“Tight Binding” Method: Linear Combination of Atomic Orbitals (LCAO) W. E. Pickett (Dated: April 14, 2014) This write-up is a homemade introduction to the tight binding representation of the electronic structure of crystalline solids. This information is important in the parametrization of the band structures of real solids and for the underlying character of model Hamiltonians for correlated electron studies, as well as for other uses. Those interested in the parametrization of band structures of real materials should consult the book by D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids, (Plenum, New York,1986). For the general TB method, one should consult the original classic work by J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). I. INTRODUCTION TO TIGHT BINDING non-uniqueness is irrelevant (only the form of the THEORY resulting parameters is relevant), while if the method is intended for real electronic structure calculations, Since a crystal is made up of a periodic array the non-uniqueness often is used to make numerical of atoms, it may seem peculiar that when we think procedures as convenient as possible. of Bloch electron wavefunctions in solids it is often This potential is periodic by construction: in terms of wavy modulations that don’t pay much ~ ~ ~ attention to just where the atoms sit. Indeed, in V (~r + Ro) = Vat(~r + Ro R) − simple metals and covalent semiconductors that is a XR~ good picture: the crystal potential that enters into = Vat(~r (R~ R~ o)) the Hamiltonian is a smooth function and atomic X − − sites per se are not critical in the understanding (al- R~ though they are in the underlying description of co- = V (~r R~ ′) at − valent semiconductors). XR~ ′ There is in fact a common picture – the tight V (~r), (1.2) binding model – that is based on the “collection of ≡ atoms” viewpoint. It is most appropriate when elec- where the change of summation index R~ R~ ′ = trons move through the crystal slowly (or not at all, → R~ R~ o was made. Here R~ o is any lattice vector. as in insulators) and therefore ‘belong’ to an atom −The crystal Hamiltonian is, using units where for an appreciable time before they move on. The ¯h2/2m 1, electrons are in some sense tightly bound to the atom ≡ and only hop because staying put on a simple atom H = 2 + V (~r). (1.3) costs a bit too much energy. The TB model is not −∇ readily applicable to simple (free or nearly free elec- The potential V (~r) is periodic so H is periodic. tron) metals, but it is quite good for a wide variety of Later we will generalize the situation to cover several other solids. It is interesting that covalent semicon- atoms in the unit cell. ductors can be described well from either viewpoint. B. Periodic array of atomic orbitals A. Crystal as a collection of atoms Shouldn’t the electron wavefunction in the crys- A good approximation for the electron’s poten- tal be related to the atomic orbitals, which satisfy tial V (~r) in a crystal is the sum of atomic potentials: 2 Hatφn ( + Vat)φn = εnφn. (1.4) V (~r) = V (~r R~), (1.1) ≡ −∇ at − XR~ We might try the most simple linear combination of atomic orbitals that is periodic where the sum runs over lattice vectors. We will not worry about the considerable non-uniqueness of this Φ (~r) = φ (~r R~); (1.5) n n − decomposition. For parametrization purposes, this XR~ is this a Block function that can be put into the form D. Proceeding toward the eigenfunctions 2 ~ Φ(~r) = eik·~ru (~r)? (1.6) ~k For many solids there will be several types (s,p Indeed it is, but only for ~k = 0. We want, and need, or d) of atomic states in the valence region, and that is why we have kept the index n. In the solid these Bloch-like functions for arbitrary ~k within the first atomic states will mix with each other due to the Brillouin zone (BZ). overlap of atomic orbitals on neighboring atoms (as Problem 1: prove this statement, that this we will see). A Bloch sum of atomic orbitals itself is proposed function has Block form for, and ~ not an eigenfunction for the crystal. It is important only for, k=0. to allow the valence wavefunction in the solid to be some of each of the atomic functions, with the actual amounts to be determined by solving Schr¨odinger’s C. Bloch Sums equation. Thus