tight - binding

tight - binding

(Parte 1 de 2)

Sci Model Simul (2008) 15:81–95 DOI 10.1007/s10820-008-9108-y

Tight-binding Hamiltonian from first-principles calculations

Cai-Zhuang Wang · Wen-Cai Lu · Yong-Xin Yao · Ju Li · Sidney Yip · Kai-Ming Ho

1 Introduction

The tight-binding method attempts to represent the electronic structure of condensed matter using a minimal atomic-orbital like basis set. To compute tight-binding overlap and Hamiltonian matrices directly from first-principles calculations is a subject of continuous interest. Usually, first-principles calculations are done using a large basis set or long-ranged basis set (e.g. muffin-tin orbitals (MTOs)) in order to get convergent results, while tight-binding overlap and Hamiltonian matrices are based on a short-ranged minimal basis representation. In this regard, a transformation that can carry the electronic Hamiltonian matrix from a large or long-ranged basis representation onto a short-ranged minimal basis representation is necessary to obtain an accurate tight-binding Hamiltonian from first principles.

The idea of calculating tight-binding matrix elements directly from a first-principles method was proposed by Andersen and Jepsen in 1984 [1]. They developed a scheme which transforms the electronic band structures of a crystal calculated using a long-ranged

C.-Z. Wang (B) · Y.-X. Yao · K.-M. Ho US Department of Energy, Ames Laboratory, Ames, IA 50011, USA e-mail: wangcz@ameslab.gov

C.-Z. Wang · Y.-X. Yao · K.-M. Ho Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA

W.-C. Lu State Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun 130021, People’s Republic of China

J. Li Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA

S. Yip Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

S. Yip Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

82 C.-Z. Wang et al.

basis set of muffin-tin orbitals (MTO’s) into a much shorter-ranged tight-binding representation. In the framework of this transformation, tight-binding matrix elements can be calculated by first-principles LMTO method and casted into an effective two-center tight-binding Hamiltonian called LMTO-TB Hamiltonian [1]. More recently, an improved version of such a “downfolding” LMTO method, namely the order-N MTO [2], has also been developed which allows the LMTO-TB Hamiltonian matrix elements to be extracted more accurately from a full LMTO calculation [3].

Another approach to determine the tight-binding Hamiltonian matrix elements by firstprinciples calculations was developed by Sankey and Niklewski [4] and by Porezag et al. [5]. In their approach, the matrix elements are calculated directly by applying an effective oneelectron Hamiltonian of the Kohn-Sham type onto a set of pre-constructed atomic-like orbitals. The accuracy of the tight-binding Hamiltonian constructed in this way depends on the choice of atomic-like basis orbitals. More recently, McMahan and Klepeis [6]h aved eveloped a method to calculate the two-center Slater-Koster hopping parameters and effective on-site energies from minimal basis functions optimized for each crystal structure, in terms of k-dependent matrix elements of one-electron Hamiltonian obtained from first-principles calculations.

All of the above mentioned work was derived from a description of electronic structures using a fixed minimal basis set, except the work of McMahan and Klepeis [6]. It should be noted that while a fixed minimal basis set can give a qualitative description of electronic structures, it is too sparse to give an accurate description of the energetics of systems in varying bonding environments. A much larger basis set would be required in the first-principles calculations in order to get accurate and convergent results if the basis set is going to be kept fixed for various structures. Thus, it is clear that in order for a minimal basis set to have good transferability, it is important to focus our attention on the changes that the basis must adopt in different bonding environments.

In the past several years, we have developed a method for projecting a set of chemically deformed atomic minimal basis set orbitals from accurate ab initio wave functions [7–12]. We call such orbitals “quasi-atomic minimal-basis orbitals” (QUAMBOs) because they are dependent on the bonding environments but deviate very little from free-atom minimal-basis orbitals. While highly localized on atoms and exhibiting shapes close to orbitals of the isolated atoms, the QUAMBOs span exactly the same occupied subspace as the wavefunctions determined by the first-principles calculations with a large basis set. The tight-binding overlap and Hamiltonian matrices in the QUAMBO representation give exactly the same energy levels and wavefunctions of the occupied electronic states as those obtained by the fully converged first-principles calculations using a large basis set. Therefore, the tight-binding Hamiltonian matrix elements derived directly from ab initio calculations through the construction of QUAMBOs are highly accurate.

In this article, we will review the concept and the formalism used in generating the

QUAMBOs from first-principles wavefunctions. Then we show that tight-binding Hamiltonian and overlap matrix elements can be calculated accurately by the first-principles methods through the QUAMBO representation. By further decomposing the matrix elements into the hopping and overlap parameters through the Slater-Koster scheme [13], the transferability of the commonly used two-center approximation in the tight-binding parameterization can be examined in detail. Such an analysis will provide very useful insights and guidance for the development of accurate and transferable tight-binding models. Finally, we will also discuss a scheme for large scale electronic structure calculation of complex systems using the QUAMBO-based first-principles tight-binding method.

