26

CHAPTER 2

HIGH PRESSURE EXPERIMENTAL TECHNIQUES AND

ELECTRONIC STRUCTURE CALCULATION METHODS

In this chapter, details of the experimental techniques and

electronic structure calculation methods used in the present research work are

described. The experimental techniques which are utilized are, 1) Tri-arc

furnace for preparing the samples, 2) Diamond Anvil Cell (DAC) for

generating high pressure, 3) Ruby fluorescence technique for measuring

pressure and 4) high pressure Guinier X-ray diffractometer for high pressure

X-ray diffraction. An attempt has been made to design and develop an

innovative jig for alignment of the diamond anvils. Electronic structure

calculations were carried out using Full Potential Linear Augmented Plane

Wave method implemented with WIEN2K code.

2.1 EXPERIMENTAL TECHNIQUES

2.1.1 Sample Preparation Technique (Tri-arc Furnace)

Intermetallic compounds are mostly prepared by arc melting

process. Vacuum arc melting was originally devised for the metals such as

tungsten, tantalum and molybdenum because of their excessively high melting

points and for titanium, hafnium and zirconium because of their high

chemical reactivity. Although, several methods are involved in preparing the

samples, the standard arc-melting technique was used in the present work.

Figure 2.1 (a) and (b) show the photographic view of Tri-arc furnace and the

whole setup respectively. Tri-arc furnace consists of three electrodes equally

27

Figure 2.1 (a) and (b) Photograph of Tri-arc furnace and the whole

setup respectively (1. Tri-arc furnace, 2. Power supply, 3.

Helium gas cylinder and 4. Pirani Gauge)

(a)

1

2

3

4

(b)

28

located around the top of the furnace. It has copper hearth in a transparent

glass chamber which can be either evacuated or flushed with inert gas. Along

with this, the furnace is connected with a rectifier welding generator and a

water cooled resistor container assembly (RCA). The working principle of

Tri-arc furnace is very simple that the electric energy is converted to in

thermal energy.

For sample preparation, high purity elements were taken in the

required stoichiometric ratio. Then, in order to remove any oxide layer, the

materials were etched with 1:1 mixture of nitric (HNO3) and sulphuric

(H2So4) acid for ~ 2 minutes. It was then washed with acetone medium.

Etched materials were used for preparing the samples. The materials were

loaded on copper hearth. The chamber was evacuated and flushed with He

gas. This procedure was repeated for 4-5 times and then, He gas was allowed

to pass into chamber with constant flow rate. Then, the arc was produced by

bringing the electrodes in direct contact with copper hearth and then the arc

was taken slowly towards the sample to melt and form an ingot in the inert

atmosphere. The arc melted ingot was flipped and remelted several times for

achieving homogeneity. This melting procedure was done in the presence of

inert atmosphere to avoid impurities and compound formation due to

atmosphere. Newly formed ingot was kept in vacuum or inert gas sealed in a

quartz tube, and then the tube was kept inside a tubular furnace for annealing.

Annealing temperature was fixed at 85% of the melting temperature of that

compound and the temperature was regulated accordingly. The quartz tube

was taken out from the furnace and was broken to take out the ingot. Again

the ingot was etched with acid to remove the oxide layer and stored in the

hexane medium. This ingot was crushed in fine powder in hexane medium

using pestle and mortar. The powdered sample was used for characterization

by using Guinier X-ray diffractometer with scintillation detector. Detailed

description of Guinier X-ray diffractometer will be discussed in the following

section. The powdered samples were loaded in DAC for high pressure

X-ray diffraction measurements.

29

Using this Tri-arc furnace facility, UGa3 and URh3 were prepared

for high pressure X-ray diffraction studies. These two samples were

characterized by X-ray diffraction using a Guinier Diffractometer with a

scintillation detector and was found to be in single phase, AuCu3 type cubic

structure with lattice parameter: a = 4.251 ± 0.001 Å and 4.099 ± 0.006 Å

respectively. The X-ray diffraction patterns of UGa3 and URh3 was indexed

by using NBS-AIDS83. The URh3 diffraction data has been accepted by

ICDD as a standard data.

