Large Supercells Research Articles

High-order harmonic generation in semiconductors driven at near- and mid-infrared wavelengths

We study high order harmonics generation (HHG) in crystalline silicon and diamond subjected to near and mid-infrared laser pulses. We employ time-dependent density functional theory and solve the time-dependent Kohn-Sham equation in the single-cell geometry. We demonstrate that clear and clean HHG spectra can be generated with careful selection of the pulse duration. In addition, we simulate dephasing effects in a large silicon super-cell through displacement of atomic positions prepared by a molecular dynamics simulation. We compare our results with the previous calculations by Floss et al. [arXiv:1705.10707] [Phys. Rev. A 97, 011401(R) (2018)] on Diamond at 800 nm and by Tancogne-Dejean et al. [arXiv:1609.09298] [Phys. Rev. Lett. 118, 087403 (2017)] on Si at 3000 nm.

Intrinsic local symmetry breaking in nominally cubic paraelectric BaTiO3

Whereas at low-temperatures ferroelectrics have a well understood ordered spatial dipole arrangement, the fate of these dipole configurations in the higher temperature paraelectric (PE) phase remains poorly understood. Using density functional theory (DFT), we find that, unlike the case in nonpolar $AB{\mathrm{O}}_{3}$, perovskites such as cubic $\mathrm{BaZr}{\mathrm{O}}_{3}$ that do not lower their energy by any form of positional symmetry breaking, the origin of distribution of the $B$-site off-centering in cubic PE such as $\mathrm{BaTi}{\mathrm{O}}_{3}$ is an intrinsic, energy lowering due to symmetry breaking. Minimizing the internal energy ${E}_{0}$ of a constrained cubic phase represented by a large enough supercell to accommodate symmetry breaking already reveals the presence of a distribution of local displacements (i.e., a polymorphous network) that (i) locally mimics the symmetries of the low temperature phases, while (ii) being the precursors of what finite temperature density functional theory (DFT) molecular dynamics (MD) finds as thermal motifs when equilibrating the free energy ${E}_{0}\ensuremath{-}TS$. Analyzing the DFT-derived configurations of the PE cubic supercell by projecting its displacements onto irreducible representations reveals that it is best described as a temperature-dependent superposition of numerous modes, including ferroelectric $({{\mathrm{\ensuremath{\Gamma}}}_{4}}^{--})$ and antiferroelectric (${{M}_{2}}^{--}$ and ${{X}_{5}}^{+}$), rather than a single mode, (e.g., ${{X}_{5}}^{+}$) representing a well-defined long-range ordered configuration. This suggests that the electronic and dielectric properties of such PE phases are best calculated from a polymorphous distribution of interrelated local motifs in large supercells rather than from either purely disordered or long-range ordered models. In this respect, PE phases share a common feature with paramagnetic (PM) and paraelastic (PEL) perovskite phases whose central feature is a polymorphous distribution of local motifs---dipole moments in PE, magnetic moments in PM, and ordinary octahedral distortion modes in PEL---all computable by DFT supercells and useable in calculating electronic and magnetic properties of paraphases.

Electronic structure of intrinsic defects in c -gallium nitride: Density functional theory study without the jellium approximation

We report the first nonjellium, systematic, density functional theory (DFT) study of intrinsic and extrinsic defects and defect levels in zinc-blende (cubic) gallium nitride. We use the local moment counter charge (LMCC) method, the standard Perdew-Becke-Ernzerhoff (PBE) exchange-correlation potential, and two pseudopotentials, where the Ga $3d$ orbitals are either in the core $({d}^{0})$ or explicitly in the valence set $({d}^{10})$. We studied 64, 216, 512, and 1000 atom supercells, and demonstrated convergence to the infinite limit, crucial for delineating deep from shallow states near band edges, and for demonstrating the elimination of finite cell-size errors. Contrary to common claims, we find that exact exchange is not required to obtain defect levels across the experimental band gap. As was true in silicon, silicon carbide, and gallium arsenide, the extremal LMCC defect levels of the aggregate of defects yield an effective LMCC defect band gap that is within 10% of the experimental gap (3.3 eV) for both pseudopotentials. We demonstrate that the gallium vacancy is more complicated than previously reported. There is dramatic metastability--a nearest-neighbor nitrogen atom shifts into the gallium site, forming an antisite, nitrogen vacancy pair, which is more stable than the simple vacancy for positive charge states. Our assessment of the ${d}^{0}$ and ${d}^{10}$ pseudopotentials yields minimal differences in defect structures and defect levels. The better agreement of the ${d}^{0}$ lattice constant with experiment suggests that the more computationally economical ${d}^{0}$ pseudopotentials are sufficient to achieve the fidelity possible within the physical accuracy of DFT, and thereby enable calculations in larger supercells necessary to demonstrate convergence with respect to finite size supercell errors.

