J. Nano Ele. Phys. 3 (2011) 884. J. Phy. Conference Series (in …shodhganga.inflibnet.ac.in/bitstream/10603/7360/12/12... · 2015-12-04 · CdO 0.419 [13] 4.3 Results Carbides (1)

Chapter-4_________________________________________________________

_____________________________________________________________________Shyamkumar G. Khambholja / Ph.D. Thesis/ Physics/ S.P. University/ April-2012

111

Chapter 4

Finite Temperature

Thermophysical Properties of

Solids

J. Nano Ele. Phys. 3 (2011) 884.

J. Phy. Conference Series (in press).

Communicated in Chinese Physics Letters (IOP Pub.).

Communicated in Advances Materials Research.

4.1 Introduction 112

4.2 The quasi harmonic Debye model 113

4.3 Results 116

4.4 Conclusions 173

4.5 References 176

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112

4.1 Introduction

In the previous chapter, we have discussed results of the ground state and high

pressure structural and lattice mechanical properties of solids. In the present chapter,

we report the study of finite temperature thermophysical properties of carbides,

fluorides and oxides compounds. The detail of the calculations used to compute the

finite temperature thermophysical properties of solid compounds is given below.

The study of thermodynamic properties of materials is important from several

points of views. It is important not only to elucidate interactions among the particles

that are essential, but to extend our knowledge on their specific behaviors, when

undergoing several constraints such as high temperatures and/or high pressures. This

is particularly true, since from modern technology one always expects new advances

and innovation of materials to reach higher performances. With a realization of these

situations, a great deal of efforts has been made in recent years on the thermophysical

properties at an extreme environment. The thermal expansion of materials arises from

the anharmonicity of the interatomic potential. Such a change is accompanied by

change in elastic, vibrational and mechanical properties. There are mainly two issues

regarding the calculation of finite temperature thermophysical properties of materials.

The first one is the calculation of cohesive properties at ambient condition, which in

the present work is performed using plane wave pseudopotential density functional

theory as implemented in the Quantum ESPRESSO code [1]. The theoretical method

used for the calculation of cohesive properties at ambient condition is already

described in the previous chapter. Once the cohesive properties at ambient conditions

are obtained, the another issue is the inclusion of effect of temperature. The equation

of state (eos) and chemical potential are the two key thermodynamic properties of

solid. Theoretically, the determination of eos and chemical potential are obtained

using standard thermodynamic relations. According to standard thermodynamics, if

the system is held at fixed temperature T and hydrostatic pressure P, then the

equilibrium state is one that minimize the availability of the non-equilibrium Gibbs

energy, given by [2],

* ( ; , ) = ( ) ( ) + ( ; )vibG x p T E x pV x A x T+ (4.1)

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113

with respect to all internal configuration parameters. These configuration parameters,

gathered in the configuration vector x, include all the relevant geometric information

for the given crystal structure i.e. independent of unit cell lengths and angles of this

phase, and all the crystallographic coordinates of the atoms is non-fixed Wyckoff

positions. On the right side of above equation (4.1), the first term gives the cohesive

energy of the system. The second term gives the corresponding hydrostatic pressure

and the third term gives the vibrational Helmoltz free energy, which includes both the

vibrational contribution to the internal energy and the constant temperature condition

term. For the calculation of vibrational Helmoltz free energy, exact knowledge of

vibrational states is required. In the quasi harmonic approximation, it is given by [2],

0

1( ; ) = + ln(1 ) ( ; )

2kT

vibA x T kT e g x dω

ω ω ω∞ −

−

∫h

h (4.2)

The quasi harmonic approximation is not like the rigid harmonic approximation

[3]. But, it allows the inclusion of effect of temperature through the volume

dependence of phonon frequencies and corresponding density of states.

Thus, it is clear that for the calculation of thermodynamic properties, one requires

the exact knowledge of( )E x , ( )V x and ( ; )g x ω . Once, these quantities are obtained,

minimization of Gibbs free energy leads to all thermodynamic properties.

