Zero-point Renormalization Research Articles

Machine learning electron-phonon interactions in two-dimensional semiconducting materials: The case of zero-point renormalization

Machine learning electron-phonon interactions in two-dimensional semiconducting materials: The case of zero-point renormalization

The coexistence of high piezoelectricity and superior optical absorption in Janus Bi2X2Y (X = Te, Se; Y = Te, Se, S) monolayers.

Bismuth chalcogenide and its derivatives have been attracting attention in various fields as semiconductors or topological insulators. Inspired by the high piezoelectric properties of Janus Bi2TeSeS monolayer and the excellent optical absorption properties of the Bi2X3 (X = Te, Se, S) monolayers, we theoretically predicted four new-type two-dimensional (2D) monolayers Janus Bi2X2Y (X = Te, Se; Y = Te, Se, S) using the first principles combined with density functional theory (DFT). The thermal, dynamic, and mechanical stabilities of Janus Bi2X2Y monolayers were confirmed based on ab initio molecular dynamics (AIMD) simulations, phonon dispersion, and elastic constants calculations. Their elastic properties, band structures, piezoelectric, and optical properties were systematically investigated. It was found that Janus Bi2X2Y monolayers have a typical Mexican hat-shaped valence band edge structure and, therefore, have a ring-shaped flat band edge, which results in their indirect band gaps. The results show that Janus Bi2X2Y monolayers are semiconductors with moderate band gaps (0.62-0.98 eV at the HSE + SOC level). After considering the electron-phonon renormalization (EPR), the band gaps are reduced by less than 5% at 0 K under the zero-point renormalization (ZPR) and further reduced by approximately 10% at 300 K. Besides, Janus Bi2X2Y monolayers also exhibit excellent optical absorption properties in the blue-UV light region, with the peak values at the order of 8 × 105 cm-1. Particularly, the Janus Bi2Te2S monolayer was found to exhibit a piezoelectric strain coefficient d11 of up to 20.30 pm V-1, which is higher than that of most of the 2D materials. Our results indicate that Janus Bi2X2Y monolayers could be promising candidates in solar cells, optical absorption, and optoelectronic devices; especially, a Janus Bi2Te2S monolayer can also be an excellent piezoelectric material with great prospects in the fields of mechanical and electrical energy conversion.

Ab initio study of temperature-dependent piezoelectric and electronic properties of thermally stable GaPO4.

Gallium-phosphate (GaPO4) is one of the ultra-high thermally stable piezoelectric materials with a high critical temperature of 1206 K. Here, first principles calculations with quasi-harmonic approximation are performed to study thermal and other physical properties of α-GaPO4. For the electronic structure, we focus on the electron-phonon interaction and lattice expansion effects on the temperature-dependent band gap, which plays a significant role in zero-point renormalization. Significantly, the large piezoelectric constants e11 primarily comes from intrinsic sensitivity of Ga and O sites to axial strain, while P atoms contribute little, which remains true in other quartz-like type APO4 (A = B, Al, In). Our work provides an insight into the temperature-dependent electronic and piezoelectric properties of α-GaPO4 and motivates its applications in a high temperature environment.

High-throughput analysis of Fröhlich-type polaron models

The electron–phonon interaction is central to condensed matter, e.g. through electrical resistance, superconductivity or the formation of polarons, and has a strong impact on observables such as band gaps or optical spectra. The most common framework for band energy corrections is the Fröhlich model, which often agrees qualitatively with experiments in polar materials, but has limits for complex cases. A generalized version includes anisotropic and degenerate electron bands, and multiple phonons. In this work, we identify trends and outliers for the Fröhlich models on 1260 materials. We test the limits of the Fröhlich models and their perturbative treatment, in particular the large polaron hypothesis. Among our extended dataset most materials host perturbative large polarons, but there are many instances that are non-perturbative and/or localize on distances of a few bond lengths. We find a variety of behaviors, and analyze extreme cases with huge zero-point renormalization using the first-principles Allen-Heine-Cardona approach.

Quantum Vibronic Effects on the Electronic Properties of Molecular Crystals.