2.1.2 Mao-Bell type Diamond Anvil Cell

Diamond Anvil Cell (DAC) is a tool par excellence for achieving

ultra high pressures in the laboratory (Mao et al 2007, Bassett 2009,

Jayaraman 1983 and 1984). The original version of DAC was introduced, by

Weir et al (1959) and another group Jamison et al (1959). A very significant

step in the subsequent pressure cell design was the development of the DAC

by Mao and Bell (1978) who introduced the long body, detachable piston

cylinder assembly and employed cylindrical rockers instead of hemispherical

ones (Mao and Bell 1978). The modified home built Mao-Bell type DAC

(Deivasigamani et al 1995) was used in this work for high pressure studies.

To generate high pressure in the Mao-Bell type DAC, belleville spring-loaded

lever-arm mechanism is used (Mao and Bell 1978, Jephcoat et al 1987, Sahu

and Chandra Shekar 2007, Yousuf 1998). Figure 2.2 shows the photographic

view of home built Mao-Bell type DAC. We can reach a maximum pressure

up to 100 GPa with this home built DAC. Moreover, the modified pair of

rockers (a circular disc and a hemisphere) for easy alignment of the anvils and

new collimator design has been developed by our group (Sahu et al 2006).

Here, the anvil centering and alignment are attained by simply translating and

tilting the two tungsten carbide rockers.

30

Figure 2.2 Photograph of Mao-Bell type DAC (developed at IGCAR)

The age of DAC has crossed 50 years, the developments during

these decades brought more advanced and modern miniature cells with the

dimensions ranging from 2–10 cm in diameters and 3–7 cm in height (Smith

and Fang 2009). In the preceding chapter, the developments and applications

of DAC in various fields are discussed elaborately. The basic principle

of a DAC is very simple and is illustrated in Figure 2.3. When a metal

Figure 2.3 Schematic view of basic mechanism of DAC

31

gasket is compressed between the small flat faces (culets) of two brilliant cut

gem quality diamonds set in opposed anvil configuration, very high pressure

is generated in the gasket hole filled with pressure transmitting fluid and

sample. Generation of high pressure is again based on the ―Principle of

Massive Support‖.

There are, in general, five types of DACs reported in the literature,

namely, NBS cell, Bassett cell, Mao-Bell cell, Syassen-Holzapfel cell and

Merrill-Bassett cell (Piermarini et al 1975, Bassett et al 1967, Mao and Bell

1978, Huber et al 1977, Merrill and Bassett 1974). Several new and compact

designs have also been reported in the literature (Smith and Fang 2009). The

different kind of DACs depend on the way in which the force is generated and

also the mechanisms of anvil alignment. Table 2.1 highlights the types of

DAC and their working mechanisms. Attention is being paid to further

developments such as easy and precision alignment of diamond anvils on the

hard rockers, and the adaptability of DAC to different types of experiments in

condensed matter physics.

Table 2.1 Different types of DAC and their working mechanisms

S. No. Type of DAC Force application

1 NBS Cell Belleville Spring washer lever arm

2 Bassett Cell Threaded gland piston

3 Mao-Bell Cell Belleville Spring washer lever arm

4 Syassen-Holzapfel Cell Thread and Knee

5 Merill-Besett Cell Three-screws platen

32

2.1.3 An Innovative jig for Alignment of Diamond Anvils

In order to reach very high pressures using Mao-Bell type DAC (1)

size of the diamond culet should be minimum i.e less than 150 µm (beveled

anvils are highly preferable for ultra high pressures i.e, more than 300 GPa),

(2) the diamond anvils should be mounted symmetrically about the rocker

holes/slots, (3) the parallelism between diamond anvil culets should be

optically parallel and (4) the culets of the two diamond anvils should be

aligned optically.

The parallelism of the two diamond culets is generally carried out

in two steps. The first one is the lateral alignment, wherein the two culet faces

are made concentric and parallel visually. This is accomplished by placing the

piston and cylinder in horizontal configuration on a V-block and adjusting the

rockers with the help of the screws under a microscope. The arrangement of

piston and cylinder in horizontal configuration on a V-block is shown in

Figure 2.4. This alignment does not require the touching of the two anvils.

The second step is for attaining better concentricity and parallelism and for

this purpose, usually the observation of Newton’s fringes formed between the

diamond culet wedges is used. The fringe width and their number depend

upon the minute wedge formed between the diamond culets. This alignment

process is carried out by placing the piston over the cylinder in the vertical

configuration such that the two diamonds just touch each other. However,

aligning the two diamonds with the piston-cylinder assembly in vertical

position is risky. The alignment procedure requires that the piston be inserted

and removed from the cylinder several times until the zero fringe condition.

