Band Gap Correction Research Articles

Band-edge problem in the theoretical determination of defect energy levels: The O vacancy in ZnO as a benchmark case

Calculations of formation energies and charge transition levels of defects routinely rely on density functional theory (DFT) for describing the electronic structure. Since bulk band gaps of semiconductors and insulators are not well described in semilocal approximations to DFT, band-gap correction schemes or advanced theoretical models which properly describe band gaps need to be employed. However, it has become apparent that different methods that reproduce the experimental band gap can yield substantially different results regarding charge transition levels of point defects. We investigate this problem in the case of the (+2/0) charge transition level of the O vacancy in ZnO, which has attracted considerable attention as a benchmark case. For this purpose, we first perform calculations based on non-screened hybrid density functionals, and then compare our results with those of other methods. While our results agree very well with those obtained with screened hybrid functionals, they are strikingly different compared to those obtained with other band-gap corrected schemes. Nevertheless, we show that all the different methods agree well with each other and with our calculations when a suitable alignment procedure is adopted. The proposed procedure consists in aligning the electron band structure through an external potential, such as the vacuum level. When the electron densities are well reproduced, this procedure is equivalent to an alignment through the average electrostatic potential in a calculation subject to periodic boundary conditions. We stress that, in order to give accurate defect levels, a theoretical scheme is required to yield not only band gaps in agreement with experiment, but also band edges correctly positioned with respect to such a reference potential.

An optimal density functional theory method for GaN and ZnO

We report an optimal DFT method (bBLYP) for studying the GaN and ZnO systems. It is developed by modifying the exchange functional in the hybrid BLYP method in order to overcome the flaw of traditional DFT that often predict a rather small band gap for those semiconductors. Results show that the bBLYP method can describe not only correct band gaps of both GaN and ZnO wurtzite crystals, but also accurate properties of relevant small molecules. The application study of crystal-cut nanoparticles and nanowires reveals a new mechanism for band gap narrowing in GaN/ZnO.

First-principles study of native and extrinsic point defects in cubic boron nitride

Using first-principles total energy methods, we systematically calculated the formation energies and transition energy levels of intrinsic and extrinsic defects in cubic boron nitride (c-BN). We employed a mixed k-point sampling approach and brought the band-gap error into consideration. We find that under B-rich conditions boron vacancy (VB) and nitrogen vacancy (VN) act as the dominant compensating centers for additional dopants, while at N-rich limits, boron antisite (NB) becomes the main acceptor ‘killer’. The result is inaccessible if the impact of band-gap correction on the formation energies is ignored. Besides, all the calculated defect levels are too deep to account for the observed n- or p-type conductivity in as-grown samples. Among all the attempted dopants, substitutional Be (BeB) is the desirable acceptor in terms of lower formation energy and energy levels under N-rich conditions. However, n-type c-BN seems difficult to obtain under equilibrium conditions, because of the low solubility of dopants and the abundance of the compensating center VB3−.

Nitride Surface States Elucidated via First-Principles Band Gap Corrections

not Available.

First principles study of electronic structures of dopants in Mg2Si

ABSTRACTWe investigate the impact of various dopants (Na, Ag, Cd, Zn, Al, Ga, In, Tl, Ge, and Sn) on the electronic structure of Mg2Si by first principles calculations using a hybrid functional that does not need a band gap correction. We find that for Na and Ge in Mg2Si, the impurity-induced states do not affect the density of states at both edges of the valence band and the conduction band. Ag- and Sn affect slightly the density of states at the valence band edge, while Cd and Zn affect slightly the density of state at the conduction band edge. Al and In could modify significantly the density of states at the conduction band edge. Ga introduces states just at the bottom of the conduction band. Tl introduces states in the band gap. This study provides useful information on optimizing the thermoelectric efficiency of Mg2Si.

