L AS CAUSAS DE LAS M ODIFICACIONES P RESUPUESTARIAS

The energy band structure is formed by broadening of the electronic energy state in a crystal with a periodic potential. The band structure is dependent on the crystal composition and structure that also defines the shape of the first Bril- louin zone (Γ-point). The motion of electrons in a many-body system needs to be formulated taking into account their own interactions (pair potentials) and po- tentials from the surrounding ions in a crystalline solid. These interactions are quantum-mechanically described by solving the many-body Schr¨odinger equation, which determines the time-independent electron energies of the system:

ˆ

H|Ψ_i=E_|Ψ_i (2.23)

where ˆ_H,E, and Ψ are the Hamiltonian operator of the system, eigenvalue (electron energy), and many-body wavefunction of the electrons. Disregarding the effect of spin-orbit coupling, the many-particle Hamiltonian is given by [98]

ˆ H=P i p2 i 2mi + P j p2_j 2Mj + 1 2 P i′_,j Z_j′Zje2 4πǫ0|Rj−Rj′|− P i,j Zje2 4πǫ0|ri−Rj|+ 1 2 P i′_,i e2 4πǫ0 1 |ri−r_i′|, (2.24) wherei and j label the electrons and ions, respectively. p is the momentum opera- tor and r_i and R_j are the position of the ith electron and jth nucleus, respectively.

m, M, and ǫ0 represent the electron mass, ion mass, and the vacuum permittivity, respectively. The complex Hamiltonian, ˆ_H, can be approximated by applying ap- propriate simplifications as described below. The electrons in a given crystal system can be divided into core electrons and outer-shell electrons. The core electrons are localized around the nuclei (ion cores). Since the ions are stationary compared to the outer-shell electrons, the mass (motion) of ions is much greater (slower) than that of the electrons (i.e.M _≫m). This allows for approximating the potentials of

the ions by a time-averaged adiabatic electronic potential. Hence, the total wave function is approximated by

Ψ = ψions(R)ψe(r,R) (2.25)

whereψions(R) is the wave function of the ions moving within their ionic potentials and ψe(r,R) is the wave function of all the electrons instantaneously dependent on the ionic position. The Hamiltonian of the system is thus described by the sum of three terms:

ˆ

H = ˆ_He(ri,Rj) + ˆHions(Rj) + ˆHe−ions(ri,∆Rj), (2.26)

where ˆ_He (ri,Rj) is the Hamiltonian of the electrons in the potential from the equi-

librium position of the ions, ˆ_Hions (Rj) the Hamiltonian for the motion of ions, and

ˆ

He−ions (ri,∆Rj) is the Hamiltonian for the change in the electronic energy asso-

ciated with the displacement of the ions (∆Rj) from their equilibrium positions.

The last term is related to the normal modes of vibration of the solid system, com- monly thermal vibration of ions, thus referring to as the electron-phonon interaction. Consequently, the simplified electronic Hamiltonian can be written by

ˆ He = X i p2_i 2mi + 1 2 X i′_,i e2 4πǫ0|ri−ri′|− X i,j Zje2 4πǫ0|ri−Rj| . (2.27)

These three terms reflects the kinetic energy of electrons, the interaction of electrons, and the electron-ion interactions, respectively. To further simplify Eq. 2.27, the ap- proximation requires the electron-electron interactions to be averaged as a constant repulsive component and a small perturbation (one-electron approximation). If the electron-electron interaction is negligible, each electron interacts independently with the lattice of ions. Furthermore, it can simply be considered that each elec- tron experiences the same average potential, V(r), with the surrounding ions in their equilibrium position. As a consequence, this approximation in the Schr¨odinger equation describing the identical motion of each electron is given by

ˆ He1ψn(r) = p2_i 2mi +V(r) ψn(r) = Enψn(r), (2.28)

where ˆ_He1, ψn (r), and En are the one-electron Hamiltonian, the wave function,

needs to be a periodic function with the same translational periodicity of V in a given crystal known as a Bloch wave function

ψnk(r) =unk(r)exp(ik·r), (2.29)

where n labels the band index and k is the wavevector of the electron in the first Brillouin zone, respectively. ψnk(r) is defined by a plane wave of exp(ik·r) that

satisfies the condition

unk(r+R) =unk(r), (2.30)

where R is a primitive translation vector of the Bravais lattice. The state of the electron can be determined by plotting the electron energy, E, as a function of wavevector, k using Eq. 2.26 to give the electronic band structure of the crystal. In order to understand fundamental physical properties of oxide semiconductors, which are intrinsically or extrinsically doped, their electronic band structures, the approximations introduced above are used to formulate the energy dispersions near the centre of the Γ-point, where the band extrema calculated in the reciprocal space usually occur.