expt_bands_prb_chiang1980 ; gasb_expt_elec_eff_mass_jphyschemsol_hill1974 ; gasb_expt_hole_eff_mass_jap_heller1985 ; gasb_expt_raman_pressure_prb_cardona1984 ; iii-v_review_bandfeatures_jap_vurgaftman2001 ; insb_expt_elastic_const_physrev_slutsky1959 ; insb_expt_bandgap_t_apl_litter1985 ; insb_expt_hall_conductivity_effmass_physrev_hrostowski1955 ; small_gap_transport_advphys_zawadzki1974 ; insb_expt_phonons_prb_price1971 The effective masses are given in units of the electronic rest mass.

		$a$ (Å )	$B_0$ (GPa)	C11 (GPa)	C12 (GPa)	C44 (GPa)	$E_g$ (eV)	${\Delta }_{\mathrm{SO}}$ (eV)	$m_e^{\ast }$	$m_{\mathrm{lh}}^{\ast }$	$m_{\mathrm{hh}}^{\ast }$	${\omega }_{\mathrm{LO}}$ (cm-1)
GaSb	Calc.	6.137	55.1	92.33	39.03	45.99	0.808	0.76	0.041	0.047	0.23	230.4
	Expt.	6.09593 gasb_expt_lowT_lattconst_doklakadsssr_siroto1962	56.35 gasb_expt_elastic_bulk_mod_pressure_jap_mcskimin1968	90.82 gasb_gap_expt_elastic_consts_prb_boyle1975	41.31 gasb_gap_expt_elastic_consts_prb_boyle1975	44.47 gasb_gap_expt_elastic_consts_prb_boyle1975	0.813 gasb_expt_lowt_bandgap_jap_wu1992	0.82 gasb_expt_bands_prb_chiang1980	0.0412 gasb_expt_elec_eff_mass_jphyschemsol_hill1974	0.05 gasb_expt_hole_eff_mass_jap_heller1985	0.28 gasb_expt_hole_eff_mass_jap_heller1985	232.6 gasb_expt_raman_pressure_prb_cardona1984
InSb	Calc.	6.548	40	68.2	33.8	31.6	0.23	0.80	0.018	0.019	0.25	180.3
	Expt.	6.4794 iii-v_review_bandfeatures_jap_vurgaftman2001	48.1 insb_expt_elastic_const_physrev_slutsky1959	69.18 insb_expt_elastic_const_physrev_slutsky1959	37.88 insb_expt_elastic_const_physrev_slutsky1959	31.32 insb_expt_elastic_const_physrev_slutsky1959	0.24 insb_expt_bandgap_t_apl_litter1985	0.80 insb_expt_bandgap_t_apl_litter1985	0.015 insb_expt_hall_conductivity_effmass_physrev_hrostowski1955	0.015 small_gap_transport_advphys_zawadzki1974	0.43 small_gap_transport_advphys_zawadzki1974	196.8 insb_expt_phonons_prb_price1971