4.2 The quasi harmonic Debye model

The quasi harmonic Debye model starts with the calculation of cohesive

energy of the system at ambient condition. Before the quasi harmonic Debye model

starts, the multivariable surface has to be transformed into a ( )E V curve. In the

present work, the multivariable surface is transformed into a ( )E V curve, by

minimizing ( )E x for a set of fixed volumes. In the present work, calculation of

cohesive energy is performed using plane wave pseudopotential density functional

theory.

After the static calculation, equation (4.1) can be rewritten as,

* ( ( ); , ) = ( ( )) + + ( ( ); )opt opt vib optG x V p T E x V pV A x V T (4.3)

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114

which is a function of ( ; , )V p T only. The vibrational contribution is written in terms

of Debye model as [2],

9( ; ) = + 3 ln(1 )

8T

vibA T nkT e DT T

θθ θθ− − −

(4.4)

Here, ( )D y is the Debye integral given by,

3

30

3( ) =

1

y

x

xD y dx

y e −∫ (4.5)

Here, = yT

θ and θ is the Debye temperature and n is number of atoms. Debye

temperature in the solid is related to the average sound velocity in solids and can be

calculated from the Poisson’s ratio using the formula given by [2],

1/32 1/2 = 6 ( ) s

B

BV n f

k Mθ π σ

h (4.6)

Where, ( )f σ is given by,

1/313/2 3/22 1 1 1

( ) 3 2 + 3 1 2 3 1

fσ σσσ σ

− + + = − −

(4.7)

Poisson’s ratio is the most important input quantity in the present calculation.

Experimental values of Poisson’s ratio of carbides, fluorides and oxides are used for

the calculation of thermodynamic properties.

Adiabatic bulk modulus sB can be calculated from the static compressibility relation,

given by [2],

=

===

2

2

2

2)(

))((

))(( )( dV

VEdV

dV

VxEdVVxBxBB

optoptstaticstatics (4.8)

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By minimizing the Gibbs function, with respect to volume at given temperature, one

can obtain thermal expansion, using,

*

,

( ; , ) = 0

p T

G V p T

V

∂ ∂

(4.9)

Isothermal bulk modulus can be obtained by,

( , ) - TT

pB p T V

V

∂ = ∂ (4.10)

To simplify the process of minimization, the calculated energies are fitted to the Birch

equation of state [2]. All the calculation of the thermodynamic properties of solids are

carried out using quasi harmonic Debye model as implemented in GIBBS program,

originally created by Blanco et al [2]. In the code, three different forms of equation of

states namely, due to Vinet et al, Birch-Murnaghan et al and the spinodal are given.

Other thermodynamic properties namely, heat capacity and thermodynamic Grüneisen

parameter are calculated using following relations [2],

, /

3 / = 3 4 -

1v vib B T

TC nk D

T eθθ θ −

(4.11)

ln ( ) -

ln

d V

d V

θγ = (4.12)

More details about the algorithm of the GIBBS code can be found out in the paper by

Blanco et al [2].

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Table 4.1 List of Poisson’s ratio used in the present calculations.

Materials Poisson’s ratio References

SiC 0.216 [4]

GeC 0.16 [5]

SnC 0.21 [6]

ZrC 0.28 [7]

LiF 0.216 [8]

NaF 0.174 [9]

CaO 0.207 [10]

SrO 0.21 [11]

BaO 0.292 [12]

CdO 0.419 [13]

4.3 Results

Carbides

(1) Silicon Carbide (SiC)

Very few theoretical studies of finite temperature thermophysical properties of

SiC are available [14-17]. In an earlier work, Karch et al [14] have calculated the

thermophysical properties of 3C-SiC using density functional theory within local

density approximation (LDA) in conjunction with quasi harmonic approximation

(QHA). Vashistha et al [15] have calculated thermophysical properties of 3C-SiC

using molecular dynamics simulation. Vardachari et al [17] have calculated the

structural properties of 3C-SiC using full potential linear augmented plane wave

method (FP-LAPW). Further, they have calculated the finite temperature

thermophysical properties using harmonic approximation. We in the present work

have calculated the thermophysical properties of 3C-SiC using plane wave

pseudopotential density functional theory [1] in conjunction with quasi harmonic

Debye model [2]. The details of the ground state properties are already given in the

previous chapter. So, we report here directly the results of thermophysical properties