Real-space representation of the quasiparticle self-consistent GW self-energy and its application to defect calculations

The quasiparticle self-consistent QS$GW$ approach incorporates the corrections of the quasiparticle energies from their Kohn-Sham density functional theory (DFT) eigenvalues by means of an energy independent and Hermitian self-energy matrix usually given in the basis set of the DFT eigenstates. By expanding these into an atom-centered basis set (specifically here the linearized muffin-tin orbitals) a real space representation of the self-energy corrections becomes possible. We show that this representation is relatively short-ranged. This offers new opportunities to construct the self-energy of a complex system from parts of the system by a cut-and-paste method. Specifically for a point defect, represented in a large supercell, the self-eneregy can be constructed from those of the host and a smaller defect containing cell. The self-energy of the periodic host can be constructed simply from a $GW$ calculation for the primitive cell. We show for the case of the As$_\mathrm{Ga}$ in GaAs that the defect part can already be well represented by a minimal 8 atom cell and allows us to construct the self-energy for a 64 cell in good agreement with direct QS$GW$ calculations for the large cell. Using this approach to an even larger 216 atom cell shows the defect band approaches an isolated defect level. The calculations also allow to identify a second defect band which appears as a resonance near the conduction band minimum. The results on the extracted defect levels agree well with Green's function calculations for an isolated defect and with experimental data.

Cation Disorder and Large Tetragonal Supercell Ordering in the Li-Rich Argyrodite Li7Zn0.5SiS6.

A tetragonal argyrodite with >7 mobile cations, Li7Zn0.5SiS6, is experimentally realized for the first time through solid state synthesis and exploration of the Li–Zn–Si–S phase diagram. The crystal structure of Li7Zn0.5SiS6 was solved ab initio from high-resolution X-ray and neutron powder diffraction data and supported by solid-state NMR. Li7Zn0.5SiS6 adopts a tetragonal I4 structure at room temperature with ordered Li and Zn positions and undergoes a transition above 411.1 K to a higher symmetry disordered F43m structure more typical of Li-containing argyrodites. Simultaneous occupation of four types of Li site (T5, T5a, T2, T4) at high temperature and five types of Li site (T5, T2, T4, T1, and a new trigonal planar T2a position) at room temperature is observed. This combination of sites forms interconnected Li pathways driven by the incorporation of Zn2+ into the Li sublattice and enables a range of possible jump processes. Zn2+ occupies the 48h T5 site in the high-temperature F43m structure, and a unique ordering pattern emerges in which only a subset of these T5 sites are occupied at room temperature in I4 Li7Zn0.5SiS6. The ionic conductivity, examined via AC impedance spectroscopy and VT-NMR, is 1.0(2) × 10–7 S cm–1 at room temperature and 4.3(4) × 10–4 S cm–1 at 503 K. The transition between the ordered I4 and disordered F43m structures is associated with a dramatic decrease in activation energy to 0.34(1) eV above 411 K. The incorporation of a small amount of Zn2+ exercises dramatic control of Li order in Li7Zn0.5SiS6 yielding a previously unseen distribution of Li sites, expanding our understanding of structure–property relationships in argyrodite materials.

Role of chemical disorder on radiation-induced defect production and damage evolution in NiFeCoCr