We present a study of molecular crystals, focused on the effect of nuclear quantum motion and anharmonicity on their electronic properties. We consider a system composed of relatively rigid molecules, a diamondoid crystal, and one composed of floppier molecules, NAI-DMAC, a thermally activated delayed fluorescence compound. We compute fundamental electronic gaps at the density functional theory (DFT) level of theory, with the Perdew-Burke-Erzenhof (PBE) and strongly constrained and approximately normed (SCAN) functionals, by coupling first-principles molecular dynamics with a nuclear quantum thermostat. We find a sizable zero-point renormalization (ZPR) of the band gaps, which is much larger in the case of diamondoids (0.6 eV) than for NAI-DMAC (0.22 eV). We show that the frozen phonon (FP) approximation, which neglects intermolecular anharmonic effects, leads to a large error (∼50%) in the calculation of the band gap ZPR. Instead, when using a stochastic method, we obtain results in good agreement with those of our quantum simulations for the diamondoid crystal. However, the agreement is worse for NAI-DMAC where intramolecular anharmonicities contribute to the ZPR. Our results highlight the importance of accurately including nuclear and anharmonic quantum effects to predict the electronic properties of molecular crystals.

Weak electron-phonon renormalization effect caused by the counteraction of the different phonon vibration modes in FeS2

Fluctuations on operating temperatures of solar cells may change the electronic structures of absorption layer materials, which will have a profound influence on the photoelectric conversion efficiency. Based on the electron-phonon renormalization (EPR) method, we investigate the temperature dependence of the band gap of optoelectronic pyrite FeS2. The zero point renormalization (ZPR) on the band gap of FeS2 is less than 100 meV, while the vibration-induced band gap reduction is even less than ZPR within the temperature range 0 ∼ 600 K. The fitted Varshni coefficients have agreement with the experimental result. The relatively small reduction of the band gap by the vibrations can be rationalized by the counteraction of the different phonon modes with opposite influences on the EPR effect. By analyzing the mode-decomposed EPR, we reveal that the shortening of the S-S bond caused by the tilting of FeS6 octahedral (A g phonon mode) is responsible for the increase of the band gap. On the other hand, the change of the Fe-S bond length (T u phonon mode) reduces the band gap value. Our work reveals the theoretical understanding of the weak EPR effect in pyrite FeS2.

Effect of spin-orbit coupling on the zero-point renormalization of the electronic band gap in cubic materials: First-principles calculations and generalized Fröhlich model

The electronic structure of semiconductors and insulators is affected by ionic motion through electron-phonon interaction, yielding temperature-dependent band gap energies and zero-point renormalization (ZPR) at absolute zero temperature. For polar materials, the most significant contribution to the band gap ZPR can be understood in terms of the Fr\"ohlich model, which focuses on the nonadiabatic interaction between an electron and the macroscopic electrical polarization created by a long-wavelength optical longitudinal phonon mode. On the other hand, spin-orbit interaction (SOC) modifies the bare electronic structure, which will, in turn, affect the electron-phonon interaction and the ZPR. We present a comparative investigation of the effect of SOC on the band gap ZPR of twenty semiconductors and insulators with cubic symmetry using first-principles calculations. We observe a SOC-induced decrease of the ZPR, up to 30%, driven by the valence band edge, which almost entirely originates from the modification of the bare electronic eigenenergies and the decrease of the hole effective masses near the $\mathrm{\ensuremath{\Gamma}}$ point. We also incorporate SOC into a generalized Fr\"ohlich model, addressing the Dresselhaus splitting which occurs in noncentrosymmetric materials, and confirm that the predominance of nonadiabatic effects on the band gap ZPR of polar materials is unchanged when including SOC. Our generalized Fr\"ohlich model with SOC provides a reliable estimate of the SOC-induced decrease of the polaron formation energy obtained from first principles and brings to light some fundamental subtleties in the numerical evaluation of the effective masses with SOC for noncentrosymmetric materials. We finally warn about a possible breakdown of the parabolic approximation, one of the most fundamental assumptions of the Fr\"ohlich model, within the physically relevant energy range of the Fr\"ohlich interaction for materials with high phonon frequencies treated with SOC.