Equations

$$E_f({\mathrm{X}}^q)=$$

Full text

I Introduction

GaSb and InSb belong to the family of III-V, zinc blende structured semiconductors of interest from both a fundamental and technological point of view. The incorporation of Sb in III-V semiconducting nitrides, phosphides and arsenides results in a red shift of the band gap, opening up the possibility of pushing the frequency domain of devices based on such materials far into the infrared (IR). gasb_expt_review_jap_dutta1997 ; long-wavelength_detectors_jap_rogalski2009 ; insb_expt_thz_transmission_quantum_oscillations_prl_gogoi2017 Both GaSb and InSb have applications in long wavelength telecommunications, inassb_expt_growth_characterise_jvacscitechb_tomasulo2018 high speed microelectronics insb_expt_highspeed_fet_apl_ashley1995 ; insb_expt_quantumwell_highmob_gasb_substrate_prm_lehner2018 ; insb_expt_diode_mid_ir_semicondscitech_petrosyan2019 and optoelectronics. gasb_expt_vecsel_ingasb_qw_jcrystgrowth_paajaste2009 ; vecsels_review_jphysd_guina2017 Due to favourable lattice matching, GaSb can be used as a substrate for a wide range of ternary and quaternary III-V compounds. gainsbn_expt_lattmatch_gasb_jcrysgrth_jefferson2007 ; gainsbn_expt_lattmatch_gasb_jap_ashwin2013 ; gasb_expt_ptype_acceptors_apl_kala2015 ; gasb_expt_epilaters_optical_semicondscitech_serincan2019 The spin-orbit interaction (SOI) has a strong effect on the valence band structure of both systems, ii-v_ii-vi_expt_valence_bands_xps_prb_ley1974 ; gasb_inas_expt_heterostructure_topolins_disorder_prm_shojaei2018 ; gasb_expt_pdoped_effmass_qw_prb_karalic2019 but is more pronounced in InSb, insb_expt_rashba_split_prb_khodaparast2004 ; insb_inas_nanowires_kp_soc_prb_campos2018 which, combined with a large Landé $g$-factor (over 50), insb_expt_kp_lande_gfactor_prb_litvinenko2008 has meant that InSb has attracted considerable attention in the field of Majorana physics. insb_expt_majorana_fermions_science_mourik2012 ; insb_expt_majorana_nanolett_deng2012 Moreover, GaSb and InSb have both been demonstrated to incorporate N and Bi effectively, resulting in a reduction in band gap insbn_expt_growth_solstatelec_ashley2003 ; gasbn_expt_struct_opto_apl_veal2005 ; gasbn_expt_ftir_kp_apl_jefferson2006 ; gasbn_expt_mbe_growth_jcrysgrth_buckle2005 ; gasbn_gainsbn_expt_growth_spieproc_ashley2006 ; gasbn_lda_bandgap_bowing_apl_belabbes2006 ; gasbn-gasb_expt_QW_PL_jap_kent2007 ; gasbn_insbn_tb_bands_prb_lindsay2008 ; gasbn_expt_PL_jap_wang2009 ; gasbn_expt_growthrates_aipadv_ashwin2011 ; gasbn_gaasn_hse06_bands_prb_virkkala2012 ; gasbn_expt_optabs_3bac_apl_mudd2013 ; gasbn_expt_growth_lattconst_jphysdapplphys_aswhin2013 ; insbbi_expt_growth_bands_apl_rajpalke2014 ; gasbn_dft_lvms_prb_buckeridge2014 ; iii-v_bi_dft_electronicstates_semicondscitech_polak2015 ; insbn_expt_bandgap_dft_apl_linhart2016 in a similar manner to the more widely studied, GaAs-based dilute nitrides and bismides. dilute-nitrides_review_semicondscitech_oreilly2009 ; iii-v_review_expt_growth_doping_applphysrev_kuech2016 Alloys can be produced of GaAs, GaSb and InSb, together with the relevant nitrides and/or bismides to tune the optical and electronic properties for a variety of applications; ingaassbn_gaas_expt_alignment_apl_chang2014 ; iii-v_bi_k.p_qw_models_jap_gladysiewicz2016 ; insbnbi_calc_kp_topologicalins_qsh_newjphys_song2017 ; inassbbi_expt_growth_bandgap_apl_webster2017 ; iii-v_dilute_n_model_solar_cell_semicondscitech_kharel2019 indeed, very high efficiency tandem solar cells include an active layer composed of such an alloy. dilute-nitride_world-record_solarcell_mrsproc_jones-albertus2013

Given the importance of GaSb and InSb, there are surprisingly few studies on their intrinsic defect properties, which are key to their dopability and hence functionality in devices. As-grown GaSb has been shown to be $p$-type regardless of growth conditions, gasb_expt_ptype_hall_physrev_leifer1954 ; gasb_expt_crystal_growth_hole_conc_jphyschemsol_vandermeulen1967 ; gasb_expt_ptype_undoped_semicondscitech_haywood1988 ; insb_expt_gasb_defects_compensate_intjhighspeedelecsys_pino2004 ; gasb_expt_ptype_acceptors_apl_kala2015 ; gasb_expt_pdoped_effmass_qw_prb_karalic2019 although the acceptor concentrations can be decreased slightly by varying the V/III flux when growing with molecular beam epitaxy (MBE). gasb_expt_sdoped_mbe_compensation_jap_lee1990 ; gasb_expt_tedoped_highmob_jvacscitechb_turner1993 Gallium vacancies ($V_{\mathrm{Ga}}$) have been shown to occur in GaSb using positron annihilation spectroscopy (PAS), gasb_expt_positron_hall_shallow_acceptor_apl_ling2004 but have been ruled out as the dominant acceptor; instead, it has been inferred in further PAS studies that the gallium antisite (GaSb) is responsible for the observed $p$-type activity, gasb_expt_positron_point-defects_jap_kujala2014 ; gasbn_expt_dft_positron_defects_jap_segercrantz2015 based on earlier density functional theory (DFT) calculations using the local density approximation (LDA). gasb_lda_intrinsic_defects_jap_hakala2002 While the LDA was also used to investigate the rôle of H in GaSb, gasb_lda_so_h_defect_prb_peles2008 this approach suffers from the well-known band gap underestimation error, which is particulary problematic in narrow gap semiconductors such as GaSb and InSb. To overcome the band gap error, a subsequent study on defects in GaSb employed hybrid DFT (without including the SOI). gasb_dft_hybrid_intrinsic_defects_prb_virkkala2012 The results, however, indicated that the intrinsic defect physics would result in a semi-insulating material as-grown, in contrast to experiment. C and O impurities were instead proposed to account for the $p$-type activity.

