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In this work we derive closed expressions for the head of the frequency-dependent microscopic polarizability matrix in the projector-augmented wave ͑PAW͒ methodology. Contrary to previous applications, the longitudinal expression is utilized, resulting in dielectric properties that are largely independent of the applied potentials. The improved accuracy of the present approach is demonstrated by comparing the longitudinal and transversal expressions of the polarizability matrix for a number of cubic semiconductors and one insulator, i.e., Si, SiC, AlP, GaAs, and diamond ͑C͒, respectively. The methodology is readily extendable to more complicated nonlocal Hamiltonians or to the calculation of the macroscopic dielectric matrix including local field effects in the random phase or density functional approximation, which is demonstrated for the previously mentioned model systems. Furthermore, density functional perturbation theory is extended to the PAW method, and the respective results are compared to those obtained by summation over the conduction band states.

We present a comparative full-potential study of generalized Kohn-Sham schemes (gKS) with explicit focus on their suitability as starting point for the solution of the quasiparticle equation. We compare $G_0W_0$ quasiparticle band structures calculated upon LDA, sX, HSE03, PBE0, and HF functionals for exchange and correlation (XC) for Si, InN and ZnO. Furthermore, the HSE03 functional is studied and compared to the GGA for 15 non-metallic materials for its use as a starting point in the calculation of quasiparticle excitation energies. For this case, also the effects of selfconsistency in the $GW$ self-energy are analysed. It is shown that the use of a gKS scheme as a starting point for a perturbative QP correction can improve upon the deficiencies found for LDA or GGA staring points for compounds with shallow $d$ bands. For these solids, the order of the valence and conduction bands is often inverted using local or semi-local approximations for XC, which makes perturbative $G_0W_0$ calculations unreliable. The use of a gKS starting point allows for the calculation of fairly accurate band gaps even in these difficult cases, and generally single-shot $G_0W_0$ calculations following calculations using the HSE03 functional are very close to experiment

The phonon dispersion relations of diamond and graphite are calculated using an ab initio force constant method. The force constants are calculated via a self-consistent supercell approach in the local-density approximation in terms of the Hellmann-Feynman forces induced by the displacement of a single atom in the supercell. For diamond our ab initio results are in very good agreement with the neutron inelastic scattering and Raman data. For graphite we find good agreement with the neutron data for the low-energy modes as well as with the reflections electron energy loss spectroscopy (REELS) and optical data at higher energies. Significant differences to the predictions of semi-empirical models appear.

The physical properties of InN crystals are known rather poorly, since the existing growth techniques have not produced epitaxial layers of good quality [1,2]. Even a key parameter of InN -the band gap E g -has not been firmly established so far. E g values of 1.8 eV to 2.1 eV have usually been estimated from the absorption spectra obtained on polycrystalline and nanocrystalline hexagonal InN [3][4][5][6]. No data on the band-to-band photoluminescence (PL) of InN are available in the literature. Recently an improved growth technique has made it possible to obtain single-crystalline InN layers [7]. Optical measurements on these InN layers have shown some strong differences from absorption data reported earlier [8]. In the present work the electronic structure of singlecrystalline InN layers was carefully studied by means of optical absorption, PL, and photoluminescence excitation (PLE) spectroscopy as well as by ab initio calculations. Our results revealed for hexagonal InN a band gap of about 0.9 eV, which is much smaller than the values of 1.8 eV to 2.1 eV reported previously.Single-crystalline InN epilayers were grown on (0001) sapphire substrates either by plasma-assisted molecular-beam epitaxy (PAMBE) [7] or metalorganic molecular-beam epitaxy (MOMBE) [9] and were characterized by many techniques. Only hexagonal symmetry, with no traces of other polymorphs, was established by X-ray analysis in all the samples. For characterization the symmetric (0002) and asymmetric ð11 2 24Þ Bragg reflexes were used. From these data the lattice constants in the InN layers were found to be c ¼ 5.7039 A and a ¼ 3.5365 A. The narrow profiles of q and q-2q scans at the (0002) reflex (250-300 arcsec and 50-60 arcsec, respectively) indicate a good crystalline quality. Polarized Raman spectra of InN show agreement with the selection rules for the hexagonal symmetry. The Raman phonon line widths correspond to a well-ordered crystal lattice [9,10]. Atomic force microscopy measurements did not reveal any columnar structure in the samples studied. According to the Auger data, the oxygen concentration did not exceed 0.1%. The Hall concentration of electrons n ranged from 9 Â 10 18 to 1.2 Â 10 19 cm -3 in the best samples, and their mobility was found to be as high as m $ 1900 cm 2 V -1 s -1 .The absorption coefficient a(w) for PAMBE-and MOMBE-grown InN samples at 300 K is shown in Fig. 1. The layer thickness was measured by means of scanning electron microscopy. The aðwÞ spectra were calculated from the transmission spectra with corrections for multiple reflections. It can be seen that the edge absorption rapidly reaches values of a(w) > 5 Â 10 4 cm À1 , which is typical of direct band-gap crystals. The inset in Fig. 1 shows that the absorption coefficient can be described by the relation a(w) $ ( hw -E g ) 1/2 usually applicable to allowed direct interband transitions. From the measurement of the absorption edges it can be concluded that the E g phys. stat. sol. (b) 229, No. 3, R1-R3 (2002)