This brings in the risk of breaking the diamonds. Moreover, this alignment

can take several hours to complete. To prevent the breaking of expensive

diamonds and to save the alignment time, an attempt has been made to design

and fabricate a jig which simplifies the alignment process enormously.

33

Figure 2.4 Photograph of the arrangement of piston and cylinder in

horizontal configuration on a V-block

2.1.3.1 Design and operation of the diamond anvil alignment jig

The schematic of the diamond anvil alignment jig is shown in

Figure 2.5. The total height of the jig is 90 mm, into which the piston and

cylinder assembly has to be placed as shown in Figure 2.6. It is made of

stainless steel. The outer diameter of the jig is around 50 mm and the inner

diameter is 34 mm. Two numbers of 6.1 mm diameter holes are provided at a

position 30 mm from the top of the jig. These holes are used for holding the

cylinder in a vertical position with the help of Teflon plugs. An 8 mm

diameter hole is provided in the bottom plate of the jig for allowing light from

the bottom. Two reflecting mirror pieces are placed by the side of the 8 mm

hole for viewing the bottom of the cylinder.

The visual alignment of the diamonds is done by adjusting the

rockers, while the piston cylinder assembly is kept on a V-block in horizontal

configuration. Then the cylinder of the DAC alone is inserted in the jig from

the top and clamped by the two Teflon plugs from the side holes matching

34

with the holes of the cylinder exactly. Subsequently the piston is inserted

inside the cylinder slowly and placed such that the two diamond culets just

touch each other. Once the piston is inserted inside the cylinder, there is no

need to remove it until the alignment is completed. Piston diamond will be

touching the cylinder diamond while doing this alignment. There is hardly

any relative motion between the two because the cylinder is clamped by the

Teflon plugs. The height of the cylinder is about 60 mm, and there is a gap of

~ 30 mm between the bottom of the cylinder and the bottom plate of the jig.

The rocking cylinder diamond is mounted on the hemispherical cylinder

rocker and the translating piston diamond is mounted on a disc rocker

(Figure 2.6). Whereas the piston rocker is adjusted and held rigidly by four

grub screws, the hemispherical cylinder rocker sitting in a hemispherical

cavity at the bottom of the cylinder is held rigidly by a locking plate. The

locking plate is adjusted by three socket-head-cap-screws accessible from the

bottom of the cylinder (Figure 2.6). It is thus very convenient to view the

screws at the bottom of the cylinder by the reflecting mirrors and adjust the

screws with an allen key. The alignment is done while viewing the Newton’s

fringes formed between the diamond culets under the microscope.

Transmission light will be coming through the bottom hole of the jig. The

adjustment of the screws is carried out until the Newton’s fringes disappear

and a uniform color appears between the diamond culets.

Figure 2.7 (a) and 2.7 (b) show achievement of the parallelism

between the culets. The diamond anvil alignment jig avoids multiple insertion

of the piston in the cylinder, there by avoiding greatly the damage to the

diamonds. It takes hardly about 5 minutes to complete the alignment. At

present, this jig is working successfully for aligning the diamond anvils.

35

Figure 2.5 Schematic view of diamond anvil alignment jig

Figure 2.6 Photograph of the alignment jig (1. Piston, 2. Cylinder,

3. Teflon plug, 4. Mirrors)

36

Figure 2.7 (a) and (b) Achievement of the parallelism between the culets

2.1.4 Pressure Calibration – Ruby Fluorescence Technique

Pressure measurement can be classified into two categories:

primary and secondary. (1) Primary methods are based on fundamental

equations relating pressure to any other physical quantities. (2) Secondary

methods are based on the systematic variation of any physical property of a

material with pressure. Primary methods are based on Mercury columns and

pressure balances are restricted to very small pressures (~1 MPa and 2.5 GPa

respectively). The common secondary methods for measuring high pressures

are: equation of state (EOS), fixed point method and Ruby fluorescence

method. In practice, pressure measurements are possible only with secondary

methods (Sahu and Chandra Shekar 2007).