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of 3C-SiC. The value of Poisson’s ratio is taken from the Reference [4] and is shown

in Table 4.1. Figure 4.1 shows the calculated equilibrium lattice constants at various

temperatures along with experimental and other theoretical results. The experimental

results [18] are available in the limited range of temperature. On the other hand,

results of molecular dynamic simulation [15] are available upto the melting point

( ≈ 3100 K) of 3C-SiC. The presently calculated value of lattice constants shows a

little deviation of only from the experimental results. On the other hand, it shows very

good agreement with the molecular dynamic simulation results of Vashistha et al [15].

Moreover, the trend of the variation of lattice constant with temperature is also similar

to that of molecular dynamics results. The presently calculation of lattice constants

also indicate that at high temperatures, anharmonic effects plays prominent role.

Figure 4.1. Calculated equilibrium lattice constant of SiC at various temperatures

(full line), experimental results (dots) [18], molecular dynamics results (dots with

dashed line) [15] and results of Tang and Yip (open circle) [19].

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Figure 4.2. Thermal equation of state of ZB-SiC at ambient condition (full line), 500

K (dashed line), 1000 K (dotted line) and experimental results (filled

dots) [20].

Figure 4.2 shows the presently calculated thermal equation of state (eos) at

various temperatures along with experimental results [20] at zero temperature. All the

features of thermal expansion at finite temperatures are observed in the Figure 4.2. A

good agreement is observed. Figure 4.3 shows the presently calculated phonon

dispersion relation at ambient condition and at finite temperatures. At finite

temperatures, phonon dispersion is calculated in a quasi harmonic way. At finite

temperatures, phonons in 3C-SiC becomes soft along all symmetry directions.

However, the decrease in phonon frequencies with temperatures is not much high.

This indicates that the temperature coefficient of elastic constants should be small, as

the phonon frequencies along [100] and [110] planes are directly related to the elastic

stiffness constants. Figure 4.4 shows the quasi harmonic phonon density of states (p-

dos) of 3C-SiC at ambient conditions and at finite temperatures. At finite

temperatures, the principal peak in the p-dos shifts towards left (lower frequency

side). Also, the height of principal peak decreases with temperature, which indicates

that at finite temperatures mean frequency of vibration decrease, indicating weak

correlation among atoms at finite temperatures.

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Figure 4.3. Phonon dispersion relation along major symmetry directions for 3C-SiC

at ambient condition (full line), T=500 K (dashed line) and in T=1100 K (dotted line).

Figure 4.5 represents the calculated Gibbs free energy function at various

temperatures. It is observed that it decreases with temperature, as expected. Figure 4.6

shows the presently calculated isothermal and adiabatic bulk modulus at various

temperatures along with the corresponding results of Karch et al [14]. The trend of

variation of bulk modulus with temperature for presently calculated results and results

due to Karch et al [14] is nearly same. The deviation in the presently calculated results

and the results of Karch et al [14] may be attributed to the different treatment of ionic

motion at finite temperature and especially the thermodynamic Grüneisen parameter

as well as the use of different exchange-correlation functional. Figure 4.7 shows the

temperature dependent Debye temperature of 3C-SiC. We have used the classical

approach of quasi harmonic Debye model for calculating the thermal contribution of

lattice ions. As a result, we do not observe any quantum effect present at low

temperatures ( DT θ< ).

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Figure 4.4. Calculated quasi harmonic phonon density of states of SiC.

0 K (full line), 500 K (dashed line) and 1100 K (dotted line).

Figure 4.5. Temperature variations of Gibbs function of SiC.

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Figure 4.6. Presently calculated Isothermal (full line) and adiabatic (dashed line) bulk

modulus of SiC along with corresponding results of Karch et al [14] (filled and open

dots respectively).

