Generalized Wannier Functions Research Articles

Wannier functions for lattices in a magnetic field

This paper considers the definition and properties of Wannier functions for Bloch electrons in a magnetic field. When the quantized Hall conductance of a band is non-zero, conventional Wannier functions with good localization properties cannot be constructed. The difficulty can be overcome by slightly broadening the definition of the Wannier function: the generalized Wannier functions are a good basis set, and well localized. They are generated by applying magnetic translations to a set of |N| fundamental Wannier states: if the number of flux quanta per unit cell is (a rational number), and the Hall conductance integer is M, then N satisfies Mq + Np = 1. Unlike conventional Wannier functions, the definition of these Wannier states depends upon the choice of the basis vectors for lattice translations. The paper gives the transformation properties of the Wannier functions induced by reassignment of the primitive-lattice basis vectors.

Maximally localized generalized Wannier functions for composite energy bands

We discuss a method for determining the optimally-localized set of generalized Wannier functions associated with a set of Bloch bands in a crystalline solid. By ``generalized Wannier functions'' we mean a set of localized orthonormal orbitals spanning the same space as the specified set of Bloch bands. Although we minimize a functional that represents the total spread sum_n [ <r^2>_n - <r>_n^2 ] of the Wannier functions in real space, our method proceeds directly from the Bloch functions as represented on a mesh of k-points, and carries out the minimization in a space of unitary matrices U_mn^k describing the rotation among the Bloch bands at each k-point. The method is thus suitable for use in connection with conventional electronic-structure codes. The procedure also returns the total electric polarization as well as the location of each Wannier center. Sample results for Si, GaAs, molecular C2H4, and LiCl will be presented.

Density functional theory for systems of very many atoms

AbstractThe standard Kohn‐Sham formulation of density functional theory (DFT) is limited, for practical reasons, to systems of less than about 50‐100 atoms. The computational effort scales as N, where Nat is the number of atoms and 2 &lt; α &gt; 3. (By comparison, conventional configuration interaction methods are limited to 5‐10 atom systems.) This article deals with the prospect of practical methods that scale linearly in Nat and may thus allow calculations for systems of 103‐104 atoms. The physical reason (“near‐sightedness”) for linear scaling is presented. Implementations of linear scaling DFT by the use of generalized Wannier functions or the one‐particle density matrix are discussed. © 1995 John Wiley &amp; Sons, Inc.

Electron dynamics in graded nanostructures

In this paper I describe recent work, jointly with M. Geller, treating the dynamics of electrons in semiconducting alloys, such as Al c Ga 1− c As, whose compositions, characterized by c, is a slowly varying function of position: | ▿ c( r)| b ⪡ 1, where b is a characteristic lattice spacing. The mathematical framework is provided by the so-called generalized Wannier functions (GWF) a n, l ( r) where n and l are band and site labels (for composite bands an additional label v is needed). The GWFs are generalizations to non-periodic systems of the traditional Wannier functions for periodic systems. We show that the eigenstates and frequency-dependent electric response functions can be completely described in terms of two quantities, ε n ( k; c) and p nn′ ( k; c), the energy and momentum matrix elements of the uniform alloys of concentrations c, ranging over the values of c in the non-uniform system under consideration.

Generalized Wannier function and renormalization of Harper's equation

Harper's equation, a model for Bloch electrons in a magnetic field, has a band spectrum when the dimensionless magnetic field beta is a rational number p/q. This paper considers the definition of generalized Wannier functions, which can be used to represent the Bloch bands of the spectrum of the rational Harper equation by means of a von Neumann lattice. One representation of these Bloch bands can be extended to the irrational case, and taking matrix elements in this basis leads to a renormalization-group transformation acting on the Hamiltonian. The results in the present paper considerably extend a previous analysis of this renormalization-group transformation, in that the formalism is suitable for systematic calculations of the renormalized Hamiltonian, and that the transformation preserves the rotational symmetry of the Harper Hamiltonian in phase space.