Understanding chemical disorder in many concentrated solid solution alloys (CSAs) at the levels of electrons and atoms has attracted increasing attention as a path forward to reveal and identify underlying mechanisms for extraordinary mechanical properties and improved radiation tolerance. Single-phase NiFeCoCr CSA is a common base for many high-entropy alloys (HEAs) that have shown improved mechanical strength and radiation tolerance. In this study, defect production and damage evolution in NiFeCoCr under ion irradiation at room temperature to dose over 20 dpa are determined using ion channeling technique along both <100> and <110> directions utilizing multiple probing beam energies. The results obtained from the multi-axial and multi-energy channeling analysis are compared with those previously obtained for Ni crystals irradiated under similar conditions. The influence of chemical complexity on defect production and clustering at early-stage under room temperature irradiation up to dose of 1 dpa is discussed based on positron annihilation spectroscopy results. Defect structure evaluation in Ni and NiFeCoCr is also discussed based on transmission electron microscopy results over a prolonged irradiation at both room and elevated temperatures. Compared with chemically complex NiFeCoCr, larger dislocation loops thus less lattice strain are expected to form in pure Ni. Moreover, the role of chemical disorder in this CSA is also investigated based on ab initio calculations using large supercells. To understand the impact of chemical complexity effect on defect structure evolution, this integrated research effort attempts to link the relatively large charge redistribution due to difference in valence electron counts resulting from alloying different 3d transition metal elements, moderate lattice distortion arising from similar adaptable atomic size, and notable suppressed or delayed damage evolution in NiFeCoCr.

First-principles study of radiation defects in silicon

The study of the properties of defects in silicon forming under irradiation condition has been carried out for many years, however, many open questions remain. Particularly, there are not comprehensive results for variety of radiation centers calculated at the same level of theory. For example the previously calculated transition levels and formation energies show a large scatter. In this work, we focus on the important radiation-induced defect complexes in Si: double vacancy, vacancy-oxygen, vacancy-phosphorus. Additionally, the phosphorus–vacancy-oxygen complex was studied. Formation energy, charge transition levels and binding energy were calculated from first-principles using the hybrid exchange–correlation functional HSE06. Spin-polarized calculations and large supercells allows us to obtain charge transition levels which agree well with experimental measurements.

The AiiDA-Spirit Plugin for Automated Spin-Dynamics Simulations and Multi-Scale Modeling Based on First-Principles Calculations

Landau-Lifshitz-Gilbert (LLG) spin-dynamics calculations based on the extended Heisenberg Hamiltonian is an important tool in computational materials science involving magnetic materials. LLG simulations allow to bridge the gap from expensive quantum mechanical calculations with small unit cells to large supercells where the collective behavior of millions of spins can be studied. In this work we present the AiiDA-Spirit plugin that connects the spin-dynamics code Spirit to the AiiDA framework. AiiDA provides a Python interface that facilitates performing high-throughput calculations while automatically augmenting the calculations with metadata describing the data provenance between calculations in a directed acyclic graph. The AiiDA-Spirit interface thus provides an easy way for high-throughput spin-dynamics calculations. The interface to the AiiDA infrastructure furthermore has the advantage that input parameters for the extended Heisenberg model can be extracted from high-throughput first-principles calculations including a proper treatment of the data provenance that ensures reproducibility of the calculation results in accordance to the FAIR principles. We describe the layout of the AiiDA-Spirit plugin and demonstrate its capabilities using selected examples for LLG spin-dynamics and Monte Carlo calculations. Furthermore, the integration with first-principles calculations through AiiDA is demonstrated at the example of γ–Fe, where the complex spin-spiral ground state is investigated.

Electronic and optical properties of SixGe1−x−ySny alloys lattice-matched to Ge

We present a combined experimental and theoretical analysis of the evolution of the near-band gap electronic and optical properties of Si$_{x}$Ge$_{1-x-y}$Sn$_{y}$ alloys lattice-matched to Ge and GaAs substrates. We perform photoreflectance (PR) and photoluminescence (PL) measurements on Si$_{x}$Ge$_{1-x-y}$Sn$_{y}$ epitaxial layers grown via chemical vapour deposition, for Si (Sn) compositions up to $x =$ 9.6% ($y =$ 2.5%). Our measurements indicate the presence of an indirect fundamental band gap, with PL observed $\approx$ 200-250 meV lower in energy than the direct $E_0$ transition identified by PR measurements. The measured PL is Ge-like, suggesting that the alloy conduction band (CB) edge is primarily derived from the Ge L-point CB minimum. Interpretation of the PR and PL measurements is supported by atomistic electronic structure calculations. Effective alloy band structures calculated via density functional theory confirm the presence of an indirect fundamental band gap, and reveal the origin of the observed inhomogeneous broadening of the measured optical spectra as being alloy-induced band hybridisation occurring close in energy to the CB edge. To analyze the evolution of the band gap, semi-empirical tight-binding (TB) calculations are employed to enable calculations for large supercell sizes. TB calculations reveal that the alloy CB edge is hybridized in nature, consisting at low Si and Sn compositions of an admixture of Ge L-, $\Gamma$- and X-point CB edge states, and confirm that the alloy CB edge retains primarily Ge L-point CB edge character. Our experimental measurements and theoretical calculations confirm a direct transition energy close to 1 eV in magnitude for Si and Sn compositions $x =$ 6.8 - 9.6% and $y =$ 1.6 - 2.2%.