Calculated and empirical results (using experimental specific heat) of temperature

dependent Debye temperature are reported by Vashistha et al [15]. They have

reported their results using well known low temperature expression of Debye

temperature in terms of specific heat, given by ( )34 / 5

12 DBv TkNC θπ= . They have

calculated specific heat using vibrational density of states. In our work, we have

directly estimated temperature dependence of Debye temperature in a quasi harmonic

way using classical approach. The results of Vashistha et al [15] are limited up to

1000 K (near to normal Debye temperature) beyond which, we believe that the Debye

temperature should decrease with temperature, due to decrease in phonon frequency

with temperature. Thus, our results provide an estimate of temperature variation of

Debye temperature at high temperatures (DT θ> ). Our presently computed values of

specific heats at constant volume and pressure are plotted in Figure 4.8 along with the

molecular dynamics results of Vashistha et al [15]. A good agreement of presently

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computed Cv is found with the molecular dynamics results. Our presently calculated

Cv is slightly higher compared to molecular dynamics results. One reason behind this

may be that we have calculated the temperature variation of specific heat using

temperature dependent Debye temperature, in order to include the effect of quasi

harmonic calculation on specific heat. Similar behavior is found for the Cp, which has

been calculated using temperature dependent thermodynamic Grüneisen parameter.

Nevertheless, overall a good agreement is achieved. Figure 4.9 shows the calculated

thermodynamic Grüneisen parameter at various temperatures. It is observed that

thermodynamic Grüneisen parameter increases parabolically with temperature.

Figure 4.7. Presently calculated Debye temperature of SiC as a function

of temperature.

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123

Figure 4.8. Calculated specific heat of SiC at constant volume (full line) and at

constant pressure (dashed line) along with the corresponding molecular dynamics

results of Vashistha et al [15] (filled and open circles respectively).

Figure 4.9. Calculated thermodynamic Grüneisen parameter of SiC at various

temperatures.

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(2) Germanium Carbide (GeC)

Compared to SiC, GeC is less studied. The phonon dispersion relation for SiC

is reported experimentally [21] but phonons in GeC are not reported, experimentally.

Work on structural properties of GeC at ambient and at high pressure is performed so

far by various researchers. However, studies on finite temperature thermophysical

properties are less. Sekkal et al [22] have performed molecular dynamics study using

Tersoff potential and calculated various thermodynamic properties namely Debye

temperature, thermal expansion coefficient, heat capacity and Grüneisen parameter.

Their results showed that physical properties of SiC and GeC systems are much

influenced by presence of C atoms, compared to Si and Ge atoms. In the present

work, we report the finite temperature thermophysical properties of zinc blende (ZB)

Gec using density functional theory [1] in conjunction with quasi harmonic Debye

model [2]. The value of the Poisson’s ratio is taken from Reference [5]. Figure 4.10

shows the calculated values of relative equilibrium lattice constants (/ oa a ) at various

temperatures. No experimental or any other theoretical data for the comparison are

available in literature. In the absence of any such data, presently calculated data shall

work as a useful set of data in future. Figure 4.10 also shows the corresponding

relative equilibrium lattice constants of 3C-SiC along with the results of Vashistha et

al [15]. It is observed that unlike 3C-SiC, the value of relative lattice constants

increases sharply with temperature, which indicates that ZB-GeC is a soft material

compared to 3C-SiC. Figure 4.11 shows the presently calculated thermal eos at

various temperatures. For comparison, no experimental or any theoretical data are

available in literature. However, since we have calculated this property using density

functional theory in conjunction with quasi harmonic Debye model, it will serve as a

reliable set of data for further research.

Figure 4.12 shows the calculated phonon frequencies in ZB-GeC at various

temperatures. It is observed that similar to ZB-SiC, in ZB-GeC also, phonon

frequencies become soft along all principal symmetry directions. Figure 4.13 shows

the computed quasi harmonic p-dos at elevated temperatures. The principal peak

shifts towards lower frequency side. Also, height of the principal peak is also affected

by temperature. Figure 4.14 shows the calculated Gibbs free energy function at

various temperatures. It is observed that it decreases with temperature as expected.