Carrier thermalization and zero-point bandgap renormalization in halide perovskites from the Urbach tails of the emission spectrum

We develop techniques to study the temperature dependent localization, thermalization, and the effects of phonon scattering on the excitons in halide perovskites from the analysis of the emission spectra. The excitonic Urbach edge, when inferred from the low energy tails of the temperature dependent luminescence spectra, is shown to be sensitive to the electron distribution and thermalization. A method to observe the Urbach focus is devised for halide perovskites where the temperature dependence of the excitonic gap is anomalous. The value of the zero-point bandgap renormalization is inferred to be about 33 meV. This small value of the bandgap renormalization rules out the formation of small polarons and points to weak electron–phonon coupling. The experiments are performed on the nanosheets of the archetypal halide perovskite, CsPbBr3.

Zero-point renormalization of the band gap of semiconductors and insulators using the projector augmented wave method

We evaluate the zero-point renormalization (ZPR) due to electron-phonon interactions of 28 solids using the projector-augmented-wave (PAW) method. The calculations cover diamond, many zincblende semiconductors, rock-salt and wurtzite oxides, as well as silicate and titania. Particular care is taken to include long-range electrostatic interactions via a generalized Fr\"ohlich model, as discussed in Phys. Rev. Lett. 115, 176401 (2015) and Phys. Rev. B 92, 054307 (2015). The data are compared to recent calculations, npj Computational Materials 6, 167 (2020), and generally very good agreement is found. We discuss in detail the evaluation of the electron-phonon matrix elements within the PAW method. We show that two distinct versions can be obtained depending on when the atomic derivatives are taken. If the PAW transformation is applied before taking derivatives with respect to the ionic positions, equations similar to the ones conventionally used in pseudopotential codes are obtained. If the PAW transformation is used after taking the derivatives, the full-potential spirit is largely maintained. We show that both variants yield very similar ZPRs for selected materials when the rigid-ion approximation is employed. In practice, we find however that the pseudo version converges more rapidly with respect to the number of included unoccupied states.

Zero-point lattice expansion and band gap renormalization: Grüneisen approach versus free energy minimization

The zero-point lattice expansion (ZPLE) is a small variation of the lattice parameters induced by the presence of phonons in a material compared to the static lattice picture. It contributes significantly to the zero-point renormalization (ZPR) of the band gap energy, but its consequences have not been investigated as thoroughly as those stemming from electron-phonon interactions. In the usual first-principles approach, one evaluates the ZPLE by minimizing the $T=0$ K Helmholtz free energy. In this work, we show that the formalism based on the Gr\"uneisen parameters, which commonly neglects zero-point effects, can be efficiently used to compute the ZPLE for both isotropic and anisotropic materials at a much lower computational cost. We systematically test this formalism on 22 cubic and wurtzite materials, and we obtain excellent agreement with free energy minimization results for both the ZPLE and the resulting band gap ZPR. We use our results to validate an empirical expression estimating the ZPLE-induced ZPR, and we unveil its sensitivity to the temperature range involved in estimating the ZPLE from experimental data. Our findings finally reveal that the ZPLE contribution to the band gap ZPR can reach from 20% to more than 80% of the electron-phonon interaction contribution for heavier or more ionic materials, including materials containing light atoms. Considering both contributions on an equal footing is thus essential should one attempt to compare theoretical ZPR results with experimental data.

Combined First-Principles Calculations of Electron-Electron and Electron-Phonon Self-Energies in Condensed Systems.

We present a method to efficiently combine the computation of electron-electron and electron-phonon self-energies, which enables the evaluation of electron-phonon coupling at the G0W0 level of theory for systems with hundreds of atoms. In addition, our approach, which is a generalization of a method recently proposed for molecules [J. Chem. Theory Comput. 2018, 14, 6269-6275], enables the inclusion of nonadiabatic and temperature effects at no additional computational cost. We present results for diamond and defects in diamond and discuss the importance of numerically accurate G0W0 band structures to obtain robust predictions of zero point renormalization (ZPR) of band gaps, and of the inclusion of nonadiabatic effects to accurately compute the ZPR of defect states in the band gap.

Quantum Confinement of Electron-Phonon Coupling in Graphene Quantum Dots.