Pressure inside the DAC can be calibrated by using either EOS

method or Ruby fluorescence method. In this work, ruby fluorescence method

was used for pressure calibration. In the Ruby fluorescence method, a small

chip (~ 5 - 10 たm) of Ruby (Al2O3:Cr3+ (0.05 %)) is introduced in the pressure

chamber and the fluorescence is excited by the wavelength 514.5 nm from

Ar-ion laser. The fluorescence is detected with an optical spectrometer. The

Ruby fluorescence spectrum contains two well defined peaks (R1 and R2)

which shift to higher wave lengths with increasing pressure. The pressure

(a)

(b)

37

dependence of the Ruby R1 (692.7 nm) and R2 (694.2 nm) shifts are used to

determine pressure. Figure 2.8 shows a typical Ruby fluorescence spectra

obtained in the laboratory. The shifts of the Ruby fluorescence lines have

been calibrated against standard substances to construct a pressure scale. The

shifts are linear up to ~ 30 GPa at the rate of 0.365 nm / GPa (or 753 m-1 /

GPa in terms of wave number) (Piermarini et al 1975, Barett et al 1973).

The calibration up to 200 GPa is given by the relation (Bell et al

1986):

P(TPa) = 0.3808 [ 1 + ( hそ/そ)5 – 1 ] (2.1)

Apart from Ruby, there are several other materials like Sm2+:

SrB4O7, Sm2+: BaFCl, Sm2+: SrFCl, Eu2+: LaOCl and Eu3+: YAG which are

used as optical pressure sensors.

Figure 2.8 Typical Ruby fluorescence spectra

38

2.1.5 High Pressure X-ray Diffraction Technique

X-ray diffraction (XRD) technique is a principal technique to study

the unknown structures of the known material. In the high pressure

experiments, the investigation of phase transition in materials is a very

demanding task and it requires a high precision XRD set up. It is well known

that a Guinier diffractometer provides high resolution XRD data with a better

S/N ratio in the case of samples at NTP. Hence diamond-anvil with high-

pressure X-ray diffractometer was used in the Guinier geometry for all the

experiments (Sahu et al 1995). The brief description of the instrumentation for

the diffractometer is given below.

The schematic and photograph of the Huber-Guinier diffractometer

setup used is shown in Figures 2.9 and 2.10 respectively. It is in the vertical

configuration and in symmetric transmission mode. The incident X-ray beam

is obtained from an 18 kW rotating anode X-ray generator (RAXRG) with a

Mo target. The curved quartz-crystal monochromator is of Johansson- Guinier

type with A = 118 mm and B = 3.55 mm, with its reflecting surface parallel to

the crystallographic (10┆1) plane and is mounted on the X-ray tube shield. The

X-ray beam from the monochromator enters the DAC after passing through a

beam reducer attached to the diffractometer. The Mao-Bell-type DAC is

secured on to a multiple stage which has X, Y, Z and tilt movements for

alignment with respect to the incident X-ray beam. The DAC is positioned in

such a manner that the sample inside lies on the Seeman-Bohlin circle of

114.6 mm in diameter. On this circle, either a position sensitive detector

(PSD) or a scintillation detector can be mounted. The PSD is a linear gas

filled type of 50 mm in length (model OED-50, Braun, Germany). A mixture

of 90% argon and 10% methane is used as the PSD gas in the continuous flow

mode at the rate 0.2 l/h at 0.75 MPa pressure. A platinum wire is used as the

anode in the PSD. The specified positional resolution of the PSD is 100 たm.

39

This corresponds to an angular resolution of 0.05° and hd/d =0.005.

For the Seeman-Bohlin circle employed here, the PSD can cover an angle of

10° in a single scan. Since the maximum angular opening in the DAC is 20°,

two positional scans are necessary to record a full HPXRD spectrum. In the

Guinier geometry, the maximum allowable し range is -45° to +45°. But in the

present setup, it can cover the range of -30° to +25° due to the mechanical

limitations imposed by the DAC mounting stage. While no slit has been used

for the PSD, a variable linear slit along with a soller slit are used for the

scintillation detector to reduce the noise. It was found convenient to use the

scintillation detector for obtaining XRD patterns of samples at NTP, and the

PSD for HPXRD data. The scintillation detector cannot be used for the latter

purpose due to very weak diffraction intensity. Angular scans with this can be

carried out with a minimum step of 0.001°. The motor drive control for the

detector movement, control of the detector electronics, data acquisition and

analysis, etc., are computerized.

