Project in Physics and Astronomy, 15C: Tight Binding Modelling of Materials
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Project in Physics and Astronomy, 15c: Tight Binding Modelling of Materials Raquel Esteban Puyuelo October 28, 2015 Abstract In this project, a parametrized tight binding (TB) code is developed in order to describe the essential physics of real materials using a minimal model. We have used this code to compute the band structure of different materials where the primary inputs are the hopping parameters, obtained from a Nth order muffin-tin orbital (NMTO) method. The code has been tested for a single atom in a unit cell having only one effective orbital (Li) as well as for many atoms having more than one orbital (NiS and IrO2). A successful reproduction of the density functional theory band structure for all these three systems implies that this code can be applied in general to any real material. We have also analyzed the effect of various nearest neighbor interactions on the electronic structure of these systems. Contents 1 Introduction 2 2 Theory 2 2.1 Description of the Tight Binding Model . .2 2.2 Examples . .3 2.2.1 One-dimensional chain: single s-band . .3 2.2.2 One dimensional chain with two atoms . .4 2.2.3 Three dimensional example: Lithium . .5 2.2.4 Summary . .5 2.3 Extraction of the Tight Binding parameters . .6 2.3.1 Density Functional Theory . .6 2.3.2 Fitting of TBM parameters . .7 3 Code operation 7 3.1 Inputs . .7 3.2 Working scheme . .7 4 Applications 8 4.1 Li .................................................8 4.2 NiS.................................................9 4.3 IrO2 ................................................ 10 5 Conclusion 11 6 Appendix 13 1 Tight Binding Modelling of Materials Raquel Esteban Puyuelo 1 Introduction model), but it can be easily extended to more com- plex cases. The localized basis set is φ(~r − R~ j), th where R~ j is the position of the j atom and φ(~r) Density functional theory is one of the most popular is the orbital wavefunction. In its usual form, the theories nowadays for the description of electronic TBM is a single electron model where each electron structure properties of various materials. However, is described by the Hamiltonian these calculations require large amounts of comput- 2 X ing time in order to achieve the accuracy needed for H = − ~ r2 + V (~r − R~ ): (2.1) l 2m l l j the appropriate description of these complex sys- j tems. In addition, the results of the first-principles calculations are often not so straight forward to in- It can be rewritten in the following way: terpret. Thus, it is needed to have a simple model, 2 X H = − ~ r2 + V (~r − R~ ) + V (~r − R~ ): which contains the essential physics necessary for l 2m l l l l j a description of complex systems, and also provide j6=l (2.2) the liberty to analyze the effect of various param- eters separately which primarily dictate the nature This Hamiltonian has two parts: the first and sec- of the systems. 0 ond can be regarded as on-site terms (Hl ), while the last accounts for the interactions between site l In view of that, I have developed a code in the and its neighbors. As it is very small, it is usually framework of parametrized tight-binding method treated as a perturbation (δV ), so the Hamiltonian taking the parameters from the NMTO downfolding 0 results in Hl = Hl + δV . The wavefunction of the calculation. This code has been tested successfully TBM Hamiltonian must obey the Bloch’s theorem for various kinds of real systems. The thesis has ~ ~ X k·Rj ~ been organized as follows. In section-II, I discuss ~k(~r) = e φ(~r − Rj) (2.3) ~ the theory of the TB model. Section-III has been Rj devoted in describing the code, Section-IV includes in the form of one atom/unit cell. If there are two all the test cases, followed by summary in section-V. (A and B), it takes the form A B (~r) = a + b ; (2.4) ~k ~k ~k ~k ~k 2 Theory where α X ~k·R~ α (~r) = e j φ (~r + ~δ − R~ ): (2.5) ~k α j ~ 2.1 Description of the Tight Binding Rj Model The goal of this method is to calculate the band structure k by solving the Schrödinger equation The tight binding approach to the electronic band H = ; (2.6) structure calculation is used extensively in con- ~k ~k ~k ~k densed matter physics and has as its starting point where H~k is the Fourier transform of the Hamilto- the wavefunctions of the free atoms. nian in the real space appearing in previous equa- tions. The matrix equation is achieved by multiply- In