The calculation of electron transfer probabilities in slow ion-metal surface collisions

We consider the interaction of an atomic ion colliding with a solid metal surface, resulting in electronic charge transfer and neutralization of the ion. The coupling of electronic and nuclear degrees of freedom in extended systems is described with a general methodology based on an eikonal representation of nuclear motions, and density matrices for the electronic degrees of freedom. We show how one can computationally deal with two very different time scales in the differential equation for the density operator by separating there the contributions of electronic relaxation and of driving forces. The boundary conditions of the solid perturbed by the ion are described by introducing electronically adiabatic and diabatic regimes. Electron transfer at short distances is calculated in a basis of localized surface states, and partitioning the density matrix into primary and secondary regions, to derive a reduced equation for its primary part. A substantial part of this article details the construction of generalized Wannier functions used in the calculation of electronic energy integrals. Results are presented for Hamiltonian matrix elements and for the time-evolution of atomic populations, in a model appropriate to the Na + + W(110) collisions with energies between 1.0 and 100.0 au. The calculated time-evolving populations provide a unique insight on the dynamics of electronic screening and transfer during collisions.

Theory of generalized Wannier functions for nearly periodic potentials.

There is a great interest in the properties of electrons in nanostructures with superposed potentials varying slowly in space, or with slowly graded composition, or both. The quantum mechanics of these electrons, moving in nearly periodic potentials, is conveniently described in a basis of localized generalized Wannier functions, similar to the conventional Wannier functions for strictly periodic potentials. This has led us to consider generalized Wannier functions for (i) weakly perturbed and (ii) compositionally graded crystals. We first construct generalized Wannier functions for uniform crystals with an everywhere weak perturbation. The corrections to an unperturbed Wannier function of a given band may be chosen to involve only the unperturbed Wannier functions from other bands. We then consider crystals with slowly varying composition, and construct generalized Wannier functions from the Wannier functions of uniform crystals.

Density functional/Wannier function theory for systems of very many atoms

Conventional theoretical methods for calculating ground-state energies of multi-atom systems in general become impractical when the number of atoms, N a, exceeds ∼ 10 2, because the computing time rises with N α a, where 2 < α < 3. Using so-called generalized Wannier functions in conjunction with density functional theory, we develop a procedure for the direct calculation of the GWFs (not requiring single particle eigenfunctions) whose computing time behaves as N a. The method allows the transfer of GWFs from one physical or chemical system to another.

A time‐dependent molecular orbital approach to electron transfer in ion–metal surface collisions

AbstractA time‐dependent molecular orbital method has been developed to study charge transfer in collisions of ions with metal surfaces at energies between 1 and 100 au. A set of localized basis functions consisting of generalized Wannier functions for the surface and s‐ and p‐atomic functions for the ion, is used to separate the system into primary and secondary regions. An effective Hamiltonian and time‐dependent equations for the electron density matrix are obtained in the primary region, where most charge transfer occurs. The equations for the electron density matrix are solved with a linearization scheme. The method is suitable to study atomic orbital orientation for collisions of ions and surfaces. A model calculation for Na+ + W(110) collisions with a prescribed trajectory is presented. The interaction potentials between the W(110) surface and Na+ 3s and 3p orbitals are calculated from Na+ pseudopotentials. Results show that the yield of neutralized atoms in 3p states changes as the collision energy is lowered.

