Lattice Wannier Functions Research Articles

Universality of Abelian and non-Abelian Wannier functions in generalized one-dimensional Aubry-André-Harper models

Within a Dirac model in $1+1$ dimensions, a prototypical model to describe low-energy physics for a wide class of lattice models, we propose a field-theoretical version for the representation of Wannier functions, the Zak-Berry connection, and the geometric tensor. In two natural Abelian gauges we present universal scaling of the Dirac Wannier functions in terms of four fundamental scaling functions that depend only on the phase $\ensuremath{\gamma}$ of the gap parameter and the charge correlation length $\ensuremath{\xi}$ in an insulator. The two gauges allow for a universal low-energy formulation of the surface charge and surface fluctuation theorem, relating the boundary charge and its fluctuations to bulk properties. Our analysis describes the universal aspects of Wannier functions for the wide class of one-dimensional generalized Aubry-Andr\'e-Harper lattice models. In the low-energy regime of small gaps we demonstrate universal scaling of all lattice Wannier functions and their moments in the corresponding Abelian gauges. In particular, for the quadratic spread of the lattice Wannier function, we find the universal result $Za\ensuremath{\xi}/8$, where $Za$ is the length of the unit cell. This result solves a long-standing problem providing further evidence that an insulator is only characterized by the two fundamental length scales $Za$ and $\ensuremath{\xi}$. Finally, we discuss also non-Abelian lattice gauges and find that lattice Wannier functions of maximal localization show universal scaling and are uniquely related to the Dirac Wannier function of the lower band. In addition, via the winding number of the determinant of the non-Abelian transformation, we establish a bulk-boundary correspondence for the number of edge states up to the bottom of a certain band, which requires no symmetry constraints. Our results present evidence that universal aspects of Wannier functions and of the boundary charge are uniquely related and can be elegantly described within universal low-energy theories.

Electron-Phonon Interaction via Electronic and Lattice Wannier Functions: Superconductivity in Boron-Doped Diamond Reexamined

We present a first-principles technique for investigating the electron-phonon interaction with millions of k points in the Brillouin zone, which exploits the spatial localization of electronic and lattice Wannier functions. We demonstrate the effectiveness of our technique by elucidating the phonon mechanism responsible for superconductivity in boron-doped diamond. Our calculated phonon self-energy and Eliashberg spectral function show that superconductivity cannot be explained without taking into account the finite-wave-vector Fourier components of the vibrational modes introduced by boron, as well as the breaking of the diamond crystal periodicity induced by doping.

Generation of lattice Wannier functions via maximum localization

A method is presented for generating approximate lattice Wannier functions (LWFs) for lattice dynamics problems, using the dynamical matrix for a supercell as input. The lattice Wannier functions fit selected phonon frequencies and eigenvectors exactly, are orthonormal, and are optimized to be maximally localized. The method easily generalizes to the case where LWF centered on more than one distinct chemical species are desired, as well as to the case of solid solutions. The method is successfully applied to a one-dimensional toy model.

Flat branches and pressure amorphization

After summarizing the phenomenology of pressure amorphization (PA), we present a theory of PA based on the notion that one or more branches of the phonon spectrum soften and flatten with increasing pressure. The theory expresses the anharmonic dynamics of the flat branches in terms of local modes, represented by lattice Wannier functions, which are in turn used to construct an effective Hamiltonian. When the low-pressure structure becomes metastable with respect to the high-pressure equilibrium phase and the relevant branches are sufficiently flat, transformation into an amorphous phase is shown to be kinetically favored because of the exponentially large number of both amorphous phases and reaction pathways. In effect, the critical-size nucleus for the first-order phase transition is found to be reduced to a single unit cell, or nearly so. Random nucleation into symmetrically equivalent local configurations characteristic of the high-pressure structure is then shown to overwhelm any possible domain growth, and an ``amorphous'' structure results.

Optimized local modes for lattice-dynamical applications

We present a new scheme for the construction of highly localized lattice Wannier functions. The approach is based on a heuristic criterion for localization and takes the symmetry constraints into account from the start. We compare the local modes thus obtained with those generated by other schemes and find that they also provide a better description of the relevant vibrational subspace.

