2.2 Tight Binding Band Structure Calculations
Bulk Hamiltonian
The Schrödinger equation in the bulk of any crystalline solid can be written
as H˜bulk kn(r) =Ekn kn(r) (2.8)
where the eigenstate wave functions kn(r)correspond to Bloch states at the wave vector k of the band n defined over all real space spanned by the vector r.Ekn are the corresponding eigenenergies that form the bulk band dispersion. These ground state solutions are usually found byab initio density functional theory (DFT) and they include core levels as well as the conduction band which is of interest for understanding the electronic properties of the material. By down-folding a subset of thesenbands onto a basis of maximally localized Wannier functions (MLWFs)'R↵(r), where Ris a lattice site and↵is a band-like index, it is possible to construct an effective HamiltonianHbulkfor the conduction band, where the corresponding Schrödinger equation is now
Hbulk kn(r) =⇠kn kn(r) (2.9) with eigenstate wave functions kn(r)and eigenvalues⇠kn that should, to a good approximation, reproduce the band structure defined byEkn.
Hbulk is an effective tight binding Hamiltonian that can be written in the form
Hbulk= X
k↵↵0
t↵↵k 0c†k↵ck↵0 (2.10) where the operator c†k↵(ck↵) creates (annihilates) a state |⌥k↵i which is related to the MLWFs by a change of basis. Thus the wave functions of the basis used in Eq. (2.10) are related to the maximally localized Wannier functions by
⌥k↵(r) = 1 pN
X
R
eik·R'R↵(r) (2.11) where N is the number of unit cells in the crystal, which is equal to the number ofkpoints in the sum in Eq. (2.10).
The matrix elements of the effective conduction band Hamiltonian are defined by
t↵↵k 0 =X
R
t↵↵R0e ik·R. (2.12) These are Fourier transforms of the transfer integrals (or colloquially "hop-ping parameters")t↵↵R 0 that describe the kinetic energy cost for an electron to tunnel along the lattice vectorRfrom the MLWF state ↵to the state
↵0. In practice, the nature of the MLWF are such thatt↵↵R 0 quickly becomes negligible for largeR.
Hbulk in the form of Eq. (2.10) is a matrix that is block diagonal ink.
The size of the matrix is determined by the number of MLWF states{↵} at each lattice site. In the case of STO the conduction band is derived from the 3d t2g manifold and the MLWF states can be identified with thedxy, dyz anddxz atomic orbitals. In this case↵becomes an orbital index and the basis can be written as
{↵}={xy", xy#, yz", yz#, xz", xz#} (2.13)
where"#indicates the spin character of the Wannier function andxy,yz or xz indicate thet2g orbital character of the Wannier function. The spin index is necessary if the DFT calculation used as a starting point includes spin-orbit interactions. Therefore in STO the diagonal blocks ofHbulk for eachk are6⇥6matrices with matrix elementst↵↵k 0. These6⇥6 matrices must be diagonalized at each kpoint in the Brillouin zone to find the eigenenergies
⇠knwhere the band indexnruns from 1 to 6. The corresponding eigenvectors have six components with the coefficients{ kn↵ }, that describe the relative contribution by each state of orbital character↵to the total wave function.
The wave functions of the eigenstates of Eq. (2.10) have the form
kn(r) =X
The charge density can be calculated from the wave functions using n3D(r) = |e|X
kn
f(⇠kn)| kn(r)|2 (2.16) wheref(⇠kn)is the Fermi function which ensures that the sum in Eq. (2.16) only counts occupied states.n(r)has the periodicity of the lattice and some structure within the unit cell. The mean bulk densityn¯ is found by averaging over all space which, by utilizing the orthogonality properties of MLWFs, gives the intuitive result
2.2 Tight Binding Band Structure Calculations
whereV is the volume of the discrete lattice. For a cubic lattice with lattice constanta,V =a3N .
Quasi-2D Supercell Hamiltonian
It is possible to make a simple extension of the above tight-binding formalism, such that the Hamiltonian is quasi-2D rather than 3D. This quasi-2D Hamil-tonian will be useful for modelling the subband structure of two-dimensional systems at oxide surfaces.
The lattice vectorRcan be separated into two contributionsR=Rk+z wherezdenotes the component of the lattice vector perpendicular to the surface and Rk the in-plane lattice vector. If the Fourier transform of Eq. (2.12) is performed only overRk such that the matrix elements of the Hamiltonian have the form
t↵↵kkzz0 0 =X
Rk
t↵↵Rk0zz0e ikk·Rk (2.20) the corresponding quasi-2D supercell Hamiltonian is given by
Hbulk= X
kk↵↵0zz0
t↵↵kkzz0 0c†k
k↵zckk↵0z0 . (2.21) By retaining the explicit sums overzandz0Eq. (2.21) is block diagonal inkk such that the resulting band dispersion is only defined in a 2D Brillouin zone.
At each kpoint it is now necessary to diagonalize matrices of dimension 6L⇥6L where L is the number of layers included along the z axis of the crystal. The6⇥6diagonal elements of these blocks are given by the Hamiltonian of a single layer of the crystal and the off diagonal matrix elements describe the interlayer coupling. There are6Leigenvalues⇠kkn for everykk point in the 2D Brillouin zone. The eigenvectors have length6L whose coefficients{ k↵zkn}describe the layer-resolved orbital composition of the wave functions
kkn(r) =X
↵z
↵z
kkn⌥kk↵z(r). (2.22) with the normalization condition
X
↵,z
| ↵,zkkn|2= 1. (2.23) ForL! 1the supercell Hamiltonian Eq. (3.5) produces6Lsolutions which are degenerate with the solutions of Eq. (2.10) atkz= 0. In practice
the supercell sizeLis finite and the hopping parameters must be truncated at the limits of the supercell slab such that
t↵↵Rk0zz0 =
(t↵↵Rk0z z0 for z, z’ in the bulk.
0 for z and/or z’ in vacuum. (2.24) This produces artefacts in the band structure due to the effective confinement of the system in an infinite potential well of widthL.
In analogy to Eqs.(2.16 - 2.19), but with a spatial average only over the thexyplane and one unit cell in thez direction, it is possible to determine the layer resolved density resulting from this quasi 2D Hamiltonian using
n3D(z) = |e| A
X
kkn
f(⇠kkn)X
↵
| k↵zkn|2 (2.25)
HereA=a2Nk is the area of the surface of the crystal withNk the number of points in the 2Dkgrid.
The explicit form of the matrix elements of Eq. (2.12) and Eq. (2.20) is discussed for a simplified model Hamiltonian of STO in Appendix B.