One of the most used approaches in simulating materials is the tight-binding approximation. When using this method in a material simulation, it is necessary to compute the eigenvalues and eigenvectors of the Hamiltonian describing the system. In general, the system possesses few explicit symmetries. Due to them, the problem has many degenerate eigenvalues. The ambiguity in choosing a orthonormal basis of the invariant subspaces, associated with degenerate eigenvalues, will result in eigenvectors which are not invariant under the action of the symmetry operators in matrix form. A meaningful computation of the eigenvectors needs to take those symmetries into account. A natural choice is a set of eigenvectors, which simultaneously diagonalizes the Hamiltonian and the symmetry matrices. This is possible because all the matrices commute with each other. The simultaneous eigenvectors and the corresponding eigenvalues will be in a parametrized form in terms of the lattice momentum components. This functional dependence of the eigenvalues is the dispersion relation and describes the band structure of a material. Therefore it is important to find this functional dependence in any numerical computation related to material properties.
Deep Dive into An Example of Symmetry Exploitation for Energy-related Eigencomputations.
One of the most used approaches in simulating materials is the tight-binding approximation. When using this method in a material simulation, it is necessary to compute the eigenvalues and eigenvectors of the Hamiltonian describing the system. In general, the system possesses few explicit symmetries. Due to them, the problem has many degenerate eigenvalues. The ambiguity in choosing a orthonormal basis of the invariant subspaces, associated with degenerate eigenvalues, will result in eigenvectors which are not invariant under the action of the symmetry operators in matrix form. A meaningful computation of the eigenvectors needs to take those symmetries into account. A natural choice is a set of eigenvectors, which simultaneously diagonalizes the Hamiltonian and the symmetry matrices. This is possible because all the matrices commute with each other. The simultaneous eigenvectors and the corresponding eigenvalues will be in a parametrized form in terms of the lattice momentum components.
Tight-binding (TB) is a method used to investigate the electronic structure of a large class of solid materials [1]. When used in conjunction with numerical simulations, this method introduces several simplifications that reduce the complexity of the description of the material. Every solid material is constituted of atomic nuclei that identify a lattice and are the source of potential energy. On other hand, the nucleus-nucleus and electron-electron interactions are neglected. The TB model assumes that the electrons are tightly bound to their corresponding nuclei, implying that their wave functions are localized. Furthermore, atoms interact weakly only through their valence electrons.
Since the electrons are moving independently, the Hamiltonian H of the system is given as a sum of the kinetic energies of the electrons (p 2 i /2m) and the potentials due to the nuclei ∑ i V (r i -R n ), with R n being the positions of the nuclei in three-dimensional space r i . Thus:
where N e denotes the number of considered electrons and N s the number of lattice sites of the crystal [2,3]. To find the eigenstates of this system, a linear combination of the atomic orbitals (LCAO) ∑ i ṽi φ i is used as an ansatz. The atomic orbitals φ i are the eigenstates of the Hamiltonian for an isolated atom and ṽi the coefficients to be computed with the constraint that | ∑ i ṽi φ i | = 1. Since the overlap of the atomic orbitals of neighboring atoms is assumed to be small, they are treated as orthonormal, i.e. their inner products are (φ i , φ j ) = δ i j . Using this property and the LCAO as an ansatz, one obtains the following eigenproblem:
where H ∈ C K×K is the Hamiltonian in the basis of the atomic orbitals. The quantity v n ∈ C K is a vector of coefficients ṽi of the LCAO and the eigenvalue e n ∈ R is the associated energy level. Note that the Hamiltonian is hermitian and therefore the eigenvalues are real. The entries of the Hamiltonian are given by
arXiv:0910.5434v1 [cs.NA] 28 Oct 2009
where β k is the result of an overlap integral between neighboring electronic orbitals and the underlying lattice potential [3]. In the simplest case of equal atoms and only nearest neighbor interaction, the expressions in Eq. ( 3) simplify to β k = -t δ k, +1 + δ k+1, and α k = α, where α and t are constants [3]. Since t represents the interaction between neighboring atoms, it is often called the hopping term.
The quantum mechanical problem of finding the electron wave function is therefore reduced to the solution of a finite dimensional eigenproblem. Having computed the eigenvalues and eigenvectors, we aim at expressing them in terms of the lattice momentum components k = (k 1 , k 2 , k 3 ). Eventually, the whole set of eigenvalues can be seen as a function of k, called the dispersion relation. This is an important relation from which we can determine a large set of physical properties of a material [2,3]. Therefore determining this relation numerically is our final goal.
In this section we construct a simple example. While it can be solved analytically, we show that it can also be accurately solved numerically. Consider a two-dimensional rectangular lattice of equal atoms, as shown in Figure 1 (left). The dark colored atoms constitute our N-by-N lattice structure and the brighter atoms represent the use of periodic boundary conditions. Each atom in the structure interacts with its four nearest neighbors. The interaction is given by the hopping term t as discussed above. The Hamiltonian H ∈ R N 2 ×N 2 of the system has the form
The matrix C ∈ R N×N is circulant. It is equivalent to the Hamiltonian for the one-dimensional lattice of N identical atoms with periodic boundary conditions and nearest neighbor interactions. It has the same structure as H in Eq. ( 4) with C and D replaced by the scalars α and -t, respectively. The matrix D ∈ R N×N is diagonal with all elements equal to -t.
The eigenvalues and eigenvectors of H can be expressed in closed form (see [4,5]). Since H is block-circulant with circulant symmetric blocks, its normalized eigenvectors
where r, s = 0, 1, . . . , N -1. The parameters ρ r and ξ s are the N-th roots of unity. The eigenvectors v n form an orthonormal basis [4]. The corresponding eigenvalues are
All eigenpairs (e n , v n ) are parametrized by the quantities r and s. The index n of the pair can be defined as any bijective function n = f (r, s).
When defining k x and k y as k x := 2πr/N and k y := 2πs/N, respectively, Eq. ( 6) describes the dispersion relation e(k x , k y ). This relation yields all the allowed energies for possible momenta k = k2 x + k 2 y . Because of the periodicity of the crystal, both energy and momentum are quantized [2,3]. As N → ∞, the dispersion relation reveals the band structure of the crystal. Therefore it is important to identify this relation. In more complicated cases for which no analytical solution is available, it is important to compute the dispersion relation through a nu
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