Tight-binding Hamiltonian from first-principles calculations 83

2 Quasi-atomic minimal-basis-sets orbitals

The method to project the QUAMBOs from the first-principles wave functions has been described in detail in our previous publications [7–12]. Some of the essential features of the method will be reviewed here using Si as an example. If the Si crystal structure contains N silicon atoms and hence 4N valence electrons in a unit cell, the total number of minimal sp3 basis orbitals per unit cell will be 4N. In our method, the 4N QUAMBOs (Aα) are spanned by 2N occupied valence orbitals which are chosen to be the same as those from the first-principles calculations, and by another 2N unoccupied orbitals which are linear combinations of a much larger number of unoccupied orbitals from first-principles calculations. The condition for picking such 2N unoccupied orbitals is the requirement that the resulting QUAMBOs deviate as little as possible from the corresponding 3s and 3p orbitals of a free

Si atom (A0α). The key step in constructing the above mentioned QUAMBOs is the selection of a small subset of unoccupied orbitals, from the entire virtual space, that are maximally overlapped with the atomic orbitals of the free atom A0 α.

Suppose that a set of occupied Bloch orbitals φµ(k,r)( µ = 1,2,..., nocc (k)) and virtual orbitals φν(k,r)( v = nocc (k)+ 1, nocc (k)+2 ,..., nocc (k)+ nvir (k)), labeled by band µ or ν, and wave vector k, have been obtained from first-principles calculations using a large basis set, our objective is to construct a set of quasi-atomic orbitals Aα(r − Ri) spanned by the occupied Bloch orbitals φµ(k,r) and an optimal subset of orthogonal virtual Bloch orbitals ϕp(k,r)

where

The orthogonal character of ϕp(k,r)g ives ∑

T∗ νp(k)Tνq(k) = δpq,i nw hich T is a rect- angular matrix which will be determined later.

The requirement is that Aα should be as close as possible to the corresponding free atom side condition 〈Aα|Aα〉= 1. Therefore the Lagrangian for this minimization problem is

k,µ where

k,µ

84 C.-Z. Wang et al. For this optimized Aα, the mean-square deviation from A0α is

It is clear from Eqs. (5)a nd (6) that the key step to get quasi-atomic minimal-basis-set orbitals is to select a subset of virtual orbitals ϕp(k,r) which can maximize the matrix trace

The maximization can be achieved by first diagonalizing the matrix

i,α for each k-point, where ν and ν′ run over all unoccupied states up to a converged upper cutoff. The transformation matrix T which defines the optimal subset of virtual Bloch orbitals

ϕp(k,r)( p = 1,2,,np(k)) by Eq. (2) is then constructed using the ∑

k np(k) eigenvectors with the largest eigenvalues of the matrixes Bk, each of such eigenvectors will be a column of the transformation matrix T.G iven ϕp(k,r), the localized QUAMBOs are then constructed by Eqs. (4)a nd (5). As one can see from the above formalism development that the key con- cept in this QUAMBO construction is to keep the bonding states (occupied state) intact and at the same time searching for the minimal number of anti-bonding states (which are usually not the lowest unoccupied states) from the entire unoccupied subspace. The bonding states that kept unchanged and the anti-bonding states constructed from the unoccupied states can form the desirable localized QUAMBOs.

Figure 1 shows the s-a nd p- like QUAMBOs of Si in diamond structure with different bond lengths of 1.95 Å, 2.35 Å and 2.75 Å, and in fcc structure with bond lengths of 2.34 Å, 2.74 Å, and 3.14 Å, respectively. The QUAMBOs are in general non-orthogonal by our construction as discussed above. One can see that the QUAMBOs constructed by our scheme are indeed atomic-like and well localized on the atoms. These QUAMBOs are different from the atomic orbitals of the free atoms because they are deformed according to the bonding environment. It is clear that the deformations of QUAMBOs are larger with shorter interaction distances. When the bond length increases to be 2.75 Å, the QUAMBOs are very close to the orbitals of a free atom.