Figure 2.9 Schematic view of Huber-Guinier diffractometer

40

Figure 2.10 Photograph of Huber-Guinier diffractometer

2.2 ELECTRONIC STRUCTURE CALCULATIONS

In general, the electronic structure of a material relates the crystal

structure with its physical and chemical properties. However, electronic

structure calculations are basis to understand the behavior of materials under

extreme conditions like pressure and temperature. Both the experimentalists

and theoreticians are looking for electronic structure calculations to know

more about the properties of matter and nature of bonding in materials.

In many body problems, the nuclei and electrons are treated as

electromagnetically interacting point charges. The exact many-particle

Hamiltonian for this system is:

41

(2.2)

Mi is the mass of the nucleus at Ri

. The electrons have mass me and

are positioned atir . The first and second term correspond to the kinetic

energy operator of the nuclei and of the electrons respectively. The last three

terms describe the coulomb interaction between electrons and nuclei, between

electrons and other electrons, and between nuclei and other nuclei. In order to

find out the eigenstates of this system, the corresponding Schrödinger

equation has to be solved.

),(),( rRErRH (2.3)

Due to the high degree complexity, some approximations are made

as illustrated below.

2.2.1 Born-Oppenheimer Approximation

Nuclei mass is much heavier than the mass of electrons. Hence, it is

reasonable to assume that the nuclei to be stationary and the electrons to be in

instantaneous equilibrium. Due to this assumption, kinetic energy of nuclei

ji

ji

ji

jiji RR

ZZe

rre

2

0

2

08

18

1

i i ji

ji

i

ei

R

rR

ZemM

rH i

,

2

0

22

41

22

i i jiji

i

ei

R

rRZe

mMrH i

,

2

0

22

41

22 Vext

V

T e

T N

VNN

42

term disappears and the nucleus-nucleus interaction term reduces to constant

in the equation (2.2).

VextVTH

(2.4)

The remaining terms belong to kinetic energy of electron gas, the

potential energy due to electron-electron interaction and the potential energy

of the electrons in the potential of the nuclei respectively. Here, the electrons

and nuclei are treated separately. This decoupling motion of the electronic and

nuclear motion is known as the Born-Oppenheimer or adiabatic

approximation.

2.2.2 Density Functional Theory

Although, the above approximation simplifies the quantum many

body problem, but still it is too complex to solve for large systems. In order to

reduce this equation further, some more approximations have to be made. A

historically very important method is Hartree-Fock method (HF). This HF

approximation performs very well for atoms and molecules. However, it is

quite difficult to solve this for solids. One more important and powerful

method is Density Functional Theory (DFT) established by Hohenberg and

Kohn (1964). Detailed description and excellent reviews are available in the

literature (von Barth 1982, Jones and Gunnarsson 1989).

2.2.2.1 Theorems of Hohenberg and Kohn

Hohenberg and Kohn established two theorems as:

First theorem: There is a one-to-one correspondence between the

ground-state density r of a many-electron system (atom, molecule, solid)

and the external potentialVext

. An immediate consequence is that the ground-

43

state expectation value of any observable O

is a unique functional of the

exact ground-state electron density:

O O p (2.5)

Second theorem: For O

being the Hamiltonian H

, the ground state

total energy functional EVH

ext

is of the form.

VE extVT

Vext

rdrr VF extHK (2.6)

where the Hohenberg-Kohn density functional F HK is universal for any

many-electron system. EVext

reaches its minimal value (equal to the

ground state total energy) for the ground state density corresponding toVext.

The detailed description and proof of the theorems of Hohenberg and Kohn

(1964) are not discussed in this chapter.

2.2.2.2 The Kohn-Sham equations

A set of eigenvalue within density functional theory are called

Kohn-Sham equations (Kohn and Sham 1965). These Kohn-Sham equations

were published in 1965. Here, they have rewritten the Hohenberg-Kohn

functional in the following way

0HK H x CV V VF T (2.7)

where T 0 is the functional for the kinetic energy of a non-interacting gas,

VH stands for Hartree contribution, which describes the interaction with field

44

obtained by averaging over positions of the remaining electrons. VX for

exchange contribution, VC for correlation contribution and VXC

is the

exchange-correlation energy functional. The total energy E of the system as a

functional of the charge density can be written as:

EVext

= T 0 + VH

+ VXC + Vext

(2.8)

This equation interpreted as the energy functional as the energy

functional of non-interacting electron gas, subject to two external potentials

VXC andVext

.