There are even fewer studies of the defect properties of InSb. The material can be made $n$- or $p$-type depending on growth conditions, while temperature ($T$) dependent studies have been employed to study variations in the $n$-type carrier concentration, Fermi energy and mobilities in order to elucidate various defect properties. insb_expt_hall_conductivity_effmass_physrev_hrostowski1955 ; insb_expt_fermi_energy_t_pssb_zukotynski1970 ; insb_expt_intrinsic_carrierconc_jap_chen1972 ; insb_expt_carrierconc_effmass_vs_t_hall_jphyschemsol_oszwaldowski1988 ; insb_expt_gasb_defects_compensate_intjhighspeedelecsys_pino2004 ; insb_expt_growth_sb_antisite_jcrystgrowth_jin2011 A computational study using DFT with the LDA indicated that the antimony antisite (SbIn) would dominate in Sb-rich growth conditions; insb_dft_inp_inas_lda_bulk_110_defects_prb_hoglund2006 by varying growth conditions, it was suggested that the formation of this defect could be suppressed in epitaxially grown thin films. insb_expt_growth_sb_antisite_jcrystgrowth_jin2011 Furthermore, it has been proposed that the formation of indium vacancies as well as SbIn can account for observed changes in the electronic properties of InSb grown in varying conditions. insb_expt_mbe_growth_chinphyslett_zhao2017 To our knowledge, no comprehensive study on the intrinsic defects in InSb using hybrid DFT has yet been performed.

In this Paper, we use hybrid DFT, including the SOI, to investigate the dominant native point defects in both GaSb and InSb. As noted above, the SOI strongly affects the dispersion of the upper valence bands in both systems; therefore, depending on the composition of the particular defect states, it can have a significant effect on the defect formation energies. We tune the fraction of exact exchange in the hybrid functional to reproduce only the band gaps, and justify this approach by computing a range of bulk properties of both systems, demonstrating close agreement with experiment for the structural, electronic, elastic and lattice vibrational properties. Our results show that GaSb will be $p$-type when grown in Sb-poor conditions, but may be semi-insulating under Sb-rich conditions. InSb, in contrast, will be $n$-type under Sb-poor conditions and $p$-type under Sb-rich conditions. From our computed defect formation energies, we determine self consistent Fermi energies and equilibrium carrier and defect concentrations as a function of $T$, by imposing the constraint of charge neutrality, calculating concentrations that agree well with experiment. Moreover, by introducing fixed concentrations of fully ionised dopants into the self-consistent Fermi energy calculation, we investigate donor and acceptor compensation by native defects in both systems. We find that, while InSb can be easily $n$- or $p$-doped, GaSb cannot be effectively $n$-doped under Sb-poor conditions. We provide the first comprehensive study of intrinsic disorder in GaSb and InSb using relativstic hybrid DFT which helps to elucidate the defect properties and dopability of both systems under equilibrium conditions.

The rest of the paper is structured as follows: In Section II, we describe our computationaly methodology. We present our results in Section III and summarize our main findings in Section IV.

II Calculations

To calculate the bulk and defect properties of GaSb and InSb, 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 Heyd-Scuseria-Ehrnzerof (HSE06) hybrid density functional hse06_functional_jphyschem_heyd2006 for electron exchange and correlation with the projector augmented wave method paw_physrevb50_blochl1994 to model the interaction between core and valence electrons (including $3d$ and $4d$ states among the 13 valence electrons in the cases of Ga and In, respectively, and five valence electrons for As). Spin-orbit interactions were included in all calculations. spin-orbit_vasp_prb_hobbs2000 The proportion $\alpha$ of exact exchange in the hybrid functional was set to $\alpha=0.335$ ($\alpha=0.31$) for GaSb (InSb) in order to reproduce the fundamental gap (see below). The total energy of the zinc blende primitive cell was calculated at a series of constant volumes, using a 400 eV plane wave cut off and a 12$\times$12$\times$12 $\Gamma$-centred Monkhorst-Pack monkhorst_pack_prb_monkhorst1976 k-point mesh (a finer 14$\times$14$\times$14 k-point grid was used when computing the density of states (DOS)), 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 $B_{0}$ was derived using this approach. The zone-centre longitudinal phonon frequencies ($\omega_{\mathrm{LO}}$) were calculated using the frozen phonon approach, 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. Electron ($m^{*}_{\mathrm{e}}$), light hole ($m^{*}_{\mathrm{lh}}$) and heavy hole ($m^{*}_{\mathrm{hh}}$) effective masses were calculated by fitting quadratic functions to the energy dispersion within 1 meV of the appropriate band extremum. For the hole masses, derived from the valence bands where the dispersion is non-spherical, we took an average of the values obtained for the different cartesian directions.

Defect calculations were performed using the supercell approach with a 64-atom $2\times 2\times 2$ expansion of the conventional cubic cell, which has been shown to be suitably converged previously. gasbn_dft_lvms_prb_buckeridge2014 ; gasb_lda_so_h_defect_prb_peles2008 ; gasb_dft_hybrid_intrinsic_defects_prb_virkkala2012 ; my_strain_paper ; gaasn_hetdevice_mob_prb_buckeridge2011 ; gaasn_si_vib_modes_solstatcomm_buckeridge2010 The formation energy of defect X in charge state $q$, $E_{f}(\mathrm{X}^{q})$, was determined through calculation of the heat of formation of the relevant defect reaction: gaas_lda_nativedefects_prl_zhang1991 ; defect-calcs_review_revmodphys_freysoldt2014