Atomic structure, electronic structure, and band offsets at Ge:GeO:GeO2 interfaces

The band gaps of GeO2 and GeO are calculated to be 6.1 and 4.0 eV, respectively, using hybrid density functionals that do not require band gap corrections. The conduction band offsets for Ge:GeO2 and Ge:GeO interfaces are calculated to be 0.8 and 0.4 eV, respectively, relatively small and similar to those found by photoemission. The atomic structure of solid GeO is found to have threefold coordinated Ge and O sites with planar oxygen sites.

Critical evaluation of the LDA + U approach for band gap corrections in point defect calculations: The oxygen vacancy in ZnO case study

AbstractThe LDA + U approach can be used essentially as a shift potential to open up the band gap of a semiconductor. This approach was previously applied to the oxygen vacancy in ZnO by Paudel and Lambrecht (PL) [Phys. Rev. B 77, 205202 (2008)]. Here, we review the results of that approach and introduce additional refinements of the LDA + U model. Good agreement is obtained with recent hybrid functional calculations on the position of the ${\varepsilon} (2 + /0)$ transition state. A comparison of various approaches on the oxygen vacancy in ZnO is provided. The relevance of the one‐electron levels to the experiments is discussed.

Discontinuities of the Chemical Potential in Reduced Density Matrix Functional Theory

Abstract Using the discontinuity of the chemical potential as a function of excess charge, the fundamental gaps for finite systems and the band gaps of extended solids are determined within reduced density matrix functional theory. We also present the necessary and sufficient conditions for the one-body reduced density matrix of a system with fractional charge to be ensemble N-representable. The performance of most modern day reduced density matrix functionals is assessed for the gaps and the correlation energy of finite systems. Our results show that for finite systems the PNOF, BBC3, and power functionals yield very accurate correlation energies while for a correct description of the fundamental gap the removal of self-interaction terms is essential. For extended solids we find that the power functional captures the correct band gap behavior for conventional semiconductors as well as strongly correlated Mott insulators, where a gap is obtained in absence of any magnetic ordering.

Role of intrinsic defects in ferromagnetism of SnO2 : First-principles calculations

The effects of tin interstitial (Sni), oxygen vacancy (VO), and Sni + VO defect pair on the electronic structure and magnetic properties of undoped SnO2 are investigated by means of density functional calculations. Only single positively charged O vacancies V can induce local magnetic moments in bulk SnO2. The magnetic coupling between two V calculated by generalized gradient approximation (GGA) is ferromagnetic. Self-consistent bandgap correction, which is achieved by adding a Coulomb U on O-2s orbital, results in the full occupation of the spin-up gap state of V. Consequently, the magnetic coupling calculated by GGA + US becomes antiferromagnetic, which shows that a self-consistent bandgap correction is essential for the correct description of the magnetism in widegap SnO2. The results indicate that O vacancy in bulk SnO2 cannot induce ferromagnetism, which suggests that the atoms or defects located at the surface or substrate interface may play a key role in turning the ferromagnetism observed in undoped SnO2.

Structural and magnetic properties of MnN and ScN binaries and their ScN:Mn diluted magnetic semiconductors and Mn xSc 1−xN alloys

Direct supercell approach calculations of the magnetic exchange interactions in Mn-doped ScN were carried out in the local spin density approximation by using the muffin-tin-orbital Green's function method. We employed the full-potential linearized-augmented plane-wave (FP-LAPW) method as used in the Wien2k code and WIEN97 package applied to a two-atoms unit cell. We found that magnetic interactions are long range interactions, affected by the randomness, band gap corrections, and carrier concentrations. In addition, we carried out calculations on the structural and magnetic properties of Mn-doped ScN diluted magnetic semiconductors and Mn x Ga 1 −x N and Mn x Sc 1 −x N alloys. Using total energy minimization approach we found that the global energy minimum of MnN is obtained for zinc-blende structure. If the compound is compressed by 6% the energy minimum corresponds to the rock-salt structure in disagreement with the experimentally observed slightly tetragonal distorted rock-salt structure, known as θ phase. An isostructural phase transition for Mn x Sc 1 −x N alloys from zinc-blende phase to hexagonal phase was observed to occur for Mn concentration x=0.1 at a hydrostatic pressure of 18 GPa. Our results on the magnetization of GaN:Mn DMS calculated at a magnetic field B=1.5 T are also presented.