The corresponding Kohn-Sham Hamiltonian:

H KS = T

0 + VH

+ VXC

+ Vext

V

Vem ext

xci

e

rdrr

r

42 0

22

2

(2.9)

First, the exchange-correlation energy EXC is not known precisely,

and second, the kinetic energy term must be created in terms of charge

density. According to the Kohn-Sham theorem, the exact ground state density

r of an N-electron system is

rrri

N

ii

1

*

(2.10)

where the single particle wave functions ri

are the N lowest-energy

solutions of the Kohn-Sham equations.

iiiKSH (2.11)

45

To obtain ground state density of the many-body system the

Schrödinger like single particle equation must be solved. The Kohn-Sham

equation proves to be a practical tool to solve many-body problems.

2.2.2.3 Exchange-correlation functional

In order to solve the Kohn-Sham equation, exchange-correlation

functional should be known. To have exact expression for exchange-

correlation functional, we need some approximations.

A widely used approximation is the Linear Density Approximation

(LDA) (von Barth and Hedin 1972, Ceperley and Alder 1980); it defines the

exchange-correlation functional as:

rdrrXC

LDA

XCV (2.12)

Here, rXC

stands for the exchange-correlation function for

the homogenous electron gas with interacting electron gas and is numerically

known from Monte-Carlo calculations. This postulate states that the

exchange-correlation energy due to a particular density r could be found

by dividing the material in infinitesimally small volumes with constant

density. Each such volume contributes to the total exchange correlation

energy by an amount equal to the exchange-correlation energy of an identical

volume filled with a homogenous electron gas. That has the same overall

density as the original material has in this volume.

Some improvements made on LDA, and this approximation called

as Generalized Gradient Approximation (GGA) (Perdew and Wang 1986,

1992, Perdew et al 1996). Here, the exchange-correlation contribution of

every infinitesimal volume not only dependent on local density in that

volume, but also on the density in the neighboring volumes,

46

rdrrrXC

GGA

XCV , (2.13)

GGA usually performs better than LDA, but in the case of LDA a

unique rXC

is available. In GGA, there is some freedom to incorporate

the density gradient, and therefore several versions of GGA exist. Moreover,

many versions of GGA contain free parameters which have to be fitted to

experimental data.

2.2.2.4 The full potential linear augmented plane wave method

In order to solve the Kohn-Sham equations that resulted from DFT,

a suitable basis set should be introduced.

The Full Potential Linear Augmented Plane Wave (FP-LAPW)

method (Slater 1937, Andersen 1975, Singh 1994, Koelling and Arbman

1975) is like most energy-band methods, with the procedure for solving the

Kohn-Sham equations for the ground state density, total energy, and (Kohn-

Sham) eigenvalues of a many electron system by introducing a basis set

which is especially adapted to the problem.

Here, the basic idea behind the method is like this: the region far

away from nuclei, the electrons are more or less free. These free electrons are

described by plane waves. The wave functions near atomic nuclei are

described by atomic like functions. Because the electrons near to the nuclei

behave quite as they were in a free atom. Hence, the space is divided in two

regions where the different basis expansions are used, (i) Non-overlapping

spheres and (ii) interstitial region. Around each atom a sphere with a radius

R is drawn. Such a sphere is called as muffin tin sphere. The remaining

space outside the sphere is called as interstitial region which is shown in

Figure 2.11.

47

Figure 2.11 Partitioning of the unit cell into atomic spheres (I) and an

interstitial region (II)

An LAPW basis function has the similar form as an APW basis

function, but the part of the basis function in the muffin tin region, the

augmentation has been performed and the final definition for LAPW:

1

1 i k K r

k

K ,k K ,k K l

l ,m i l l ,m i l m i M ,T ,l ,m l

r IVr

, ,

e

U r SA r E B r E Y rU

(2.14)

where ErU li

,

1 is the normal way out of the radial Schrödinger equation for

energy E l and spherical part of the potential inside the sphere. k

stands for a

vector in the first Brillouin zone, K

a reciprocal lattice vector, V is the unit

cell volume and rY i

l

m

are spherical harmonics with {l, m}. AKk

ml

,

,

and

BKk

ml

,

,

are expansion coefficients.