[TABLE]

where $E_{tot}(\mathrm{X}^{q})$ ($E_{tot}(\mathrm{bulk})$) is the total energy of the defect-containing (pure bulk) supercell, $E_{\mathrm{VBM}}$ is the energy at the valence band maximum (VBM), $E_{F}$ is the Fermi energy (introduced as a parameter), $\Delta$ is the energy required to align the electrostatic potential in the defect supercell with that of bulk and $E_{\mathrm{c}}$ is a correction term to account for supercell errors such as image charge interactions and, where applicable, erroneous band filling by delocalised carriers. To calculate $\Delta$ and $E_{\mathrm{c}}$, we follow the procedure outlined by Lany et al., lany-zunger_correction_prb_lany2008 which has been shown to result in corrections closely matched to those derived from full solutions to Poisson’s equation. charge_correction_poisson_solve_jchemphys_durrant2018 $n_{i}$ is the number of species $i$ that is added to ($n_{i}>0$) or removed from ($n_{i}<0$) the supercell to form $X$, and $\mu_{i}$ is the chemical potential of species $i$, taken with reference to the calculated standard state energies $E_{i}$ so that $\mu_{i}=E_{i}+\Delta\mu_{i}$. cplap_cpc_buckeridge2014 The values of $\Delta\mu_{i}$ can vary depending on the environmental conditions in thermodynamic equilibrium, but are contstrained by the relation $\Delta\mu_{\mathrm{M}}+\Delta\mu_{\mathrm{Sb}}=\Delta H[\mathrm{MSb}]$, where M$=$Ga or In and $\Delta H[\mathrm{MSb}]$ is the heat of formation of MSb; we calculate $\Delta H[\mathrm{GaSb}]=-0.507$ eV and $\Delta H[\mathrm{InSb}]=-0.470$ eV, which are in reasonable agreement with the experimental values of -0.433 eV and -0.316 eV, respectively, crc_handbook_89th_2008 particularly taking into account that the experimental values correspond to room $T$, while the calculations are done at the athermal limit (one would expect the heats of formation to become more negative by $\sim 0.05$ eV crc_handbook_89th_2008 at 0 K). 111If we were to use the experimental heats of formation, there would be no significant difference in our conclusions. We calculate the $E_{f}[\mathrm{X}]$ at two extremes: Sb rich, where $\Delta\mu_{\mathrm{Sb}}=0$ eV, corresponding to an excess of Sb in the growth environment and absence of pure In, and Sb poor, the opposite extreme, where $\Delta\mu_{\mathrm{Sb}}=\Delta H[\mathrm{MSb}]$.

From the calculated defect formation energies and DOS, we used the code SC-FERMI scfermigithub ; lafeo3_dft_defects_chemmater_taylor2016 ; taas_dft_nonstoich_prb_buckeridge2016 ; sc-fermi_accepted_compphyscom_buckeridge2019 to determine the equilibrium carrier and defect concentrations. SC-FERMI employs Fermi-Dirac statistics to calculate the concentrations, which are functions of $E_{F}$. With the constraint of overall charge neutrality in the system, a self-consistent $E_{F}$ can be derived at any temperature and consequently so can the electron ($n_{0}$), hole ($p_{0}$) and defect ($[\mathrm{X}]$) concentrations. Moreover, the charge neutrality constraint can be exploited in order to introduce fixed concentrations of ionised impurities, and the equilibrium carrier and defect concentrations recalculated in the presence of such impurities. In such a way, one can analyse ionised donor and acceptor compensation. In our calculations we neglect the temperature dependence of the free energies of defect formation due to the high computational cost in determining the associated vibrational entropy; one would expect the free energies to change by $\sim 0.1-0.2$ eV over the temperature range we employ, but including such changes would not affect significantly the conclusions we draw from our results.

III Results

III.1 Bulk properties

In Table 1, we show our calculated lattice parameter $a$, $B_{0}$, elastic constants C11, C12 and C44, band gap $E_{g}$, spin-orbit split off energy $\Delta_{\mathrm{SO}}$, $m^{*}_{\mathrm{e}}$, $m^{*}_{\mathrm{lh}}$, $m^{*}_{\mathrm{hh}}$ and $\omega_{\mathrm{LO}}$ for GaSb and InSb, compared with experiment. gasb_expt_lowT_lattconst_doklakadsssr_siroto1962 ; gasb_expt_elastic_bulk_mod_pressure_jap_mcskimin1968 ; gasb_gap_expt_elastic_consts_prb_boyle1975 ; gasb_expt_lowt_bandgap_jap_wu1992 ; gasb_expt_bands_prb_chiang1980 ; gasb_expt_elec_eff_mass_jphyschemsol_hill1974 ; gasb_expt_hole_eff_mass_jap_heller1985 ; gasb_expt_raman_pressure_prb_cardona1984 ; iii-v_review_bandfeatures_jap_vurgaftman2001 ; insb_expt_elastic_const_physrev_slutsky1959 ; insb_expt_bandgap_t_apl_litter1985 ; insb_expt_hall_conductivity_effmass_physrev_hrostowski1955 ; small_gap_transport_advphys_zawadzki1974 ; insb_expt_phonons_prb_price1971 As described above, the $\alpha$ used in the hybrid functional was chosen to reproduce the band gap at low $T$. From Table 1, however, we see that the hybrid DFT approach reproduces very well the experimental structural, elastic, and lattice vibrational properties of both materials, while the energy dispersion derived properties are also well reproduced. The only significant discrepancies occur for InSb, particularly in $B_{0}$ and $\omega_{\mathrm{LO}}$, indicating a slightly softer lattice in the calculation compared with experiment. The calculated $m^{*}_{\mathrm{hh}}$ for InSb is significantly lower than the experimental value, but this discrepancy may be due to difficulties in measuring this property accurately. Overall, the agreement between the calculated values and experiment is satisfactory, and indicates that our DFT approach is appropriate.

