Final Project: Si band structure using Quantum Espresso

Theory

Inside Quantum Espresso software package there is a sub-package called PWSCF (Plane Wave Self Consistent Field) that we will use in order to calculate electronic structures. The theory used here is DFT (density functional theory) with plane wave basis sets and pseudopotentials.

DFT

The H-K (Hohenberg-Kohn) theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates. It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron density. Using Kohn-Sham DFT, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. In many body electronic structure calculations, the atoms are seen as fixed (Born-Oppenherimer approximation) and the stationary electronic state is described by a wave function satisfying the many electron Schrodinger equation. Please see this reference for a full derivation of Kohn-Sham equation. Here we start from KS equations on a crystal:

(1)

 

Is the kinetic energy

Is the non-local pseudopotential. It is smooth practical effective potential that reproduces the effect of the nucleus plus core electrons on valence electrons).

Is the Coulomb potential or Hartree term describing the electron-electron Coulomb repulsion.

Is the exchange correlation potential are relatively simple functions of the local charge density at point r.

Bloch States: is periodical over reciprocal space and is reciprocal over the real space.

Running special points and directions () in the irreducible first Brillouin zone is the basis for BAND DIAGRAMS.

We need to use a basis function in order to calculate KS equation:

(2)

 

Where is a basis function at the reciprocal cell. After minimizing the energy with as the variational parameters the KS equations transform into the following and it is solved for every:

(3)

 

Where

(4)

 

And the KS equations takes the form of an Eigen equation: Hc = SCe when sampling (q) points along special directions on the BZ1 we get the band diagrams: .The total energy for example (E) is produced by integration for all. (Monkhorst-Pack special points methods is often used for this).

For practical computational purposes it is necessary to use Plane Waves (PW). The wave vector of the periodic PWs is:

(5)

 

Where is the parallelepiped cell and this discrete mesh of allowed k-points is distributed uniformly in the reciprocal space. Each k point can be associated with a small parallelepiped:

Figure 1: plane waves and unit cell used to practically calculate. (Ref 3)

Plane waves are delocalised, periodic basis functions

Plenty of them are needed but the operations are simple

FFT (fast fourier-transform) is used to switch between real and reciprocal space

Orthogonal

Independent of atomic positions

Naturally periodic

 

Pseudopotential are used in order to replace all-electron problem with a Hamiltonian with an effective potential that reproduces the effect of the nucleus plus core electrons on valence electrons. It should reproduce the necessary physical properties of the full problem at the reference state. We will use them in QE and there is a database for different materials. Using just PWs like above is not so practical because we need for example, 250000 PWs for a diamond 1s state wave function with lattice parameter a0= 6.74 [a.u] (for details see this reference).

Brillouin zones or first BZ is the common name for the Wigner-Seitz primitive cell of the reciprocal lattice. An example is the simple square lattice construction and an important point to check is that every Brillouin zone has the same volume and any BZ can be mapped back to the first zone by using just primitive translations.

Figure 2: building the Brillouin zone (Ref2)

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The rotational symmetry determines the equivalence between different positions within the Brillouin zone and special symmetry points receive particular names like is used to designate the origin of the reciprocal cell.

The BZ1 for the FCC (Face Centered Cubic lattice) is shown in the next picture with his special symmetry points:

Figure 3: First Brillouin zone (ref 2)

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