Exploring metastable states in UO2 using hybrid functionals and dynamical mean field theory

A detailed exploration of the f-atomic orbital occupancy space for UO2 is performed using a first principles approach based on density functional theory (DFT), employing a full hybrid functional within a systematic basis set. Specifically, the PBE0 functional is combined with an occupancy biasing scheme implemented in a wavelet-based algorithm which is adapted to large supercells. The results are compared with previous DFT + U calculations reported in the literature, while dynamical mean field theory is also performed to provide a further base for comparison. This work shows that the computational complexity of the energy landscape of a correlated f-electron oxide is much richer than has previously been demonstrated. The resulting calculations provide evidence of the existence of multiple previously unexplored metastable electronic states of UO2, including those with energies which are lower than previously reported ground states.

Thermodynamics and kinetics of H adsorption and intercalation for graphene on 6H-SiC(0001) from first-principles calculations

Previous experimental observations for H intercalation under graphene on SiC surfaces motivate the clarification of configuration stabilities and kinetic processes related to intercalation. From first-principles density-functional-theory calculations, we analyze H adsorption and intercalation for graphene on a 6H-SiC(0001) surface, where the system includes two single-atom-thick graphene layers: the top-layer graphene (TLG) and the underling buffer-layer graphene (BLG) above the terminal Si layer. Our chemical potential analysis shows that in the low-H coverage regime (described by a single H atom within a sufficiently large supercell), intercalation into the gallery between TLG and BLG or into the gallery underneath BLG is more favorable thermodynamically than adsorption on top of TLG. However, intercalation into the gallery between TLG and BLG is most favorable. We obtain energy barriers of about 1.3 and 2.3 eV for a H atom diffusing on and under TLG, respectively. From an additional analysis of the energy landscape in the vicinity of a step on the TLG, we assess how readily one guest H atom on the TLG terrace can directly penetrate the TLG into the gallery between TLG and BLG versus crossing a TLG step to access the gallery. We also perform density functional theory calculations for higher H coverages revealing a shift in favorability to intercalation of H underneath BLG and characterizing the variation with H coverage in interlayer spacings.

Characterization of the negatively charged NV defect through the spin density distribution and the hyperfine coupling constants

The spin density function and hyperfine coupling constants of the NV− defect (a vacancy with one nitrogen atom as a first neighbor) in its triplet ground state, are computed by using hybrid functionals, a Gaussian type all electron basis set, the supercell scheme (216 atoms before the defect formation), and the CRYSTAL code. The charged defect has been simulated by using the charge compensated (CC) scheme, in which a background of positive charge is added to restore the neutrality of the unit cell. The local geometry, charge and spin distribution are also reported and discussed. The EPR constants (Fermi contact, the hyperfine coupling and electric field gradient tensors) of the N (both 14N and 15N) and of two C (for the 13C isotope) atoms around the vacancy are in excellent agreement with the recent experimental results by Felton et al. (Phys. Rev. B, 79,075 203, 2009). The comparison is extended to the many other experimental data collected in the last 30 years, to simpler functionals (LDA and PBE), larger supercells (512 atoms), and to recent simulations. It is shown that the present results are much closer to experiment than previous theoretical studies, with a quantitative, rather than qualitative, agreement.

Accessing Structural, Electronic, Transport and Mesoscale Properties of Li-GICs via a Complete DFTB Model with Machine-Learned Repulsion Potential.