In Fig. 1, we show our hybrid-DFT-computed band structures of GaSb and InSb compared with experimental values determined using angle-resolved photoemission spectroscopy (ARPES) and, for the case of GaSb, reflectance measurements. gasb_expt_bands_prb_chiang1980 ; gasb_insb_etc_expt_valencebands_arpes_prb_williams1986 ; insb_expt_arpes_synch_prb_middelmann1986 ; insb_arpes_111_jphyscondmat_kim1996 For GaSb, we have also calculated band energies using the fully self consistent $GW$ approach, as implemented in VASP, vasp_gw_paw_prb_shishkin2006 ; vasp_gw_selfconsis_prb_shishkin2007 ; vasp_gw_vertex_prl_shishkin2007 including the SOI. As these calculations are computationally expensive, we have not determined the dispersion along the high symmetry path in the Brillouin zone with as small a grid spacing as we have for the hybrid DFT calculations. The band structure is similar in both cases to GaAs, yuandcardona2 with the VBM and conduction band minimum (CBM) both occuring at the $\Gamma$ point, and a splitting of the 6-fold degenerate upper valence bands into 4-fold and 2-fold degenerate bands, the latter forming the spin-orbit split-off bands. For both systems, the hybrid DFT approach reproduces the band structure well, apart from the lower-lying Sb s states (at about -11 eV), which are deeper than either experiment or the $GW$ results. The bands near the VBM and the conduction band minimum (CBM), however, are very well reproduced. These bands are the most significant for defect state formation.

III.2 Defects in GaSb

Our calculated formation energies of intrinsic defects in GaSb are shown in Fig. 2 as a function of $E_{F}$, referenced to the VBM, for Sb-poor and Sb-rich conditions. GaSb dominates in Sb-poor conditions; it has a formation energy under 1 eV and is negatively charged for all values of $E_{F}$ within the band gap, with an adiabatic transition from the $-$ to $2-$ state, $(-/2-)$, occurring at $E_{F}=0.16$ eV above the VBM. Such a low energy, negatively charged defect indicates an intrinsically $p$-type material, as is observed experimentally. gasb_expt_ptype_hall_physrev_leifer1954 ; gasb_expt_crystal_growth_hole_conc_jphyschemsol_vandermeulen1967 ; gasb_expt_ptype_undoped_semicondscitech_haywood1988 ; gasb_expt_ptype_acceptors_apl_kala2015 ; gasb_expt_pdoped_effmass_qw_prb_karalic2019 All other defects have formation energies of at least 1 eV higher than GaSb for $E_{F}$ within the band gap. Previous calculations by Hakala et al., using DFT-LDA, gasb_lda_intrinsic_defects_jap_hakala2002 and Virkkala et al., gasb_dft_hybrid_intrinsic_defects_prb_virkkala2012 using hybrid DFT, both found that GaSb had the lowest formation energy for $E_{F}$ in the upper half of the band gap, but predicted compensation by Ga interstitials (Ga${}_{i}^{+}$), resulting in an insulating material. The LDA calculations did not include the SOI nor any correction for the band gap underestimation, while the hybrid DFT calculations did not include the SOI and used higher convergence criteria than those we employ; gasb_dft_hybrid_intrinsic_defects_prb_virkkala2012 their results contradict the experimentally observed $p$-type activity of undoped GaSb.

In Sb-rich conditions, we find that $E_{f}(\mathrm{Ga}_{\mathrm{Sb}})$ increases significantly, while $E_{f}(V_{\mathrm{Ga}})$ and $E_{f}(\mathrm{Sb}_{\mathrm{Ga}})$ both decrease, so that the lowest energy defects are SbGa for $E_{F}<0.36$ eV and $V_{\mathrm{Ga}}$ for $E_{F}>0.42$ eV, with GaSb having the lowest energy for $E_{F}$ between these ranges. As SbGa are positively charged and GaSb and $V_{\mathrm{Ga}}$ negatively charged for $E_{F}$ within the band gap, these defects self compensate and one would expect $E_{F}$ to remain trapped roughly mid-gap, resulting in an intrinsically insulating material (we note that the formation energy of Gai is also low in this range of $E_{F}$ and we expect that this defect will play a minor rôle in the self-compensation mechanism). These formation energies suggest significant concentrations of $V_{\mathrm{Ga}}$ will be present, in agreement with PAS studies, gasb_expt_positron_hall_shallow_acceptor_apl_ling2004 ; gasb_expt_positron_point-defects_jap_kujala2014 ; gasbn_expt_dft_positron_defects_jap_segercrantz2015 ; gasb_expt_positron_vsb_instable_prb_segercrantz2017 but the insulating nature contradicts the $p$-type activity of GaSb observed in many differently produced samples. It may be the case that, in non-equilibrium growth techniques, formation of the compensating SbGa may be suppressed, which would result in a $p$-type material where the hole concentration arises from the ionisation of $V_{\mathrm{Ga}}$ and GaSb. gasb_expt_sdoped_mbe_compensation_jap_lee1990 ; gasb_expt_tedoped_highmob_jvacscitechb_turner1993 Our results for Sb-rich conditions agree qualitatively with those of Virkkala et al., gasb_dft_hybrid_intrinsic_defects_prb_virkkala2012 although they did not predict that the $V_{\mathrm{Ga}}$ would become the lowest energy defect for any value of $E_{F}$ within the band gap. Comparisons with the LDA calculations of Hakala et al. gasb_lda_intrinsic_defects_jap_hakala2002 are more difficult, as they only reported formation energies for SbGa in the neutral state. We note, however, that they also found $V_{\mathrm{Ga}}$ to be the lowest energy defect close to the conduction band minimum (CBM).