Silicon crystallized in the usual cubic (diamond) lattice structure has dominated the electronics industry for more than half a century. However, cubic silicon (Si), germanium (Ge) and SiGe-alloys are all indirect bandgap semiconductors that cannot emit light efficiently. Accordingly, achieving efficient light emission from group-IV materials has been a holy grail 1 in silicon technology for decades and, despite tremendous efforts 2-5 , it has remained elusive 6 . Here, we demonstrate efficient light emission from direct bandgap hexagonal Ge and SiGe alloys. We measure a sub nanosecond, temperature insensitive radiative recombination lifetime and observe a similar emission yield to direct bandgap III-V semiconductors. Moreover, we demonstrate how by controlling the composition of the hexagonal SiGe alloy, the emission wavelength can be continuously tuned in a broad range, while preserving a direct bandgap. Our experimental findings are shown to be in excellent quantitative agreement with the ab initio theory. Hexagonal SiGe embodies an ideal material system to fully unite electronic and optoelectronic functionalities on a single chip, opening the way towards novel device concepts and information processing technologies.Silicon has been the workhorse of the semiconductor industry since it has many highly advantageous physical, electronic and technological properties. However, due to its indirect bandgap, silicon cannot emit light efficientlya property that has seriously constrained potential for applications to electronics and passive optical circuitry 7-9 . Silicon technology can only reach its full application potential when heterogeneously supplemented 10 with an efficient, direct bandgap light emitter.The band structure of cubic Si, presented in Fig. 1a is very well known, having the lowest conduction band (CB) minimum close to the X-point and a second lowest * These authors contributed equally to this work. † Correspondence to E.P.A.M.(e.p.a.m.bakkers@tue.nl).minimum at the L-point.As such, it is the archetypal example of an indirect bandgap semiconductor, that, notwithstanding many great efforts 3-6 , cannot be used for efficient light emission.By modifying the crystal structure from cubic to hexagonal, the symmetry along the 111 crystal direction changes fundamentally, with the consequence that the L-point bands are folded back onto the Γ-point. As shown in Fig. 1b, for hexagonal Si (Hex-Si) this results in a local CB minimum at the Γ-point, with an energy close to 1.7 eV 11-13 . Clearly, Hex-Si remains indirect since the lowest energy CB minimum is at the M-point, close to 1.1 eV. Cubic Ge also has an indirect bandgap but, unlike Si, the lowest CB minimum is situated at the L-point, as shown in Fig. 1c. As a consequence, for Hex-Ge the band folding effect results in a direct bandgap at the Γ-point with a magnitude close to 0.3 eV, as shown in the calculated band structure in Fig. 1d 14 .To investigate how the direct bandgap energy can be tuned by alloying Ge with Si, we calculated the band structures of He...

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