Computational methods used in real life : From SRM KTR (Semiconductors for Year 1)

1. 21PYB102J-SEMICONDUCTOR PHYSICS AND COMPUTATIONAL
METHODS
Topics in unit – 1 related to computational materials science
Dr. PJJ

2. Methods for calculating band structure
The computational solid state physics is a very fast growing area of research. Modern methods for calculating the electronic band structure
of solids allow predicting many important properties of solids.
Tight-binding approximation
Tight-binding method uses atomic orbitals as basis wave functions.
Energy band generation occurred by stacking up several atoms together.
To understand the band structure, The approximation which starts from the wave functions of the free atoms is known as the tight
binding approximation or the LCAO (the linear combination of atomic orbitals) approximation.
In general this approximation is quite good to describe the inner electronic shells of atoms and relatively localized bands. However, it is not
very good for the description of the conduction electrons themselves.
This is because the origin of the free electrons is completely different to the nature of the localized atomic states. Nevertheless, using tight-
binding approximation it is possible to reproduce accurately the band structure of many solids including metals. Moreover, it is possible
to describe reasonably well transport properties of metals as well.

4. We assume that the basis set is orthogonal, i.e.
(rpa'm' l(f)am) =om·m o
a 'a .
This leads to the following equation for the coefficients
2
a
where
(6)
(7)

6. The cellular (Wigner-Seitz) method
The cellular method was the earliest method employed in band calculations by Wigner and Seitz. It
was applied with success to the alkali metals, particularly to Na and K.
Within the cellular method, we divide the crystal into unit cells; each atom is centered at the
middle of its cell, as shown in Fig.
Such a cell, known as the Wigner-Seitz (WS) cell, is constructed by drawing bisecting planes
normal to the lines connecting an atom A, say, to its neighbors, and "picking out" the volume
enclosed by these planes. (The procedure for constructing the WS cell is
analogous to that used in constructing the Brillouin zone in k-space.) For Na, which has a bcc
structure, the WS cell has the shape of a regular dodecahedron.
The Schrödinger equation whose solution we seek is
In order to find the solution of Eq., we assume that the electron, when in a particular cell, say
A, is influenced by the potential of the ion in that cell only. The ions in other cells have a negligible
effect on the electron in cell A because each of these cells is occupied, on the average, by another
conduction electron which tends to screen the ion, thereby reducing its potential drastically

7. Wigner-Seitz

8.  look at your graph axis.
 is it in distance? -> Wigner-Seitz cell
is it e-k (momentum) space? -> Brillouin zone
 Other/ map of the world:
do they quote crystal orientations or planes, such as (110) -> Wigner-Seitz cell
is there a mention about cleaving? -> Winger-Seitz cell
Do they talk about lattice parameters (a,b,c), strain &stress, angles -> Wigner-Seitz cell
 is there a mention of X-rays? -> Brillouin zone
do they talk about (accumulated) energy, waves, or wavelength? -> Brillouin zone
do you have no clue what’s going on? -> probably Brillouin zone.
 In a somewhat simplistic analogy, a Brillouin zone is to a Wigner-seitz cell as a musical graphical equalizer bar representation
is to the vibration of a string that makes that music.
How can I differentiate between Wigner-Seitz cells and Brillouin zones?

9. The augmented-plane wave (APW)method
The APW method was developed by Slater in 1937. The APW method is based on assuming a muffin-tin potential. The potential is that of a free
ion at the core, and is strictly constant outside the core.
The wave function for the wave vector k is now taken to be
where r0 is the core radius. Outside the core the function is a plane wave because the potential is constant there.
Inside the core the function is atom-like, and is found by solving the appropriate free-atom Schrödinger equation.
Also, the atomic function in above eqn is chosen such that it joins continuously to the plane wave at the surface
of the sphere forming the core; this is the boundary condition here.
a physically-motivated approximation
to the shape of the potential a solid:
spherical regions around each
nucleus and flat potential in between
the spheres.