Basis functions defined in the formula are infinitely large and two

more parameters have to be introduced to limit these sizes. The first one is

lmax, which controls the infinite sum over angular momenta l. This summation

will be truncated at lmax. The second parameter is Kmax, which determines the

48

size of the basis set. Only those basis functions with k that satisfies the

condition K are introduced in the basis set. As a consequence, lmax and Kmax

control the accuracy of the calculation. Well converged basis is obtained for

RMT - Kmax = 7 - 9 for most systems.

2.2.3 WIEN2K

In WIEN2K, the calculation generally starts with structure file

which contains the crystallographic information such as space group, lattice

parameters and atomic positions. Here, the Muffin tin radii can be calculated

by WIEN itself and is system specific. The working process of WIEN2K code

can be separated in two parts. The first one processes input file while the

second one performs a self consistent calculation.

In the first part the parameters in the input file are optimized for

economizing the computational time and optimizing the lattice parameter. For

initializing the calculation with input file, the following parameters are

optimized: K points, Rmt*Kmax and Gmax and lattice parameters (volume). The

general procedure followed for optimizing these parameters are given in the

flowcharts 1& 2. Flowchart – 1 explains the procedure for optimizing K

points, Rmt*Kmax and Gmax and the flowchart – 2 elucidates the optimization of

lattice parameters or volume. After completing the task of finding the

optimum values of these parameters, the calculations are started afresh with

optimized lattice parameters, K points, Rmt*Kmax and Gmax.

In the second part, the process is divided into several subroutines

which are repeated again and again until the convergence is reached. This Self

Consistent Field (SCF) runs through five modules. The description of five

modules are discussed below.

49

Structure Gen

* Initialize

calculation

SCF cycle

** Reproducible

Energy (Check File: . outputm)

Op

tim

ized

Rm

t*K

max

Structure Gen

* Initialize

calculation

SCF cycle

** Reproducible

Energy (Check File: . outputm)

Op

tim

ized

k-p

oin

ts

Structure Gen

* Initialize

calculation

SCF cycle

** Reproducible

Energy (Check File: . outputm)

No

Yes

No

n r

epro

du

cib

le

ener

gy

No

No

n r

epro

du

cib

le

ener

gy

Yes

No

n r

epro

du

cib

le

ener

gy

Yes

No

* While optimizing k-points;

Vary k-points from (8* 8* 8 to

25* 25* 25). Fix Rmt*Kmax value (default

minimum value is 7). Fix Gmax value (default value is

12).

** until get reproducible energy, the k-

points varied from lowest to larger k-

mesh.

* While optimizing Rmt*Kmax; Optimized k-points are used and

fixed that value for all the

calculations. vary Rmt*Kmax value (from 7 -

10). Fix Gmax value (default value is

12).

** until get reproducible energy,

Rmt*Kmax value varied from 7 -10.

* While optimizing Gmax; Optimized k-points are used and

fixed that value for all the

calculations. Optimized Rmt*Kmax value was

used for all the calculations. Vary Gmax value (default value is

12). In this case, I have kept Gmax

default value as a fixed value for all

the calculations.

** until get reproducible energy, Gmax

value varied from 12 to 14.

Optimization of calculation parameters

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50

Optimization of lattice parameters or volume

* The file #case.outputeos gives the data of volume and energy for different pressures. The

volume which corresponds to the minimum pressure (almost zero pressure) is considered as an

equilibrium volume (Relaxed lattice parameters can be derived from the equilibrium volume).

StructGen

Initialize calc.

Run SCF

*Optimize (V, c/a) Insert change in volume in

various percentages (check file: case. outputeos)

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51

(1) LAPW0 starts with calculating Coulomb and the exchange

correlation potential, (2) LAPW1 calculates the eigenvalues and

eigenfunctions of the valence states, (3) LAPW2 calculates the valence

density from the Fermi energy which separates the filled states from unfilled

states, (4) This subroutine calculates the core electrons from the results of

total core density and (5) The last subroutine mixes the old, core and valence

densities. After completion of these five modules, it checks the convergence

between old and new density until it reaches the consistent density. Once the

convergence is over, with these files it is easy to calculate electron density

plots, density of states (DOS), X-ray spectra, optical data and band structure

by using Wien. In this, calculations were made for the DOS, electron density

plots and band structure.