the tight-binding model (TBM) of electronic ing from the left with ∗ ~k structures, single-electron wave functions are ex- panded in terms of atomic orbitals, centered around ~vyH^ = ~vyS^ ~v ; (2.7) ~k ~k ~k ~k ~k ~k each atom. Thus this method is closely related to the linear combination of atomic orbitals (LCAO) where ~vy = (a∗; b∗); (2.8) method. ~k ~k ~k the Hamiltonian matrix is The simplest model consists in assuming that there D E Hα,β = α(~r) H β(~r) (2.9) is only one orbital in each atomic site (single band ~k ~k ~k ~k October 28, 2015 2 Project in Physics and Astronomy, 15c Tight Binding Modelling of Materials Raquel Esteban Puyuelo and the overlap matrix is the simplest case, where each atom in the position D E n has only one s-type symmetrical orbital which Sα,β = α(~r) β(~r) : (2.10) ~k ~k ~k interacts with the s-orbitals of the atoms located The overlap corrections are often neglected as we in sites n − 1 and n + 1 (only nearest neighbors will do in this project. interactions) to form the crystal states. In general, the expression for the Hamiltonian is X ~ ~ ~ Hα,β = eik·(Rl−Rm)× ~k R~ mR~ l D α ~ β ~ E Figure 1: One dimensional chain with one atom φ (~r + δα − R~ l) H φ (~r + δβ − R~ k) : (2.11) ~k ~k in the basis Using translational invariance and 2.2 the Hamilto- nian results in: ~ ~ α,β X ik·Rl In order to calculate the band structure of the sys- H~ = N e × k tem we need to solve (2.6) applying the TB ap- R~ l proximation explained above. In this case, we only D α 0 β ~ ~ E φ (~r) H + δV φ (~r + δα − δβ − R~ l) = ~k ~k consider the nearest neighbors interactions. Hence, N(αsαβ + tαβ); (2.12) the hopping interactions are zero unless n0 = n − 1 ~k ~k or n0 = n + 1: the overlap matrix in: X ~ ~ αβ X ikRl H ≈ [Hat(n) + U(n; n − 1) + U(n; n + 1)] : s~ = e × k n ~ Rl (2.16) D α ~ β ~ E (~r + δα − R~ l) (~r + δβ − R~ j) (2.13) Also, the atomic Schrödinger equation in (2.6) can ~k ~k be written in Dirac’s notation and the hopping parameters can be expressed as X ~ ~ H j i = " j i ; (2.17) tαβ = eikRl × at n n ~k ~ Rl where En = " is the energy of the atomic electron, D α ~ β ~ E which is known. (~r + δα − R~ l) δV (~r + δβ − R~ j) (2.14) ~k ~k The energy dispersion can be calculated as following In practice, this eigenvalue problem should be ad- if we assume the orthogonality of the wavefunction dressed using the diagonalization of the matrix and use the expressions in (2.17) and (2.16): (H − ) at each point in the reciprocal space in ~k ~k X the path along which we want to calculate the band Ek = hΨkj H jΨki = hΨkj Hat(n) jΨki structure, with n X X X + hΨkj [U(n; n − 1) + U(n; n + 1)] jΨki (2.18) H ≈ H (n) + U 0 ; (2.15) at n;n n n hn;n0i In the first neighbors approximation the interaction where hn; n0i means that only nearest-neighbor in- terms are nonzero in the following cases: teraction U is considered (it is easily extensible to ( further neighbors). −t (n00 = n; n0 = n ± 1) h nj U(n; n ± 1) j ni = 0 otherwise (2.19) 2.2 Examples Then, taking in the Bloch factors the dispersion re- lation results in the following expression, which is 2.2.1 One-dimensional chain: single s-band plotted in figure 2: −ika +ika Ek = " − t(e + e ) = " − 2t cos(ka): Consider an infinite linear chain formed by identical (2.20) atoms separated by a distance a. Here we consider October 28, 2015 3 Project in Physics and Astronomy, 15c Tight Binding Modelling of Materials Raquel Esteban Puyuelo Again, we want to calculate the band structure by solving a Schrödinger equation and we will use the Tight Binding method assumptions: 1. The Hamiltonian is as 2.23 2. The atomic problem is known: Hat γn(x) = Eγn γn(x) (2.24) 3. The atomic wavefunction γn is localized in each site. Figure 2: The band structure of a single atom 1D chain with 1s orbital is a single s-type band that 4. The crucial TB assumption (Cγ are unknown): behaves as a cosine function. The TBM parameters have been chosen to be " = 2 and t = 1 X ikna X Ψk(x) = e Cγ γn(x) (2.25) in this plot. n γ 2.2.2 One dimensional chain with two To determine the dispersion relation we multiply atoms the Schrödinger equation from the left by each of the two different atomic orbitals and integrate: In this case we have two different atoms (we intro- h Anj H jΨki = Ek h An jΨki (2.26) duce the index γ which can be either A or B) in each h Bnj H jΨki = Ek h Bn jΨki (2.27) unit cell and the distance between an atom and the In the tight binding approach, only the on-site next atom of the same kind is a.