Accurate prediction of defect properties in density functional supercell calculations

The theoretical description of defects and impurities in semiconductors is largely based on density functional theory (DFT) employing supercell models. The literature discussion of uncertainties that limit the predictivity of this approach has focused mostly on two issues: (1) finite-size effects, in particular for charged defects; (2) the band-gap problem in local or semi-local DFT approximations. We here describe how finite-size effects (1) in the formation energy of charged defects can be accurately corrected in a simple way, i.e. by potential alignment in conjunction with a scaling of the Madelung-like screened first order correction term. The factor involved with this scaling depends only on the dielectric constant and the shape of the supercell, and quite accurately accounts for the full third order correction according to Makov and Payne. We further discuss in some detail the background and justification for this correction method, and also address the effect of the ionic screening on the magnitude of the image charge energy. In regard to (2) the band-gap problem, we discuss the merits of non-local external potentials that are added to the DFT Hamiltonian and allow for an empirical band-gap correction without significantly increasing the computational demand over that of standard DFT calculations. In combination with LDA + U, these potentials are further instrumental for the prediction of polaronic defects with localized holes in anion-p orbitals, such as the metal-site acceptors in wide-gap oxide semiconductors.

Effects of nonlocality on second-harmonic generation in bulk semiconductors

Calculations of the second-harmonic susceptibility tensor ${\ensuremath{\chi}}^{abc}(\ensuremath{-}2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega})$ are presented for bulk semiconductors within both the $\mathbf{v}\ensuremath{\cdot}\mathbf{A}$ and the $\mathbf{r}\ensuremath{\cdot}\mathbf{E}$ gauges. The description of the semiconductor states incorporates the ``scissors'' Hamiltonian commonly used to obtain the correct band gap. The nonlocality of the scissors correction leads to terms in ${\ensuremath{\chi}}^{abc}(\ensuremath{-}2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega})$ not considered before within a sum-over-states approach to the $\mathbf{v}\ensuremath{\cdot}\mathbf{A}$ gauge. Using this expression, we show that the results of the two gauges give the same result for ${\ensuremath{\chi}}^{abc}(\ensuremath{-}2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega})$, within very good numerical accuracy. As part of the derivation, we clarify the well-known result for the linear optical response which states that the scissors correction rigidly shifts the spectrum along the energy axis, keeping the line-shape intact. The calculation is presented for GaAs using an all-electron scheme and a pseudopotential scheme.

Structural and Magnetic Properties of Mn-Based Diluted Magnetic Semiconductors and Alloys

Direct supercell approach calculations of the magnetic exchange interactions in Mn-doped ScN was carried out in the local spin density approximation by using the muffin-tin-orbital Green's function method. We found that magnetic interactions are long range interactions and affected by the randomness, band gap corrections, and carrier concentrations. Using total energy minimization approach we found that the global energy minimum of MnN is obtained for zinc-blende structure. If the compound is compressed by 6%, the energy minimum corresponds to the rock-salt structure in disagreement with the experimentally observed tetragonal distorted rock-salt structure, known as -phase. An isostructural phase transition for alloys from MnN -phase to -ScN phase was found to occur at a hydrostatic pressure of 18 GPa. We predict above room temperature for Mn concentrations of about 10% in ScN : Mn system.