Lattice instabilities, anharmonicity and phase transitions in PbZrO3 from first principles

An effective Hamiltonian for the antiferroelectric transition in PbZrO3 is constructed from first-principles density-functional-theory total-energy and linear-response calculations through the use of a localized, symmetrized basis set of “lattice Wannier functions''. The effective Hamiltonian consists of three-component vector degrees of freedom at the Pb sites and two-component vector degrees of freedom at the oxygen sites. This effective Hamiltonian allows us to analyze the structural energetics and identify the role of various physical contributions. We find that there exists a strong competition between sets of coupled unstable modes in producing low energy states. Two such groups of instabilities found from the effective Hamiltonian are seen to be relevant to the experimental ground and intermediate states.

Ab initio statistical mechanics of the ferroelectric phase transition inPbTiO3

An effective Hamiltonian for the ferroelectric transition in PbTiO3 is constructed from first-principles density-functional-theory total-energy and linear-response calculations through the use of a localized, symmetrized basis set of ``lattice Wannier functions.'' Explicit parametrization of the polar lattice Wannier functions is used for subspace projection, addressing the issues of LO-TO splitting and coupling to the complementary subspace. In contrast with ferroelectric BaTiO3 and KNbO3, we find significant involvement of the Pb atom in the lattice instability. Monte Carlo simulations for this Hamiltonian show a first-order cubic-tetragonal transition at 660 K. Resulting temperature dependence of spontaneous polarization, c/a ratio and unit-cell volume near the transition are in good agreement with experiment. Comparison of Monte Carlo results with mean field theory analysis shows that both strain and fluctuations are necessary to produce the first order character of this transition.

Ferroelectric phase transitions from first principles

An effective Hamiltonian for the ferroelectric transition in PbTiO 3 is constructed from first principles density-functional-theory total-energy and linear-response calculations through the use of a localized, symmetrized basis set of ‘lattice Wannier functions’. Preliminary results of Monte Carlo simulations for this system show a first-order cubic-tetragonal transition at 660 K. The involvement of the Pb atom in the lattice instability and the coupling of local distortions to strain are found to be particularly important in producing the behavior characteristics of the PbTiO 3 transition. Factors affecting the distribution of local distortions above T c are discussed. Further applications of this method to a variety of systems and structures are proposed for the first principles study of finite temperature structural properties in individual materials.

Localized basis for effective lattice Hamiltonians: Lattice Wannier functions.

A systematic method is presented for constructing effective Hamiltonians for general phonon-related structural transitions. The key feature is the application of group theoretical methods to identify the subspace in which the effective Hamiltonian acts and construct for it localized basis vectors, which are the analogue of electronic Wannier functions. The results of the symmetry analysis for the perovskite, rocksalt, fluorite and A15 structures and the forms of effective Hamiltonians for the ferroelectric transition in $PbTiO_3$ and $BaTiO_3$, the oxygen-octahedron rotation transition in $SrTiO_3$, the Jahn-Teller instability in $La_{1-x}(Ca,Sr,Ba)_xMnO_3$ and the antiferroelectric transition in $PbZrO_3$ are discussed. For the oxygen- octahedron rotation transition in $SrTiO_3$, this method provides an alternative to the rotational variable approach which is well behaved throughout the Brillouin zone. The parameters appearing in the Wannier basis vectors and in the effective Hamiltonian, given by the corresponding invariant energy expansion, can be obtained for individual materials using first- principles density-functional-theory total energy and linear response techniques, or any technique that can reliably calculate force constants and distortion energies. A practical approach to the determination of these parameters is presented and the application to ferroelectric $PbTiO_3$ discussed.

The origin of the α, β, γ blue no-phonon transitions in ZnO:Cu-A deep-level problem

The first satisfactory assignment is given for the three closely spaced zero-phonon lines which appear near 2.86 eV in the absorption and photoluminescence excitation spectra of Cu-doped ZnO crystals. The zero-phonon lines, designed alpha , beta , gamma , represent charge-transfer type transitions in which a hole is excited from the Cu 3d shell to form a bound state with a neutral deep-acceptor-like configuration, i.e. Cuzn2+(3d9) to (CuZn1+(3d10))h. The three transitions terminate on different excited hole states which derive from the special nature of thMe wurtzite valence band structure. A qualitative description of these excited hole wave-functions, consistent with the magneto-optical data, is obtained using a Wannier-function basis for the upper valence bands of ZnO and including a small admixture of Cu impurity 3d orbital character in the bound state. It is shown that the energies of the bound hole states in the excited configuration (Cu1+)h are determined by the Wannier-function energies and by spin-orbit interaction with the heavy Cu impurity core, whereas the hole g-factors and the polarisations of the alpha , beta , gamma , lines reflect mainly the symmetry properties of the lattice Wannier functions.