From our computed defect formation energies and total DOS, we have calculated the self-consistent $E_{F}$ and equilibrium carrier and defect concentrations by applying the constraint of overall charge neutrality to our system. The results are shown in Fig. 3(a) over the $T$ range below the melting point (985 K crc_handbook_89th_2008 ). It is worth noting here that, when varying $T$ in this analysis and for the case of InSb below we do not take into account the variation in band gap, which can be substantial for these narrow gap semiconductors. Indeed, at room temperature the band gap reduces by 86 meV for GaSb gasb_expt_review_jap_dutta1997 and 67 meV for InSb, insb_expt_bandgap_t_apl_litter1985 compared with their extrapolated 0 K values. Such reductions are a result of thermal expansion and increased electron-phonon coupling, the modelling of which is beyond the scope of this study on defects in both systems. Including the experimental variation in $E_{g}$ with $T$ in our calculations is not straightforward, as the defect transition levels vary with $T$ in a non-trivial manner. If we do include just the experimental $E_{g}$ variation, we calculate slightly different electron and hole concentrations which do not alter our conclusions significantly. As modelling temperature effects on the defect formation and transition levels is beyond the scope of the current work, we present our analysis below with the band gap fixed for all temperatures studied. We expect that, at higher $T$, where the band gap is reduced and consequently the electron and hole concentrations increased, compensating defect formation energies will also be lowered as vibrational entropy contributions to the free energy become more significant, so that the changes in concentrations will approximately cancel each other.

From our analysis we find that, in Sb-poor conditions, GaSb is $p$-type with hole concentrations $p_{0}$ of $\sim 10^{16}-10^{18}$ cm*-3* for $400<T<800$ K. The source of the $p_{0}$ is the formation and ionisation of GaSb; $p_{0}$ is equal to $2[\mathrm{Ga}_{\mathrm{Sb}}]$, which is consistent with the dominant charge state of GaSb being $2-$, but at $T\approx 800$ K the concentrations become close to being equal, as $E_{F}$ moves closer to the VBM where the $-$ state dominates. These calculated hole concentrations are lower by about an order of magnitude than those seen in experiment; gasb_expt_crystal_growth_hole_conc_jphyschemsol_vandermeulen1967 ; gasb_expt_ptype_undoped_semicondscitech_haywood1988 the discrepancy may be due to unwanted impurities such as C that can be introduced during experimental growth, which are not accounted for here. $p_{0}$ and $[\mathrm{Ga}_{\mathrm{Sb}}]$ are also about an order magnitude lower than those computed by Hakala et al., gasb_lda_intrinsic_defects_jap_hakala2002 which can be attributed to their lower value of $E_{f}(\mathrm{Ga}_{\mathrm{Sb}}^{2-})$. The difference in formation energies is probably due to a combination of the difference in functional and in the more crude image charge corrections used in their much earlier work. In Sb-rich conditions, we find that $E_{F}$ remains trapped at about 0.4 eV above the VBM over the range of $T$ investigated, due to the self-compensating defect physics, whereby the combined concentration of $\mathrm{Sb}_{\mathrm{Ga}}^{+}$, $\mathrm{Sb}_{\mathrm{Ga}}^{2+}$ and $\mathrm{Ga}_{i}^{+}$ equals that of $V_{\mathrm{Ga}}^{-}$, $V_{\mathrm{Ga}}^{2-}$ and $\mathrm{Ga}_{\mathrm{Sb}}^{2-}$, with the individual proportions depending on $T$. Consequently, the electron concentration $n_{0}$ is equal to $p_{0}$ and the material is intrinsically insulating. This insulating nature is rarely seen experimentally; again, unwanted $p$-type impurities not included in this study, as well as non-equilibrium defect formation, expected to be important in samples grown epitaxially where kinetics dominate, gasb_expt_ptype_undoped_semicondscitech_haywood1988 ; gasb_expt_pdoped_effmass_qw_prb_karalic2019 may account for the discrepancy.