10. The pseudopotential method
Pseudopotential method is distinguished by the manner in which the wave function is chosen.
We seek a function which oscillates rapidly inside the core, but runs smoothly as a plane wave
in the remainder of the open space of the WS cell.
where φk is a plane-wave-like wave function and φα is an atomic function. The sum over a extends over all the atomic shells
which are occupied. For example, in Na, the sum extends over the 1s, 2s, and 2p shells. The coefficients cα are chosen such that
the function φk, representing a 3s electron, is orthogonal to the core function cα
The function Φk has the features we are seeking:
Away from the core, the atomic functions φα are negligible, and thus Φk ~ φk, a plane wave. At the core, the atomic
functions are appreciable, resulting in rapid oscillations, as shown in Fig
The pseudopotential concept, (a) The actual potential and the corresponding wave function, as seen by
the electron, (b) The corresponding pseudopotential and pseudofunction.

11. Eigenvalue Equation
What is an Eigen value equation:
Let Ô be an operator which operates on functions of x. Lets denote the eigen functions of Ô as fj(x). The set
of functions fj(x), j = 1,2,3,….
And lets denote the corresponding eigen values of Ô as λj. If we know the operator Ô , we can extract the
eigen functions and eigen values by solving the equation:
Each eigen function fj(x) has one eigen value λj associated with it. i.e. eigen values and eigen functions come in pairs (fj(x),
λj). Typically many solutions to this equation: (f1(x), λ1), (f2(x), λ2), (f3(x), λ3),…..
The Schrodinger equation is the basic equation for obataining the constant energy of atoms, molecules etc…
The Schrodinger equation itself is a eigen value equation with a general form
Understanding the eigen value equations in general useful for understanding the Schrodinger equation, its solutions and its
implications in solid state physics.

12. Fermi Surface:
In reciprocal lattice, the ground state of N Bloch electrons is constructed is constructed in a similar fashion as that for free
electrons, i.e. by occupying all one-electron energy levels with band energies En(k) less than EF, where EF is determined by
requiring the total number of levels with energies less than EF to be equal to the total number of electrons.
The wave vector k must be confined to a single primitive cell of the reciprocal lattice. When the lowest of these levels are
filled by a specified number of electrons, two quite distinct types of configuration can result:
1. A certain number of bands may be completely filled, all others remaining empty. Because the
number of levels in a band is equal to the number of primitive cells in the crystal (and because each level can
accommodate two electrons (one of each spin), a configuration with a band gap can arise only if the number of
electrons per primitive cell is even.
2. A number of bands may be partially filled. When this occurs, the energy of the highest occupied level, the Fermi energy EF,
lies within the energy range of one or more bands. For each partially filled band there will be a surface in k-space separating the
occupied from the unoccupied levels. The set of all such surfaces is known as the Fermi surface, and is the generalization to
Bloch electrons of the free electron Fermi sphere. The parts of the Fermi surface arising from individual partially filled bands
are known as branches of the Fermi surface. Analytically, the branch of the Fermi surface in the n-th band is that surface in k-
space determined by
En (k) = EF
Thus the Fermi surface is a constant energy surface (surfaces) in k-space.

13. Electrons in metals: free electron model
•Simplest way to represent the electronic structure of metals
•Although great simplification, works pretty well in many cases, describes many important properties of metals
•In this model, the valence electrons of free atoms become conduction electrons in crystal and travel freely
•Neglect the interaction of conduction electrons with ions of the lattice and the interaction between the conduction
electrons – a free electron gas
•Fundamental difference between the free electron gas and ordinary gas of molecules:
1)electrons are charged particles  to maintain the charge neutrality of the whole crystal, we need to include
positive ions.
This is done within the jelly model : the positive charge of ions is smeared out uniformly throughout the crystal -
charge neutrality is maintained,
no field on the electrons exerted
2)Free electron gas must satisfy the Pauli exclusion principle, which leads to important consequences.

14. Free electron gas in one dimension
Assume an electron of mass m is confined to a length L by infinite barriers Schrödinger equation for electron wave function
ψn(x):
En - the energy of electron orbital
assume the potential lies at zero  H includes only the kinetic energy 
Note: this is a one-electron equation – neglected electron-electron interactions
General solution: A sin qnx + B cos qnx
boundary conditions for the wave function:
 B = 0; qn = πn/L ; n – integer
Substitute, obtain the eigenvalues:

15. What is Hamiltonian?

16. First three energy levels and wave-functions of a free electron of
mass m confined to a line of length L:
picture from Kittel