On the basis of first-principles calculations and the special displacement method, we demonstrate the quantum confinement scaling law of the phonon-induced gap renormalization of graphene quantum dots (GQDs). We employ zigzag-edged GQDs with hydrogen passivation and embedded in hexagonal boron nitride. Our calculations for GQDs in the sub-10 nm region reveal strong quantum confinement of the zero-point renormalization ranging from 20 to 250 meV. To obtain these values we introduce a correction to the Allen-Heine theory of temperature-dependent energy levels that arises from the phonon-induced splitting of 2-fold degenerate edge states. This correction amounts to more than 50% of the gap renormalization. We also present momentum-resolved spectral functions of GQDs, which are not reported in previous contributions. Our results lay the foundation to systematically engineer temperature-dependent electronic structures of GQDs for applications in solar cells, electronic transport, and quantum computing devices.

Giant Linear and Nonlinear Excitonic Responses in an Atomically Thin Indirect Semiconductor Nitrogen Phosphide

Large linear and nonlinear optical responses can be obtained from two-dimensional semiconductors with an indirect electronic gap. Here, we demonstrate exceptionally large exciton-driven responses such as the fundamental exciton binding energy (2 eV), zero-point band-gap renormalization (200 meV), and nonlinear second and third harmonic coefficients (800 pm/V and 1.4 × 10–18 m2/V2, respectively) from an atomically thin binary group-V nitrogen phosphide (NP) semiconductor. The influence of lattice vibrations on the absorption spectra unfolds strong electronic couplings to both LA and ZO-TO phonon modes. All the linear process analyses were computed using a fully ab initio-based G0W0 + Bethe–Salpeter equation approach that also includes the electron–phonon self-energies. The nonlinear processes were instead obtained using a real-time ab initio process flow after creating a coupling between the time-dependent external electric field and the correlated electrons within the modern theory of polarization and the same electron–hole interaction level.

Capturing the Electron-Phonon Renormalization in Molecules from First-Principles.

Since the interaction between electrons and atomic nuclei can affect the electronic structure, in recent years, first-principles-based electron-phonon renormalization methods have been applied in the condensed matter physics community to account for the influence of the electron-phonon coupling in solid systems. However, little is yet known about the behavior and trends of the electron-phonon renormalization in the molecules. In this work, the method for the electron-phonon renormalization in molecules has been derived, using which, we exhaustively investigate the zero-point renormalization in 32 molecules with three different density functions. We find that the renormalization of the highest occupied molecular orbital-lowest unoccupied molecular orbital gap due to electron-vibration coupling does not relate to the atomic masses but quite relates to the electronic structure properties of the molecules.

Assessment of the Mass Factor for the Electron–Phonon Coupling in Solids

Electron–phonon (e–ph) interaction plays an important role in addressing the renormalization of the electronic structure properties. Even at 0 K, such e–ph renormalization still exists because of the zero-point motion of the nuclei, which is known as zero-point renormalization (ZPR). Cardona and Thewalt have found that, for monoatomic materials, the ZPR should be proportional to the inverse square root of the atomic mass. In order to thoughtfully examine the mass factor with the accurate all-electron first-principles approach, we calculate the ZPR of the electronic structure for 82 octet binary materials in both the zincblende and the rocksalt structures. We found that only considering the mass factor is not enough for the ZPR, especially for the investigated polar materials. Besides the mass factor, the lattice geometry, the electronic band structure properties, and the long-range polar corrections are also need to be considered.

FandMcenters in alkali halides: A theoretical study applying self-consistent dielectric-dependent hybrid density functional theory

Point defects significantly change electronic properties of alkali halides and thereby enhance their reactivity. However, both the experimental and theoretical description of defects such as the $F$ center and the $M$ center are still far from complete, in particular for the less common bromides. A self-consistent dielectric-dependent global hybrid and plane-wave approach is employed for a comparative theoretical study of the electronic properties of NaCl, KCl, NaBr, and KBr bulk and (100) surface, both perfect and defective. For these systems, a zero-point renormalization was calculated to account for electron-phonon interaction and enhance comparability with the experiment. We focus on anion vacancy defects, the so-called $F$ and $M$ centers. The methodology employed is capable of reproducing measured defect level energies, electronic band gaps, ionization energies, and electron affinities within experimental errors. A general trend of the $F$ center defect level energy with respect to the lattice parameter is found. The results for both the $F$ and the $M$ center of KCl agree with findings from magnetic resonance experiments. The defect orbitals are analyzed and virtual states of the defect electron are identified.