As we discussed above, the effective one-electron Hamiltonian matrix in the QUAMBO representation by our construction preserves the occupied valence subspace from the firstprinciples calculations so that it should give the exact energy levels and wavefunctions for the occupied states as those from first-principles calculations. This property can be seen from Fig. 2 where the electronic density-of-states (DOS) of Si in the diamond structure calculated using QUAMBOs are compared with that from the original first-principles calculations. It is clearly shown that the electronic states below the energy gap are exactly reproduced by the

Tight-binding Hamiltonian from first-principles calculations 85

1.95 Å 2.35 Å 2.75 Å

2.34 Å 2.74 Å 3.14 Å

Fig. 1 Non-orthogonal s-a nd p- like QUAMBOs in Si (a) diamond structure in the (110) plane for three bond lengths 1.95 Å, 2.35 Å and 2.75 Å, and (b) fcc structure in the (100) plane for three bond lengths 2.34 Å, 2.74 Å, and 3.14 Å

QUAMBOs, while the unoccupied states have been shifted upwards so that the energy gap between the valence and conduction states increases from ∼0.7 eV to ∼1.8 eV. This shift is expected because the QUAMBOs contain admixtures of eigenstates from the higher energy spectrum.

It should be noted that the formalism for the QUAMBOs construction discussed in the section is based on the wavefunctions from first-principles calculations using all-electrons or norm-conserving pseudopotentials [14]. The formalism for constructing the QUAMBOs from first-principles calculations using ultra-soft pseudopotential (USPP) [15] or projector augmented-wave (PAW) [16], as implemented in the widely used VASP code [17,18], is similar and has been recently worked out by Qian et al. [12]. Moreover, Qian et al. also adopt a projected atomic orbital scheme [19–21] which replaces the unoccupied subspace from the first-principles calculations in the above formula with a projection of the unoccupied part of the atomic orbitals, and improve the efficiency and stability of the QUAMBO construction procedure [12].

86 C.-Z. Wang et al.

Fig. 2 Electronic density of states of diamond Si obtained by using the QUAMBOs as basis set, compared with those from the corresponding LDA calculations using the PW basis set

E (eV)

3 Tight-binding matrix elements in terms of QUAMBOs

Once the QUAMBOs have been constructed, overlap and effective one-electron Hamiltonian matrices in representation of QUAMBOs are readily calculated from first-principles.

H in Eq. 10 can then be expressed by using the corresponding eigenvalues εn and eigenfuc- tions φn from original DFT calculations, i.e., H = ∑ n εn |φn〉〈φn|, and thus the matrix elements Hi,α,jβ can be calculated easily. Note that in our approach the electronic eigenvalues and eigenfunctions of the occupied states from first-principles calculations are exactly reproduced by the QUAMBO representation. Although the overlap and effective one-electron Hamiltonian matrices in terms of the QUAMBOs are in a minimal basis representation, the matrices obtained from our method go beyond the traditional two-center approximation. Therefore, the Slater-Koster tight-binding parameters [13] obtained by inverting such first-principles matrices are expected to be environment-dependent.

In order to examine how the overlap and hopping integrals are dependent on the environment and to see how serious the error the two-center approximation will make in traditional tight-binding approaches, we have performed calculations for 3 types (i.e, diamond, simple cubic (sc), and face-centered cubic (fcc)) of crystal structures of Si with several different bond lengths for each type of structures in order to study the tight-binding parameters in different bonding environments. Based on the overlap and effective one-electron Hamiltonian matrix elements from our QUAMBO scheme, the Slater-Koster overlap integrals sσ,sspσ,sppσ, and sppπ, and hopping integrals hssσ, hspσ, hppσ,a nd hppπ are then extracted using the Slater-Koster geometrical factors [13]. The results for the overlap and hopping integrals as a function of interatomic distance in the three different crystal structures are plotted in Figs. 3 and 4, respectively.

Figure 3 shows the overlap parameters sσ, sspσ, sppσ,a nd sppπ from different structures and different pairs of atoms, plotted as a function of interatomic distance. Note that the two- center nature of overlap integrals for fixed atomic minimal basis orbitals may not necessarily

Tight-binding Hamiltonian from first-principles calculations 87

Fig. 3 Overlap integrals as a function of interatomic distance for Si in the diamond, sc, and fcc structures hold for the QUAMBOs because QUAMBOs are deformed according to the bonding environments of the atoms. Nevertheless, the overlap parameters obtained from our calculations as plotted in Fig. 3 fall into smooth scaling curves nicely. These results suggest that the two-center approximation is adequate for overlap integrals. By contrast, the hopping parameters as plotted in Fig. 4 are far from being transferable, especially for hppσ. Even for the best case of hssσ, the spread in the first neighbor interaction is about 1 eV. For a given pair of atoms, the hopping parameters hppσ and hppπ obtained from the decompositions of different matrix elements can exhibit slightly different values, especially for the sc and fcc structures. The hopping parameters from different structures do not follow the same scaling curve. For a given crystal structure, although the bond-length dependence of hopping parameters for the first and second neighbor interactions can be fitted to separate smooth scaling curves respectively, these two scaling curves cannot be joined together to define an unique transferable scaling function for the structure. These results suggest that under the two-center approximation, it is not possible to describe the scaling of the tight-binding hopping parameters accurately.

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