A Simple Wilson Orthonormal Basis with Exponential Decay

Following a basic idea of Wilson [“Generalized Wannier functions,” preprint] orthonormal bases for $L^2 (\mathbb{R})$ which are a variation on the Gabor scheme are constructed. More precisely, $\phi \in L^2 (\mathbb{R})$ is constructed such that the $\psi _{ln } $, $l \in \mathbb{N}$, $n \in \mathbb{Z}$, defined by \[ \begin{gathered} \psi _{0n} (x) = \phi \left( {x - n} \right) \hfill \\ \psi _{in} (x) = \sqrt 2 \phi \left( {x - \frac{n} {2}} \right)\cos (2\pi lx)\,{\text{if}}\,l \ne 0,\,l + n \in 2\mathbb{Z} \hfill \\ = \sqrt 2 \phi \left( {x - \frac{n} {2}} \right)\sin (2\pi lx)\,{\text{if}}\,l \ne 0,\,l + n \in 2\mathbb{Z} + 1, \hfill \\ \end{gathered} \] constitute an orthonormal basis. Explicit examples are given in which both $\phi $ and its Fourier transform $\hat \phi $ have exponential decay. In the examples $\phi $ is constructed as an infinite superposition of modulated Gaussians, with coefficients that decrease exponentially fast. It is believed that such orthonormal bases could be useful in many contexts where lattices of modulated Gaussian functions are now used.

Modified method of variational calculation of generalized Wannier functions for a crystal with a model surface potential

Generalized Wannier functions for a crystal described by the Mathieu potential and the model surface potential are calculated by the modified variational method previously used for calculating the Wannier functions for bulk crystals. The interaction energy integrals calculated in various approximations are compared and conclusions expected to be also valid for crystals with realistic potentials are derived. The main conclusions are the following: The trial functions with exponential tails used for constructing Wannier functions give better Wannier functions than the trial functions with Gaussian tails. The approximation of assuming that the trial function for the surface layer is the only trial function which differs from bulk trial functions is a very good approximation for the case when the potential differs from the bulk crystal potential only for the surface layer.

Bandstructure matching at a heterojunction interface

An analytic theory for the matching of the bandstructure of different crystals at the interface of a heterojunction is presented. The Generalized Wannier functions serve as a basis. For a simple uniform band, the Hamiltonian matrix elements then reduce to the k-space Fourier coefficients of the bandstructure along the superlattice direction. The technique therefore accounts for non-effective mass effects, and the lower and upper valleys, and enables us to invoke both the quasi-k-space periodicity together with the spatial variations of the bandstructure. The superlattice Hamiltonian is a system of difference equations taking the form of a band matrix. A new definition of current not based on the effective-mass is introduced for this higher-order Schroedinger equation. The enforcement of the continuity of the elemental currents leads to analytic connection rules for the overlap Hamiltonian matrix elements. A maximum transparency for all energies is achieved only for geometrically related bandstructures. Non-effective-mass effects are demonstrated in resonant-tunneling structures. The technique presented offers new theoretical insights together with efficient numerical tools for the study of non-effective-mass superlattices.

Generalized Wannier functions as a way to study the electron-phonon interaction in silicon

We have developed a method of obtaining the generalized Wannier functions of a crystal distorted by the presence of a frozen phonon by taking full advantage of the infinitesimal character of the atomic displacements. In this way the screening charge is obtained for a phonon of arbitrary wave vector, allowing both for the calculation of the dynamical matrix and for the screening of the bare phonon potential. The calculated frequencies are satisfactory for the acoustic branches but are too high for the optical branches (29% at $\ensuremath{\Gamma}$). The coupling constants that describe processes among electrons at the bottom of the conduction band have been calculated. They are in good agreement with experiment.

Generalized Wannier functions at interfaces: Stacking faults in silicon

A variational method suitable to compute the generalized Wannier functions (GWF) of a crystal with defects is presented. The scheme is applied to study the structural and electronic properties of stacking faults in silicon. Since no model Hamiltonian is used, this is the first time that GWF are self-consistently determined. The minimization of the total energy causes very small relaxations of the ionic layers at the fault. Our results for the energies of formation are in satisfactory agreement with experimental data. Once we have obtained the Hamiltonian, the band structure is calculated giving some interface states. The available experimental information on states at the fundamental gap of these imperfect crystals is discussed.

Generalized Wannier functions for a model impurity in a one-dimensional lattice

The generalized Wannier function formalism is used to calculate the impurity perturbed electronic structure of a one-dimensional lattice. A comparison is made between this approach and the exact results. It is found that very satisfactory accuracy can be achieved using simple trial functions with only one variational parameter.