Lithium-graphite intercalation compounds (Li-GICs) are the most popular anode material for modern lithium-ion batteries and have been subject to numerous studies—both experimental and theoretical. However, the system is still far from being consistently understood in detail across the full range of state of charge (SOC). The performance of approaches based on density functional theory (DFT) varies greatly depending on the choice of functional, and their computational cost is far too high for the large supercells necessary to study dilute and non-equilibrium configurations which are of paramount importance for understanding a complete charging cycle. On the other hand, cheap machine learning methods have made some progress in predicting, e.g., formation energetics, but fail to provide the full picture, including electrostatics and migration barriers. Following up on our previous work, we deliver on the promise of providing a complete and affordable simulation framework for Li-GICs. It is based on density functional tight binding (DFTB), which is fitted to dispersion-corrected DFT data using Gaussian process regression (GPR). In this work, we added the previously neglected lithium–lithium repulsion potential and extend the training set to include superdense Li-GICs (LiC6−x; ) and lithium metal, allowing for the investigation of dendrite formation, next-generation modified GIC anodes, and non-equilibrium states during fast charging processes in the future. For an extended range of structural and energetic properties—layer spacing, bond lengths, formation energies and migration barriers—our method compares favorably with experimental results and with state-of-the-art dispersion-corrected DFT at a fraction of the computational cost. We make use of this by investigating some larger-scale system properties—long range Li–Li interactions, dielectric constants and domain-formation—proving our method’s capability to bring to light new insights into the Li-GIC system and bridge the gap between DFT and meso-scale methods such as cluster expansions and kinetic Monte Carlo simulations.

Parameterization of magnetic vector potentials and fields for efficient multislice calculations of elastic electron scattering.

The multislice method, which simulates the propagation of the incident electron wavefunction through a crystal, is a well established method for analysing the multiple scattering effects that an electron beam may undergo. The inclusion of magnetic effects into this method proves crucial towards simulating enhanced magnetic interaction of vortex beams with magnetic materials, calculating magnetic Bragg spots or searching for magnon signatures, to name a few examples. Inclusion of magnetism poses novel challenges to the efficiency of the multislice method for larger systems, especially regarding the consistent computation of magnetic vector potentials A and magnetic fields B over large supercells. This work presents a tabulation of parameterized magnetic (PM) values for the first three rows of transition metal elements computed from atomic density functional theory (DFT) calculations, allowing for the efficient computation of approximate A and B across large crystals using only structural and magnetic moment size and direction information. Ferromagnetic b.c.c. (body-centred cubic) Fe and tetragonal FePt are chosen to showcase the performance of PM values versus directly obtaining A and B from the unit-cell spin density by DFT. The magnetic fields of b.c.c. Fe are well described by the PM approach while for FePt the PM approach is less accurate due to deformations in the spin density. Calculations of the magnetic signal, namely the change due to A and B of the intensity of diffraction patterns, show that the PM approach for both b.c.c. Fe and FePt is able to describe the effects of magnetism in these systems to a good degree of accuracy.

First Principles Calculations of Atomic and Electronic Structure of TiAl3+- and TiAl2+-Doped YAlO3.

In this paper, the density functional theory accompanied with linear combination of atomic orbitals (LCAO) method is applied to study the atomic and electronic structure of the Ti3+ and Ti2+ ions substituted for the host Al atom in orthorhombic Pbnm bulk YAlO3 crystals. The disordered crystalline structure of YAlO3 was modelled in a large supercell containing 160 atoms, allowing simulation of a substitutional dopant with a concentration of about 3%. In the case of the Ti2+-doped YAlO3, compensated F-center (oxygen vacancy with two trapped electrons) is inserted close to the Ti to make the unit cell neutral. Changes of the interatomic distances and angles between the chemical bonds in the defect-containing lattices were analyzed and quantified. The positions of various defect levels in the host band gap were determined.

First-principles calculation of lattice distortions in four single phase high entropy alloys with experimental validation

Exceptional properties of high entropy alloys (HEAs) are attributed to the disordered random solid solution of multiple alloying elements. Despite numerous studies, the fundamental understanding at the electronic and atomic levels is still missing. We report a comparative study of four fcc HEAs of NiFeCoCr and NiFeCoCrX (X = Mn, Cu, or Pd) based on ab initio calculations using large supercells with 500 atoms in equal composition. After fully optimizing their structures using the VASP package, their electronic structure, interatomic bonding, partial charge distribution, and mechanical properties are calculated and compared, revealing the intricate interdependence among them. A novel parameter based on the quantum mechanical metric for internal cohesion, the total bond order density (TBOD), is used to interpret the calculated properties. The highest TBOD is found in Cantor alloy NiFeCoCrMn but lower in NiFeCoCrPd. The atomic radii vary depending on their local chemical environment. The resulting lattice distortions is validated experimentally in NiFeCoCrMn and NiFeCoCrPd. Moreover, modeling of Cu and Pd clustering in the supercell shows they have lower total energy in agreement with the observation of Cu-enhanced nano-participates in NiFeCoCrCu and Pd-induced concentration wave in NiFeCoCrPd HEAs.

