This article reviews the current status of lattice-dynamical calculations in crystals, using density-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calculation of the response to macroscopic electric fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodology is demonstrated with a number of applications existing in the literature.

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Phonons and related crystal properties from density-functional perturbation theory

Preface Acknowledgements Notation Part I. Overview and Background Topics: 1. Introduction 2. Overview 3. Theoretical background 4. Periodic solids and electron bands 5. Uniform electron gas and simple metals Part II. Density Functional Theory: 6. Density functional theory: foundations 7. The Kohn-Sham ansatz 8. Functionals for exchange and correlation 9. Solving the Kohn-Sham equations Part III. Important Preliminaries on Atoms: 10. Electronic structure of atoms 11. Pseudopotentials Part IV. Determination of Electronic Structure, The Three Basic Methods: 12. Plane waves and grids: basics 13. Plane waves and grids: full calculations 14. Localized orbitals: tight binding 15. Localized orbitals: full calculations 16. Augmented functions: APW, KKR, MTO 17. Augmented functions: linear methods Part V. Predicting Properties of Matter from Electronic Structure - Recent Developments: 18. Quantum molecular dynamics (QMD) 19. Response functions: photons, magnons ... 20. Excitation spectra and optical properties 21. Wannier functions 22. Polarization, localization and Berry's phases 23. Locality and linear scaling O (N) methods 24. Where to find more Appendixes References Index.

Electronic Structure: Basic Theory and Practical Methods

https://par.nsf.gov/servlets/purl/10107309

ShengBTE: A solver of the Boltzmann transport equation for phonons ☆

The adiabatic Born-Oppenheimer approximation (ABO) has been the standard ansatz to describe the interaction between electrons and nuclei since the early days of quantum mechanics. ABO assumes that the lighter electrons adjust adiabatically to the motion of the heavier nuclei, remaining at any time in their instantaneous ground state. ABO is well justified when the energy gap between ground and excited electronic states is larger than the energy scale of the nuclear motion. In metals, the gap is zero and phenomena beyond ABO (such as phonon-mediated superconductivity or phonon-induced renormalization of the electronic properties) occur. The use of ABO to describe lattice motion in metals is, therefore, questionable. In spite of this, ABO has proved effective for the accurate determination of chemical reactions, molecular dynamics and phonon frequencies in a wide range of metallic systems. Here, we show that ABO fails in graphene. Graphene, recently discovered in the free state, is a zero-bandgap semiconductor that becomes a metal if the Fermi energy is tuned applying a gate voltage, Vg. This induces a stiffening of the Raman G peak that cannot be described within ABO.

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Breakdown of the adiabatic Born-Oppenheimer approximation in graphene.

The Born-Oppenheimer approximation (BO) has proven effective for the accurate determination of chemical reactions, molecular dynamics and phonon frequencies in a wide range of metallic systems. Graphene, recently discovered in the free state, is a zero band-gap semiconductor, which becomes a metal if the Fermi energy is tuned applying a gate-voltage Vg. Graphene electrons near the Fermi energy have twodimensional massless dispersions, described by Dirac cones. Here we show that a change in Vg induces a stiffening of the Raman G peak (i.e. the zone-center E2g optical phonon), which cannot be described within BO. Indeed, the E2g vibrations cause rigid oscillations of the Dirac-cones in the reciprocal space. If the electrons followed adiabatically the Dirac-cone oscillations, no change in the phonon frequency would be observed. Instead, since the electron-momentum relaxation near the Fermi level is much slower than the phonon motion, the electrons do not follow the Dirac-cone displacements. This invalidates BO and results in the observed phonon stiffening. This spectacular failure of BO is quite significant since BO has been the fundamental paradigm to determine crystal vibrations from the early days of quantum mechanics.

Born-Oppenheimer breakdown in graphene

The microscopic quantum-mechanical expressions for the Born-von Karman force constants in an arbitrary solid, crystalline or amorphous, are derived in terms of the complete inverse dielectric function ${\ensuremath{\epsilon}}^{\ensuremath{-}1}(\mathrm{r}, {\mathrm{r}}^{\ensuremath{'}})$ of the electrons The many-body nature of the electrons is treated exactly; only the Born-Oppenheimer approximation is made. Born's translation and rotation invariance conditions are shown to be satisfied by the microscopic force constants. In the case of a perfect crystal, it is shown for the first time that the microscopic formulas recapture completely the phenomenological form of the dynamical matrix; in particular, the microscopic expression for the effective charge in an insulator is found. We prove that the charge neutrality of the system implies the "effective charge neutrality" condition and that, consequently, all acoustic-mode frequencies vanish at long wavelength. This condition may be stated as a useful property of ${\ensuremath{\epsilon}}^{\ensuremath{-}1}$ which we term the acoustic sum rule. Many results of the phenomenological theory, e.g., the generalized Lyddane-Sachs-Teller relation, carry over exactly to the microscopic theory.

Microscopic theory of force constants in the adiabatic approximation

: The expression of the force constants are derived from the energy of a neutral system of interacting nuclei and electrons. Those force constants depend only on the nuclear charges and the inverse dielectric function of the electronic system. It is verified that they fulfill the general symmetry, translational and rotational invariance properties required by the Born-Van Karman theory. When the role of the core electrons is neglected, those force constants can be evaluated in metals from the bare ionic potentials. In insulators, the existence of long wave length acoustical phonons implies that the inverse dielectric function satisfies a certain number of sum rules. The method becomes more difficult to apply, which explains the general use of phenomenological models. (Author)

Robert M. Pick

Papers

Microscopic theory of force constants in the adiabatic approximation

Microscopic theory of force constants in solids