This review provides a perspective on the use of orbital-dependent functionals, which is currently considered one of the most promising avenues in modern density-functional theory. The focus here is on four major themes: the motivation for orbital-dependent functionals in terms of limitations of semilocal functionals; the optimized effective potential as a rigorous approach to incorporating orbital-dependent functionals within the Kohn-Sham framework; the rationale behind and advantages and limitations of four popular classes of orbital-dependent functionals; and the use of orbital-dependent functionals for predicting excited-state properties. For each of these issues, both formal and practical aspects are assessed.

Orbital-dependent density functionals: Theory and applications

Wide-band-gap GaN and Ga-rich InGaN alloys, with energy gaps covering the blue and near-ultraviolet parts of the electromagnetic spectrum, are one group of the dominant materials for solid state lighting and lasing technologies and consequently, have been studied very well. Much less effort has been devoted to InN and In-rich InGaN alloys. A major breakthrough in 2002, stemming from much improved quality of InN films grown using molecular beam epitaxy, resulted in the bandgap of InN being revised from 1.9 eV to a much narrower value of 0.64 eV. This finding triggered a worldwide research thrust into the area of narrow-band-gap group-III nitrides. The low value of the InN bandgap provides a basis for a consistent description of the electronic structure of InGaN and InAlN alloys with all compositions. It extends the fundamental bandgap of the group III-nitride alloy system over a wider spectral region, ranging from the near infrared at ∼1.9 μm (0.64 eV for InN) to the ultraviolet at ∼0.36 μm (3.4 eV for GaN...

When group-III nitrides go infrared: New properties and perspectives

Kohn–Sham density functionals are widely used; however, no currently available exchange–correlation functional can predict all chemical properties with chemical accuracy. Here we report a new functional, called MN15, that has broader accuracy than any previously available one. The properties considered in the parameterization include bond energies, atomization energies, ionization potentials, electron affinities, proton affinities, reaction barrier heights, noncovalent interactions, hydrocarbon thermochemistry, isomerization energies, electronic excitation energies, absolute atomic energies, and molecular structures. When compared with 82 other density functionals that have been defined in the literature, MN15 gives the second smallest mean unsigned error (MUE) for 54 data on inherently multiconfigurational systems, the smallest MUE for 313 single-reference chemical data, and the smallest MUE on 87 noncovalent data, with MUEs for these three categories of 4.75, 1.85, and 0.25 kcal mol−1, respectively, as compared to the average MUEs of the other 82 functionals of 14.0, 4.63, and 1.98 kcal mol−1. The MUE for 17 absolute atomic energies is 7.4 kcal mol−1 as compared to an average MUE of the other 82 functionals of 34.6 kcal mol−1. We further tested MN15 for 10 transition-metal coordination energies, the entire S66x8 database of noncovalent interactions, 21 transition-metal reaction barrier heights, 69 electronic excitation energies of organic molecules, 31 semiconductor band gaps, seven transition-metal dimer bond lengths, and 193 bond lengths of 47 organic molecules. The MN15 functional not only performs very well for our training set, which has 481 pieces of data, but also performs very well for our test set, which has 823 data that are not in our training set. The test set includes both ground-state properties and molecular excitation energies. For the latter MN15 achieves simultaneous accuracy for both valence and Rydberg electronic excitations when used with linear-response time-dependent density functional theory, with an MUE of less than 0.3 eV for both types of excitations.

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MN15: A Kohn-Sham global-hybrid exchange-correlation density functional with broad accuracy for multi-reference and single-reference systems and noncovalent interactions.

We have derived consistent sets of band parameters band gaps, crystal field splittings, band-gap deformation potentials, effective masses, and Luttinger and EP parameters for AlN, GaN, and InN in the zinc-blende and wurtzite phases employing many-body perturbation theory in the G0W0 approximation. The G0W0 method has been combined with density-functional theory DFT calculations in the exact-exchange optimized effective potential approach to overcome the limitations of local-density or gradient-corrected DFT functionals. The band structures in the vicinity of the point have been used to directly parametrize a 44 k·p Hamiltonian to capture nonparabolicities in the conduction bands and the more complex valence-band structure of the wurtzite phases. We demonstrate that the band parameters derived in this fashion are in very good agreement with the available experimental data and provide reliable predictions for all parameters, which have not been determined experimentally so far.