When imposing the charge neutrality constraint to determine the self-consistent $E_{F}$, it is possible to introduce fixed concentrations of other charged defects and calculate the equilibrium carrier and intrinsic defect concentrations in their presence. In this way, one can analyse compensation of fully ionised impurities in an approximate manner. By assuming a fixed concentration of some ionised donor, $[$D${}^{+}]=10^{18}$ cm*-3*, we have calculated donor compensation in GaSb, with our results shown in Fig. 3(b). We find that, in Sb-poor conditions, rather than introducing $n$-type carriers, the donors are compensated by Ga${}_{\mathrm{Sb}}^{2-}$, so that $[$D${}^{+}]=2[\mathrm{Ga}_{\mathrm{Sb}}]$ for $T<600$ K. We see, therefore, that in Sb-poor conditions donor doping will not be effective, assuming that defect formation occurs in equilibrium. In fact, $p_{0}$ will become greater than $10^{16}$ cm*-3* at about $T=600$ K, and continues to rise with temperature as $[\mathrm{Ga}_{\mathrm{Sb}}]$ increases above the value necessary to compensate $[$D${}^{+}]$ due to thermal activation, while $E_{F}$ is pushed closer to the VBM. In Sb-rich conditions, however, we have $[$D${}^{+}]=n_{0}$ for most of the temperature range studied, so that GaSb will be doped effectively. At lower temperature, $E_{F}$ remains close to the CBM, but decreases into the band gap with increasing temperature. There is a very small dip in $n_{0}$ around $T=400$ K, which occurs as thermally induced concentrations of $V_{\mathrm{Ga}}$ compensate slightly the donors. We note that, in MBE-grown samples intentionally doped $n$-type, increasing the V/III ratio (i.e. going towards increasingly Sb-rich conditions) caused a slight increase in compensating acceptor concentrations, gasb_expt_sdoped_mbe_compensation_jap_lee1990 ; gasb_expt_tedoped_highmob_jvacscitechb_turner1993 contrary to our findings here. The effect is small and may be due to non-equilibrium defect formation and/or the presence of unwanted impurities.

In the same way, we can analyse acceptor compensation in GaSb. In Fig. 3(c), we show the equilibrium carrier and intrinsic defect concentrations in the presence of a fixed concentration of an ionised acceptor, $[$A${}^{-}]=10^{18}$ cm*-3*. The situation here is quite different to donor compensation discussed above; in both Sb-poor and Sb-rich conditions the acceptors are uncompensated and we have a $p$-type material with $p_{0}=[$A${}^{-}]$. $E_{F}$ remains close to the VBM, but moves towards mid-gap as $T$ increases, as one would expect due to $T$-induced intrinsic carrier generation. In Sb-poor conditions, for $T>600$ K, substantial concentrations of GaSb form, which further contribute to the $p$-type activity. We therefore find that GaSb can be effectively $p$-doped, whether in Sb-rich or Sb-poor conditions, a result that is consistent with experiment.

III.3 Defects in InSb

We show our calculated intrinsic defect formation energies as a function of $E_{F}$ referenced to the VBM in Fig. 4. We find that, in contrast to the case of GaSb, we have a positively charged defect, SbIn, dominating in Sb-rich conditions and a negatively charged defect, InSb ,dominating in Sb-poor conditions. Consequently, one would expect an $n$-type material if grown in Sb-rich conditions, and a (weakly, due to the relatively high formation energy) $p$-type material if grown in Sb-poor conditions. Experimentally, both $n$- and $p$-type unintentionally doped samples are routinely prepared, and InSb can be doped relatively easily with electrons or holes as majority carriers. insb_expt_hall_conductivity_effmass_physrev_hrostowski1955 ; insb_expt_fermi_energy_t_pssb_zukotynski1970 ; insb_expt_intrinsic_carrierconc_jap_chen1972 ; insb_expt_carrierconc_effmass_vs_t_hall_jphyschemsol_oszwaldowski1988 ; insb_expt_gasb_defects_compensate_intjhighspeedelecsys_pino2004 ; insb_expt_growth_sb_antisite_jcrystgrowth_jin2011 Hoglund et al. insb_dft_inp_inas_lda_bulk_110_defects_prb_hoglund2006 calculated the defect formation energies using DFT-LDA, finding results consistent with ours for Sb-rich conditions, but for the Sb-poor conditions they found that Ini would dominate, resulting in an $n$-type material, in contrast to our results. In their calculations, they found InSb to be gapless, contradicting experiment, and did not discuss corrections for this error nor for image charge interactions in their supercell model. The SbIn defect has been proposed to be a source of intrinsic $n$-type carriers in epitaxially grown InSb, but can be removed effectively by decreasing the V/III ratio, i.e. moving away from Sb-rich conditions. insb_expt_growth_sb_antisite_jcrystgrowth_jin2011 Such an observation is consistent with our calculated formation energies. Vacancies have also been proposed to be important in InSb, insb_expt_selfdiffusion_jap_kendall1969 ; insb_expt_mbe_growth_chinphyslett_zhao2017 ; insb_expt_vacancies_highT_crysrestech_morozov1986 ; insb_expt_thermoelectric_zt1_jmaterchema_xin2018 but our results show that their concentrations should be small as their formation energies are relatively high. We note that, although we have pointed out some differences between the defect physics of InSb and GaSb, some of these differences can be traced to the much lower band gap of InSb, compared with GaSb (0.23 eV vs 0.808 eV). Restricting the range of $E_{F}$ to remain less than 0.23 eV in GaSb would result in a similar transition level diagram to that of InSb. This result indicates a small valence band offset between the materials, consistent with earlier studies. ii-v_ii-vi_expt_valence_bands_xps_prb_ley1974 ; insb_gasb_calc_bandoffset_prb_magri2002 ; iii-v_review_bandfeatures_jap_vurgaftman2001