Figure 4.15 shows the presently calculated isothermal and adiabatic bulk modulus of

Chapter-4_________________________________________________________


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ZB-GeC at various temperatures. The temperature coefficient of isothermal bulk

modulus is -0.0396 GPa/K, while the temperature coefficient of adiabatic bulk

modulus is -0.0651 GPa/K at high temperatures. Figure 4.16 shows the presently

calculated temperature dependent Debye temperature of ZB-GeC at various

temperatures. The temperature coefficient of Debye temperature is -0.0912 at high

temperatures. Again, as we are using the classical approach to calculate thermal

properties, we do not observe any quantum effect at low temperatures.

Figure 4.10. Lattice constant of ZB-GeC (full line) as a function of temperature

calculated using quasi harmonic Debye model. Also, shown are the results of

equilibrium lattice constant of 3C-SiC (dashed line) and molecular dynamics results

of Vashistha et al (dots) [15].

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Figure 4.11. Calculated thermal equation of state of ZB-GeC at various

temperatures.

Figure 4.12. Quasi harmonic phonon dispersion in ZB-GeC along high symmetry

directions at ambient condition (full line), at T=1000K temperature (dotted line) and

T=1500K temperature (dashed line).

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Figure 4.13. Quasi harmonic p-dos of ZB-GeC at various temperatures. 0K (full line),

1000K (dashed line) and 1500K (dotted line).

Figure 4.14. Temperature variations of Gibbs free energy function for ZB-GeC.

Chapter-4_________________________________________________________


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Figure 4.15. Isothermal (BT) (full line) and adiabatic (Bs) (dashed line) bulk modulus

of ZB-GeC at various temperatures.

Figure 4.16. Temperature dependence of Debye temperature of ZB-GeC.

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Figure 4.17. Specific heat at constant volume (Cv) (full line) and at constant pressure

(Cp) (dashed line) for ZB-GeC.

Figure 4.18. Thermodynamic Grüneisen parameter of ZB-GeC as a function of

temperature.

Chapter-4_________________________________________________________


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Figure 4.17 shows the presently calculated specific heats of GeC at various

temperatures. Figure 4.18 shows the calculated thermodynamic Grüneisen parameter

at various temperatures. Thermodynamic Grüneisen parameter does not increase

sharply with temperature. We observe in our study that ZB-GeC is soft material, with

relatively large thermal expansion (as compared to ZB-SiC). The full set of

thermodynamic properties of ZB-GeC is reported here for the first time. As, in the

literature, no experimental or other theoretical results for finite temperature

thermodynamic properties of ZB-GeC are available; we could not compare our

results. However, presently calculated results will serve as a useful set of data for

further research in this field.

(3) Tin Carbide (SnC)

Like SiC and GeC, SnC is also a technologically important material [22].

Thermodynamic properties of ZB-SnC are not studied till date either theoretically or

experimentally. Motivated by this fact, we in the present study represent the study of

thermodynamic properties of SnC in zinc blende structure. In previous chapter, we

have reported the ground state and high pressure properties of this material. The

Poisson’s ratio for ZB-SnC is taken from Reference [6]. Figure 4.19 shows the

calculated equilibrium lattice constant at various temperatures. Figure 4.20 shows the

presently calculated thermal equation of state (eos) of ZB-SnC at 0 K, 500 K and

1000 K temperature. It is observed that the eos at different temperatures are controlled

by cold pressure. Figure 4.21 shows the presently calculated quasi harmonic phonon

dispersion relation in ZB-SnC at 0 K and 500 K temperature. At higher temperature,

the phonon frequencies in SnC become soft along all symmetry directions. Figure

4.22 shows the presently calculated quasi harmonic p-dos at various temperatures. At

high temperatures, the peak in the phonon frequencies shifts towards lower frequency

side, showing weak correlation among atoms. Figure 4.23 shows the presently

calculated temperature dependent Gibbs free energy function. It decreases with

temperature. Figure 4.24 shows the calculated isothermal and adiabatic bulk modulus

as a function of temperature. A small hump is observed at low temperatures. The

temperature coefficient of isothermal and adiabatic bulk modulus is –0.041 GPa/K

and -0.0229 GPa/K, respectively. Figures 25, 26 and 27 respectively show the

presently calculated Debye temperature, specific heats and thermodynamic Grüneisen

parameter at various temperatures. The temperature coefficient of Debye temperature