we try expressing the electron wave- function in the crystal as a bit bn,~k of each of the A much better choice of candidate for a crystal Bloch sums, wavefunction is to form the “Bloch sums” of atomic orbitals, given by ~ ψ~k(~r) = bn(k)Bn,~k(~r), (1.11) 1 ~ ~ Xn B (~r) = N − 2 eik·Rφ (~r R~). (1.7) n,~k n − XR~ where the coefficients b gives the amount of Bloch sum Bn in the crystal wavefunction. The Bloch where R~ suns over the N lattice cites in our normal- sums become the basis functions that we express ization volume. This type of linear combination is the wavefunction in terms of. Since the Bloch sums called a Bloch sum because it produces a function themselves are normalized to unity, we will want that satisfies the Bloch condition for wavevector ~k: b (~k) 2 = 1 for each ~k. n | n | 1 ~ ~ P Now we want to learn how to find the wavefunc- N 2 B (~r + R~ ) = eik·Rφ (~r + R~ R~) n,~k o n o tions. The condition is that they be solutions of the X − R~ Schr¨odinger equation i~k·R~ = e φn(~r (R~ R~ o)) − − Hψ~k = ε~kψ~k. (1.12) XR~ ~ ~ ′ ~ ik·(R +Ro) ′ The eigenvalues ε~ will be the energy bands of the = e φn(~r R~ ) k − crystal. But how do we solve for ε~ and the coeffi- X~ ′ k R ~ ~ ~ ~ ~ ′ cients bn(k)? = eik·Ro eik·R φ (~r R~ ′), n − XR~ ′ 1 ~ 2 ik·R~ o E. The matrix equation = N e Bn,~k(~r). (1.8) so In quantum mechanics generally, a good try is to ~ ~ take matrix elements (integrals between basis func- B (~r + R~ ) = eik·Ro B (~r). (1.9) n,~k o n,~k tions) and reduce the problem to a matrix equa- This result then is in Bloch form. All such Bloch tion. In this case, the thing to do is to multiply the functions are periodic in ~k from BZ to BZ. Schr¨odinger equation on the left by another Bloch sum B∗ and integrate over the crystal. Note that Problem 2: prove this statement by manip- m,~k ulating it into the Block form, and discover we have not chosen a Bloch function corresponding ′ what periodic part u~ (~r) is. to another wavevector ~k = ~k. k 6 It suffices to confine ~k in Eq. 1.7 to the 1st BZ. Problem 3: calculate <Bmk Bnk′ > to see why ′ | If the ~k point were of the form ~kr + K~ , where ~kr we simplify to k = k. See below for notation. (the reduced wavevector) is in the 1st BZ and K~ is a The result is reciprocal lattice vector, note that ~ ~ ~ ~ Hm,n(k)bn(k) = ε~k Sm,n(k)bn(k), (1.13) ~ ~ ~ ~ X X eik·R = eikr ·R (1.10) n n where for any and all lattice vectors R~. (Be sure you un- derstand why.) Thus these Bloch sums, and any ~ ∗ Hm,n(k) B ~ (~r)HB ~ (~r) functions constructed from them, are periodic from ≡ Z m,k n,k BZ to BZ. = <B H B > (1.14) mk| | nk and G. The Real Space TB Matrix Elements 3 S (~k) B∗ (~r)B (~r) The expression for the real space integral is m,n ≡ Z m,~k n,~k = <B S B (1.15)> . ∗ m.k n,k n,~k H (R~) = φ (~r)Hφ (~r R~), (1.20) | | m,n Z m n − These matrices are called the Hamiltonian matrix i.e. it indicates the amount by which the Hamil- and the overlap matrix, respectively, where m and n tonian H couples atomic orbital φ on the site at are the matrix indices. Writing the matrices implic- m the origin to the atomic orbital φ that is located at itly (without displaying the indices), the equation n site R~. Physically, H (R~) is the amplitude that an becomes m,n electron in orbital φn at site R~ will hop to the orbital φm at the origin under the action of the Hamiltonian. H(~k)b(~k) = ε~ S(~k)b(~k) (1.16) k One limit is easy to see: if R~ is so large that either | | or one or the other of the orbitals is vanishingly small where the other in nonzero, then the integral is neg- ligible. Thus we can confine ourselves to values of H(~k) ε~ S(~k) b(~k) = 0. (1.17) { − k } R~ that connect an atom to only a few near neigh- This is a linear algebra problem (generalized bors. Often the “nearest neighbor” approximation eigenvalue problem), but we have to know what the is excellent. matrices H and S are. All of this discussion applies as well to S, by just removing the Hamiltonian H from inside the inte- gral. Sm,n(R~) in fact is called the overlap of φm(~r) ~ F. The H and S matrices and φn(~r R). Note that, if the orbitals are normal- ized (as is− always possible, and is always assumed), ~ Well, we never said that this wouldn’t get a little the Sm,m(0)=1 for all m, i.e.