Thermal fluctuations and carrier localization induced by dynamic disorder in MAPbI3 described by first-principles based tight-binding model

Halide perovskites are strongly influenced by large amplitude anharmonic lattice fluctuations at room temperature. We develop a tight-binding model for dynamically disordered ${\mathrm{MAPbI}}_{3}$ based on density functional theory calculations to calculate electronic structure for finite temperature crystal structures at the length scale of thermal disorder and carrier localization. The model predicts individual Hamiltonian matrix elements and band structures with high accuracy, owing to the inclusion of additional matrix elements and descriptors for non-Coulombic interactions. We apply this model to electronic structure at length and timescales inaccessible to first-principles methods, finding an increase in band gap, carrier mass, and the subpicosecond fluctuations in these quantities with increasing temperature as well as the onset of carrier localization in large supercells induced by thermal disorder at 300 K. We identify the length scale ${L}^{*}=5$ nm as the onset of localization in the electronic structure, associated with decreasing band edge fluctuations, increasing carrier mass, and Rashba splitting approaching zero.

An efficient computational framework for charge density estimation in twisted bilayer graphene

Electronic properties such as band structure and Fermi velocity in low-angle twisted bilayer graphene (TBG) are intrinsically dependent on the atomic structure. Rigid rotation between individual graphene layers provides an approximate description of the bilayer symmetry. Upon relaxation, in-plane displacement of the atoms in low angle TBG causes a change in the symmetry through the enlargement of the AB stacking regions and the reduction in size of AA and SP stacking regions. However, the effect of this in-plane relaxation on the charge density remains unexplored, because the necessary electronic structure calculations of such large supercells of low twist angle TBG are computationally infeasible. Therefore, we develop a computationally efficient framework that enables the exploration of the charge density symmetry of the low twist angle TBG. This framework is based on the Fourier representation of the charge density which presents high intensity Bragg peaks. We find that with the decrease of twist angle, low intensity satellite peaks also become apparent. Our framework incorporates these satellite peaks which reveals transformation of symmetry in the charge density distribution from high to low twist angle TBG. One striking outcome is the demonstration of the electron localization in the AA region of low twist angle TBG. Our framework helps to explain the effect of the atomistic relaxation on the charge density distribution and thus, it provides information about exotic electronic properties of low twist angle TBG at a low computational expense.

Theoretical and experimental study of the nitrogen-vacancy center in 4H-SiC

Using first-principles calculations and magnetic resonance experiments, we investigated the physical properties of the negatively charged nitrogen-vacancy (NV) center in 4H-SiC, a promising spin qubit. Our predictive theoretical model in conjunction with experimental measurements reveals a large sensitivity to strain and symmetry. The measured and computed zero-phonon lines (ZPLs) are in agreement and show a consistent trend as a function of the defect location in the crystal. The computed ZPLs are extremely sensitive to the geometrical configurations of the ground and excited states, and large supercells with more than 2 000 atoms are required to obtain accurate numerical results. We find that the computed decoherence time of the basal NV centers at zero magnetic field is substantially larger than that of the axial configurations. Furthermore, at natural nuclear spin abundance and zero field, the Hahn-echo coherence time of one of the basal configurations is similar to that of the axial divacancy in isotopically purified SiC.

Quantum vibronic effects on the electronic properties of solid and molecular carbon

We study the effect of quantum vibronic coupling on the electronic properties of carbon allotropes, including molecules and solids, by combining path integral first principles molecular dynamics (FPMD) with a colored noise thermostat. In addition to avoiding several approximations commonly adopted in calculations of electron-phonon coupling, our approach only adds a moderate computational cost to FPMD simulations and hence it is applicable to large supercells, such as those required to describe amorphous solids. We predict the effect of electron-phonon coupling on the fundamental gap of amorphous carbon, and we show that in diamond the zero-phonon renormalization of the band gap is larger than previously reported.