As with the case of GaSb, we have calculated equilibrium carrier and defect concentrations in InSb (excluding the variation in $E_{g}$ with $T$, see the discussion above); our results are shown in Fig. 5(a) over the $T$ range below the melting point (797 K crc_handbook_89th_2008 ). Despite the dominance of positively and negatively charged defects in Sb-rich and Sb-poor conditions respectively, we find that, under either condition InSb will be insulating as-grown. This result is a consequence of the low band gap and relatively high defect formation energies; thermally induced intrinsic carrier formation will dominate as defect concentrations remain several orders of magnitude below the carrier concentrations over the relevant $T$ range (in Sb-poor conditions, $[\mathrm{In}_{\mathrm{Sb}}]$, not shown in the figure, rises above $10^{14}$ cm*-3* only for $T>700$ K). $E_{F}$ remains closer to the CBM, as the DOS at the bottom of the conduction band is much lower than that at the top of the valence band. To produce $n$- and $p$-type samples therefore, one needs to dope the material and nominally undoped samples that have substantial carrier concentrations probably have unwanted impurities present, according to our results.

In Fig. 5(b) we show the equilibrium carrier and defect concentrations in the presence of a fixed concentration of ionised donors, $[$D${}^{+}]=10^{18}$ cm*-3*. In both Sb-poor and Sb-rich conditions, we find that InSb can be donor doped effectively, resulting in $n_{0}=[$D${}^{+}]$ for much of the $T$ range. As the DOS is relatively low at the CBM, to induce the relevant electron concentration $E_{F}$ is pushed very up to the CBM (see the inset in Fig. 5(b)). No significant defect compensation is observed; indeed, we find that, for $T>400$ K, thermal ionisation increases $n_{0}$ above $[$D${}^{+}]$.

We have also analysed acceptor compensation in InSb by assuming a fixed ionised acceptor concentration, $[$A${}^{-}]=10^{18}$ cm*-3*, and computing the resultant carrier and defect concentrations; our results are shown in Fig. 5(c). In both Sb-poor and Sb-rich conditions there is no effective compensation of the acceptors by defects, indicating that InSb will be easily acceptor doped in either extreme condition. $E_{F}$ varies across the gap as $T$ increases, which induces minority carrier concentrations while also increasing the majority carrier concentration. We therefore see that InSb can be both $n$- and $p$-doped without significant compensation by intrinsic point defect formation, a result that is consistent with experiment. insb_expt_gasb_defects_compensate_intjhighspeedelecsys_pino2004 ; insb_dft_inp_inas_lda_bulk_110_defects_prb_hoglund2006 ; insb_expt_growth_sb_antisite_jcrystgrowth_jin2011

IV Summary

We have investigated the intrinsic defect physics in GaSb and InSb by computing native defect formation energies using hybrid DFT. We justify our approach by first calculating a range of bulk properties of both systems, obtaining results in good agreement with experiment. We find that, in GaSb GaSb will dominate in Sb-poor conditions, resulting in a $p$-type material, while in Sb-rich conditions self-compensation will occur and the material will be intrinsic. We confirm these inferences from the formation energy calculations by computing equilibrium carrier and defect concentrations as a function of temperature, then study donor and acceptor compensation by assuming fixed concentrations of ionised dopants. We find that GaSb can be easily $p$-doped, but in equilibrium conditions, should only be effectively $n$-doped under Sb-rich conditions. For InSb, we find that positively charged (SbIn) and negatively charged antisite defects (InSb) dominate in Sb-rich and Sb-poor conditions, respectively. By calculating equilibrium carrier and defect concentrations, however, we show that the material will be intrinsic as-grown, due to the relatively high formation energies, low band gap and consequent thermally induced carrier generation. As the concentrations of compensating defects remain low over the relevant $T$ range, InSb can be effectively $n$- and $p$-doped. Our study provides crucial information on the defect physics of GaSb and InSb, important semiconductors for a range of technological applications.

Acknowledgment

The authors acknowledge funding from EPSRC grants ED/D504872, EP/K016288/1 and EP/I01330X/1 and the European Research Council (grant 758345). The authors also acknowledge the use of the UCL Legion and Grace High Performance Computing Facilities (Legion@UCL and Grace@UCL) and associated support services, the IRIDIS cluster provided by the EPSRC funded Centre for Innovation (EP/K000144/1 and EP/K000136/1), the Thomas supercomputer via the U.K. Materials and Modelling Hub (EPSRC grant EP/P020194/1) and the ARCHER supercomputer through membership of the UK’s HPC Materials Chemistry Consortium, which is funded by EPSRC grants EP/L000202 and EP/R029431, in the completion of this work. D. O. S. and T. D. V. acknowledge membership of the Materials Design Network.