Theory of interface energies

We have developed a practical scheme for the calculation of interface energies. It combines the theory of generalized Wannier functions, the generalized recursion method for calculating local densities of states and electronic density, and the local-density-functional theory. As a first application of this method we are calculating the stacking-fault energy of nickel using tight-binding-type Wannier functions for the $d$ electrons and ignoring the effect of the $s$ electrons. The $d$-band degeneracy is fully taken into account. The method also allows one to handle charge-transfer effects: a stacking-fault perturbation potential is included and calculated self-consistently. Comparison is made between the moment scheme, the non-self-consistent scheme, and the self-consistent scheme. Earlier moment calculations, up to the fourth moment, lead to stacking-fault energies which are a factor of 3 smaller than our more complete calculations. We also find that the self-consistent refinement affects the stacking-fault energy very little. Our best theoretical result is still much smaller, by a factor of about 3-7, than the (rather uncertain) experimental energy.

Chemical bond approach to the electric susceptibility of semiconductors. II

The method previously developed to relate chemical bonds to the dielectric constant of elemental semiconductors is extended to the ionic tetrahedral III-V compounds. Expressions for the static dielectric constant and the average optical gap in terms of the generalized Wannier functions for the valence bands are obtained. Only two parameters are needed. The first is a measure of the antibonding character of the valence bands and the second is related to the ionic nature of these compounds. We have allowed the former to vary from compound to compound, while we have set the latter at a predetermined constant value, as prescribed by Coulson. We separate the energy gap into two analytical parts, homopolar and heteropolar, thus directly confirming the ionicity scale established by Phillips. Finally, Coulson's definition of ionicity is reconciled with that of Phillips.

Theory of silicon hyperfine interactions with shallow-donor electrons: A reply

A reply to the Onffroy comments on the Ivey-Mieher (IM) calculation of silicon hyperfine constants with shallow donor electrons is presented. We conclude that the Onffroy calculation using generalized Wannier functions is interesting and more accurate than the predictions of effective-mass theory but not as accurate as the IM calculation using Bloch functions. Onffroy's alternative assignment of experimental electron-nuclear-double-resonance lines to lattice sites are shown to be incorrect. The Onffroy criticism of the Ivey-Mieher treatment of the anisotropy of the effective mass is shown to be inappropriate and not significant.

Theory of shallow-donor states with a strong central-cell perturbation

Abstract : A new theory for shallow donor states with a strong central cell perturbation is presented. The impurity wave function is expanded in terms of generalized Wannier functions, equivalent to the eigenfunctions of the periodic lattice with the strong central cell perturbing potential. The single unknown parameter of this theory can be fitted by comparison with the observed binding energy. The theory is applied to the impurity states of As, P and Sb in Silicon. The Fermi contact constants for the Si29 lattice nuclei surrounding the impurity are calculated and compared to the ENDOR experiments of Hale and Mieher and to recent calculations, using a different method, by Ivey and Mieher. Some elucidations of the theory of Ivey and Mieher are offered. In contrast to the conclusions of these authors, it is found that a suitable form of the Effective Mass Theory can account for the observations within an accuracy of about 10-15%.

Inherently self-consistent procedure for obtaining electronic structure: Results for a lithium particle

The commonly used self-consistency procedure for electronic structure involves an iteration between potential and wave functions. A new method is proposed which neither requires iteration nor is based on wave functions. Its basis functions are generalized Wannier functions. The total energy is written self-consistently at the outset using the one-electron density matrix $\ensuremath{\rho}$. The ground-state $\ensuremath{\rho}$ is selected from a family of idempotent matrices as that which gives the lowest total energy. Applications are made to both periodic and nonperiodic systems. Charge densities, binding energies, and the equilibrium lattice constant are computed for a lithium particle. The results are compared with those of a recent self-consistent-field---$X\ensuremath{\alpha}$-scattered-wave calculation.