The role of Co impurities and oxygen vacancies in the ferromagnetism of Co-doped SnO 2: GGA and GGA+U studies

The electronic structure and ferromagnetic stability of Co-doped SnO 2 are studied using the first-principle density functional method within the generalized gradient approximation (GGA) and GGA+ U schemes. The addition of effective U Co transforms the ground state of Co-doped SnO 2 to insulating from half-metallic and the coupling between the nearest neighbor Co spins to weak antimagnetic from strong ferromagnetic. GGA+ U Co calculations show that the pure substitutional Co defects in SnO 2 cannot induce the ferromagnetism. Oxygen vacancies tend to locate near Co atoms. Their presence increases the magnetic moment of Co and induces the ferromagnetic coupling between two Co spins with large Co–Co distance. The calculated density of state and spin density distribution calculated by GGA+ U Co show that the long-range ferromagnetic coupling between two Co spins is mediated by spin-split impurity band induced by oxygen vacancies. More charge transfer from impurity to Co-3d states and larger spin split of Co-3d and impurity states induced by the addition of U Co enhance the ferromagnetic stability of the system with oxygen vacancies. By applying a Coulomb U O on O 2 s orbital, the band gap is corrected for all calculations and the conclusions derived from GGA+ U Co calculations are not changed by the correction of band gap.

Assessment of correction methods for the band-gap problem and for finite-size effects in supercell defect calculations: Case studies for ZnO and GaAs

Contemporary theories of defects and impurities in semiconductors rely to a large extent on supercell calculations within density-functional theory using the approximate local-density approximation (LDA) or generalized gradient approximation (GGA) functionals. Such calculations are, however, affected by considerable uncertainties associated with: (i) the ``band-gap problem,'' which occurs not only in the Kohn-Sham single-particle energies but also in the quasiparticle gap (LDA or GGA) calculated from total-energy differences, and (ii) supercell finite-size effects. In the case of the oxygen vacancy in ZnO, uncertainties (i) and (ii) have led to a large spread in the theoretical predictions, with some calculations suggesting negligible vacancy concentrations, even under Zn-rich conditions, and others predicting high concentrations. Here, we critically assess (i) the different methodologies to correct the band-gap problem. We discuss approaches based on the extrapolation of perturbations which open the band gap, and the self-consistent band-gap correction employing the $\text{LDA}+U$ method for $d$ and $s$ states simultaneously. From the comparison of the results of different gap-correction, including also recent results from other literature, we conclude that to date there is no universal scheme for band gap correction in general defect systems. Therefore, we turn instead to classification of different types of defect behavior to provide guidelines on how the physically correct situation in an LDA defect calculation can be recovered. (ii) Supercell finite-size effects: We performed test calculations in large supercells of up to 1728 atoms, resolving a long-standing debate pertaining to image charge corrections for charged defects. We show that once finite-size effects not related to electrostatic interactions are eliminated, the analytic form of the image charge correction as proposed by Makov and Payne leads to size-independent defect formation energies, thus allowing the calculation of well-converged energies in fairly small supercells. We find that the delocalized contribution to the defect charge (i.e., the defect-induced change of the charge distribution) is dominated by the dielectric screening response of the host, which leads to an unexpected effective $1/L$ scaling of the image charge energy, despite the nominal $1/{L}^{3}$ scaling of the third-order term. Based on this analysis, we suggest that a simple scaling of the first order term by a constant factor (approximately 2/3) yields a simple but accurate image-charge correction for common supercell geometries. Finally, we discuss the theoretical controversy pertaining to the formation energy of the O vacancy in ZnO in light of the assessment of different methodologies in the present work, and we review the present experimental situation on the topic.

Electronic structure of substitutional defects and vacancies in GaSe

We have investigated the nature of defect states associated with substitutional impurities (Cd, In, Sn) and both Ga and Se vacancies in GaSe using ab initio electronic structure methods within density functional theory. These calculations were done using supercell model allowing for internal atomic relaxation. Binding energies (BEs) of defects obtained in this model are compared with effective mass approximation results. Significant central cell corrections are present for most of the defects. This is consistent with charge densities associated with the defect states that show clearly their strongly localized nature. Because of the difficulties associated with LDA/GGA in giving the correct band gap in semiconductors, we have only compared the acceptor BEs with available experiments. Our theoretical results agree well with the experiment for Cd Ga and V Ga. The fundamental role played by the Ga dimers in the formation of defect states is discussed.