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Consistent set of band parameters for the group-III nitrides AlN, GaN, and InN

We report Auger recombination rates for wurtzite InGaN calculated from first-principles density-functional and many-body-perturbation theory. Two different mechanisms are examined—inter- and intra-band recombination—that affect different parts of the emission spectrum. In the blue to green spectral region and at room temperature the Auger coefficient can be as large as 2×10−30 cm6 s−1; in the infrared it is even larger. Since Auger recombination scales with the cubic power of the free-carrier concentration it becomes an important nonradiative loss mechanism at high current densities. Our results indicate that Auger recombination may be responsible for the loss of quantum efficiency that affects InGaN-based light emitters.

Auger recombination rates in nitrides from first principles

The strain fields in and around self-organized $\mathrm{In}(\mathrm{Ga})\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots (QDs) sensitively depend on QD geometry, average InGaAs composition, and the $\mathrm{In}∕\mathrm{Ga}$ distribution profile. Piezoelectric fields of varying sizes are one result of these strain fields. We study systematically a large variety of realistic QD geometries and composition profiles, and calculate the linear and quadratic parts of the piezoelectric field. The balance of the two orders depends strongly on the QD shape and composition. For pyramidal InAs QDs with sharp interfaces, a strong dominance of the second-order fields is found. Upon annealing, the first-order terms become dominant, resulting in a reordering of the electron $p$ and $d$ states and a reorientation of the hole wave functions.

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Impact of size, shape, and composition on piezoelectric effects and electronic properties of In ( Ga ) As ∕ Ga As quantum dots

We present an eight-band $\mathbf{k}∙\mathbf{p}$-model for the calculation of the electronic structure of wurtzite semiconductor quantum dots (QDs) and its application to indium gallium nitride $({\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N})$ QDs formed by composition fluctuations in ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N}$ layers. The eight-band $\mathbf{k}∙\mathbf{p}$-model accounts for strain effects, piezoelectricity and pyroelectricity, and spin-orbit and crystal-field splitting. Exciton binding energies are calculated using the self-consistent Hartree method. Using this model, we studied the electronic properties of ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N}$ QDs and their dependence on structural properties, i.e., their chemical composition, height, and lateral diameter. We found a dominant influence of the built-in piezoelectric and pyroelectric fields, causing a spatial separation of the bound electron and hole states and a redshift of the exciton transition energies. The single-particle energies as well as the exciton energies depend heavily on the composition and geometry of the QDs.

Interrelation of structural and electronic properties in InxGa1-xN/GaN quantum dots using an eight-band k · p model

The authors have studied the electronic structure of InN and GaN employing G0W0 calculations based on exact-exchange density-functional theory. For InN their approach predicts a gap of 0.7eV. Taking the Burnstein-Moss effect into account, the increase of the apparent quasiparticle gap with increasing electron concentration is in good agreement with the observed blueshift of the experimental optical absorption edge. Moreover, the concentration dependence of the effective mass, which results from the nonparabolicity of the conduction band, agrees well with recent experimental findings. Based on the quasiparticle band structure the parameter set for a 4×4k∙p Hamiltonian has been derived.

/pdf/band-gap-and-band-parameters-of-inn-and-gan-from-16hw2diw0o.pdf

Band gap and band parameters of InN and GaN from quasiparticle energy calculations based on exact-exchange density-functional theory

We have studied the electronic structure of InN and GaN employing G0W0 calculations based on exact-exchange density-functional theory. For InN our approach predicts a gap of 0.7 eV. Taking the Burnstein-Moss effect into account, the increase of the apparent quasiparticle gap with increasing electron concentration is in good agreement with the observed blue shift of the experimental optical absorption edge. Moreover, the concentration dependence of the effective mass, which results from the non-parabolicity of the conduction band, agrees well with recent experimental findings. Based on the quasiparticle band structure the parameter set for a 4x4 kp Hamiltonian has been derived.

Momme Winkelnkemper

Papers

Consistent set of band parameters for the group-III nitrides AlN, GaN, and InN

Impact of size, shape, and composition on piezoelectric effects and electronic properties of In ( Ga ) As ∕ Ga As quantum dots

Interrelation of structural and electronic properties in InxGa1-xN/GaN quantum dots using an eight-band k · p model

Band gap and band parameters of InN and GaN from quasiparticle energy calculations based on exact-exchange density-functional theory

Band gap and band parameters of InN and GaN from quasiparticle energy calculations based on exact-exchange density-functional theory