Predominance of non-adiabatic effects in zero-point renormalization of the electronic band gap

Electronic and optical properties of materials are affected by atomic motion through the electron–phonon interaction: not only band gaps change with temperature, but even at absolute zero temperature, zero-point motion causes band-gap renormalization. We present a large-scale first-principles evaluation of the zero-point renormalization of band edges beyond the adiabatic approximation. For materials with light elements, the band gap renormalization is often larger than 0.3 eV, and up to 0.7 eV. This effect cannot be ignored if accurate band gaps are sought. For infrared-active materials, global agreement with available experimental data is obtained only when non-adiabatic effects are taken into account. They even dominate zero-point renormalization for many materials, as shown by a generalized Fröhlich model that includes multiple phonon branches, anisotropic and degenerate electronic extrema, whose range of validity is established by comparison with first-principles results.

Band gap renormalization, carrier mobilities, and the electron-phonon self-energy in crystalline naphthalene

Organic molecular crystals are expected to feature appreciable electron-phonon interactions that influence their electronic properties at zero and finite temperature. In this work, we report first-principles calculations and an analysis of the electron-phonon self-energy in naphthalene crystals. We compute the zero-point renormalization and temperature dependence of the fundamental band gap, and the resulting scattering lifetimes of electronic states near the valence- and conduction-band edges employing density functional theory. Further, our calculated phonon renormalization of the $GW$-corrected quasiparticle band structure predicts a fundamental band gap of 5 eV for naphthalene at room temperature, in good agreement with experiments. From our calculated phonon-induced electron lifetimes, we obtain the temperature-dependent mobilities of electrons and holes in good agreement with experimental measurements at room temperatures. Finally, we show that an approximate energy self-consistent computational scheme for the electron-phonon self-energy leads to the prediction of strong satellite bands in the electronic band structure. We find that a single calculation of the self-energy can reproduce the self-consistent results of the band gap renormalization and electrical mobilities for naphthalene, provided that the on-the-mass-shell approximation is used, i.e., if the self-energy is evaluated at the bare eigenvalues.

Theory of the special displacement method for electronic structure calculations at finite temperature

Calculations of electronic and optical properties of solids at finite temperature including electron-phonon interactions and quantum zero-point renormalization have enjoyed considerable progress during the past few years. Among the emerging methodologies in this area, we recently proposed a new approach to compute optical spectra at finite temperature including phonon-assisted quantum processes via a single supercell calculation [Zacharias and Giustino, Phys. Rev. B 94, 075125 (2016)]. In the present work we considerably expand the scope of our previous theory starting from a compact reciprocal space formulation, and we demonstrate that this improved approach provides accurate temperature-dependent band structures in three-dimensional and two-dimensional materials, using a special set of atomic displacements in a single supercell calculation. We also demonstrate that our special displacement reproduces the thermal ellipsoids obtained from X-ray crystallography, and yields accurate thermal averages of the mean-square atomic displacements. At a more fundamental level, we show that the special displacement represents an exact single-point approximant of an imaginary-time Feynman's path integral for the lattice dynamics. This enhanced version of the special displacement method enables non-perturbative, robust, and straightforward ab initio calculations of the electronic and optical properties of solids at finite temperature, and can easily be used as a post-processing step to any electronic structure code. Given its simplicity and numerical stability, the present development is suited for high-throughput calculations of band structures, quasiparticle corrections, optical spectra, and transport coefficients at finite temperature.

Experimental determination of the bare energy gap of GaAs without the zero-point renormalization

The energy gap of simple band insulators like GaAs is a strong function of temperature due to the electron–phonon interactions. Interestingly, the perturbation from zero-point phonons is also predicted to cause significant (a few percent) renormalization of the energy gap at absolute zero temperature but its value has been difficult to estimate both theoretically and, of course, experimentally. Given the experimental evidence (Bhattacharya et al 2015 Phys. Rev. Lett. 114 047402) that strongly supports that the exponential broadening (Urbach tail) of the excitonic absorption edge at low temperatures is the manifestation of this zero temperature electron–phonon scattering, we argue that the location of the Urbach focus is the zero temperature unrenormalized gap. Experiments on GaAs yield the zero temperature bare energy gap to be 1.581 eV and thus the renormalization is estimated to be 66 meV.