Challenges for ab initio defect modeling

Supercell calculations using density functional theory with local or semi-local exchange functionals have proved to be a very successful tool in defect engineering, apparently leading to some degree of overconfidence lately. With a case study on selected defects, we demonstrate that the approximations involved in these functionals lead not only to an underestimation of the gap but also to related errors in the total energy difference of two defect configurations (be these in the same or different charge states). Sometimes the error is so serious, that even the common expectation: “DFT provides the correct ground state” is refuted. We also demonstrate, that semi-empirical hybrid exchange functionals, which reproduce the correct band gap, give good total energy differences even in these cases. In such calculations, the position of Kohn-Sham gap levels with respect to the band edges gives a good estimation of the vertical ionization energy. Based on that, we propose a simple correction scheme for checking the necessity of extended total energy calculations with a hybrid functional.

VEELS band gap measurements using monochromated electrons

With the development of monochromators for transmission electron microscopes, valence electron energy-loss spectroscopy (VEELS) has become a powerful technique to study the band structure of materials with high spatial resolution. However, artefacts such as Cerenkov radiation and surface effects pose a limit for interpretation of the low-loss spectra; also the inelastic delocalisation restricts the spatial resolution on band gap mapping. For direct semiconductors, spectra acquired at thin regions can efficiently minimize the Cerenkov effects. Examples of h-GaN spectra acquired at different thickness showed that a correct band gap onset value can be obtained for sample thicknesses up to about 60 nm. For indirect semiconductors, the correct band gap onset can be obtained in the dark-field mode when the required momentum transfer for indirect transition is satisfied. At low energy-loss range the spatial resolution of this technique, which is mainly limited by the inelastic delocalisation, can be improved by dark-field VEELS at high collection angles.

Investigation of pure and Co2+-doped ZnO quantum dot electronic structures using the density functional theory: choosing the right functional

The electronic structures of pure and Co2+-doped ZnO quantum dots (QDs) with sizes up to 300 atoms were investigated using three different density functional theory approximations: local spin density approximation (LSDA), gradient-corrected Perdew–Burke–Ernzerhof (PBE) and the hybrid PBE1 functionals with LANL2DZ pseudo-potential and associated basis set. Qualitative agreement among the three methods is found for the pure ZnO nanostructures, but only the hybrid functional reproduces the correct bandgap energies quantitatively. For Co2+-doped ZnO QDs, both LSDA and PBE incorrectly model interactions between Co2+ d levels and the valence band of ZnO, which will strongly impair predictions of dopant–carrier magnetic exchange interactions based on such computational results. Experimental observations are reproduced well in calculations at the hybrid PBE1 level of theory, making this the method of choice for exploring the magnetism of transition metal ions in ZnO QDs computationally. The qualitative features of the Co2+ 3d levels do not change appreciably with changes in cluster size over the range examined, leading to size-dependent dopant-band edge energy differences. The results presented here thus provide an experimentally calibrated framework for future ab initio descriptions of dopant–carrier and dopant–dopant magnetic exchange interactions in diluted magnetic semiconductors (DMS) nanocrystals.

Linear response theoretical study of the exchange interactions in Mn-doped ScN: Effects of disorder, band gap, and doping

Linear response calculations of the magnetic exchange interactions in Mn-doped ScN were carried out in the rigid-spin approximation by using the muffin-tin-orbital Green's function method. They show that the exchange interactions are long range and strongly affected by disorder, band gap corrections, and doping. The cluster variation method is shown to reduce the ${T}_{c}$ by about 30% as compared to the mean-field treatment. A Curie temperature saturating with Mn concentration above 20% at $\ensuremath{\sim}440\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ is predicted with a semiconducting behavior present until about 15%. Both $p$- and $n$-type doping are found to decrease ${T}_{c}$.