Chapter-4_________________________________________________________


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is -0.0245 at high temperatures. The comparative study of GeC and SnC indicates that

bulk modulus (which is related to elastic constants) of SnC decreases rapidly

compared to GeC and hence SiC. Further, the computed Debye temperature also

decreases sharply with temperature. It is observed that all the thermodynamic

properties calculated here show the characteristics features of temperature dependent

thermodynamic properties of a ceramic material. In the absence any experimental or

theoretical data for the thermodynamic properties of ZB-SnC, we could not compare

our data and hence, we could not put any concrete remark on the presently calculated

properties. It is observed in the present study that as we move from

SiC→GeC→SnC, elastic properties decrease with temperature, more rapidly. The

present results will serve as a useful set of data for further research in this field.

Figure 4.19. Calculated equilibrium lattice constants of ZB-SnC at

various temperatures.

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Figure 4.20. Thermal eos of ZB-SnC. At 0 K (full line), 500 K (dashed line) and

1000 K (dotted line).

Figure 4.21. Quasi harmonic pdc in ZB-SnC at various temperatures. At 0 K (full

line) and at 500 K (dashed line).

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133

Figure 4.22. Quasi harmonic p-dos of ZB-SnC at various temperatures. At 0 K (full

line) and at 500 K(dashed line).

Figure 4.23. Temperature dependent Gibbs free energy function for ZB-SnC.

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134

Figure 4.24. Isothermal (full line) and adiabatic (dashed line) bulk modulus of ZB-

SnC at various temperatures.

Figure 4.25. Temperature dependent Debye temperature of ZB-SnC.

Chapter-4_________________________________________________________


135

Figure 4.26. Specific heats of SnC at constant volume (full line) and at constant

pressure (dashed line).

Figure 4.27. Calculated thermodynamic Grüneisen parameter at various temperatures

for SnC.

Chapter-4_________________________________________________________


136

(4) Zirconium Carbide

Very few experimental and theoretical studies of thermodynamic properties of

cubic ZrC are available in literature [24,25]. Lawson et al [24] have studied the

thermodynamic properties of ZrC using neutron diffraction method. They have

reported thermal expansion and mean square displacement in B1-ZrC. Jun et al [25]

have studied few thermodynamic properties of ZrC using CASTEP code in

conjunction with quasi harmonic Debye model. In the present work, we report the

complete set of thermodynamic properties of B1-ZrC using plane wave

pseudopotential density functional theory [1] in conjunction with the quasi harmonic

Debye model [2]. The Poisson’s ratio of cubic ZrC is taken from Ref. [7]. Figure 4.28

shows the presently calculated equilibrium lattice constants at various temperatures

along with the experimental results of Lawson et al [24]. The presently calculated

value of equilibrium lattice constants is slightly overestimated as compared to the

experimental results. However, a good agreement is observed. Figure 4.29 shows the

presently calculated thermal eos at various temperatures. Intersection of p-V curve on

volume axis shows the volume thermal expansion and is the indication of proper

treatment of anharmonic effects. Also, one can infer from the nature of graphs for

different temperatures that eos are largely controlled by the cold pressure, which also

implies that thermal pressure is linearly increases with temperature. And its effect is

to harden the material and therefore results into the shift in p-V graph towards higher

volume side. Figure 4.30 shows the presently calculated quasi harmonic pdc of ZrC at

ambient condition and at 300 K temperature. It is observed that at finite temperatures,

the phonon frequencies along all symmetry directions become soft. Also, the rate of

change of phonon frequencies with temperature is not much high. This indicates that

the rate of change of elastic stiffness constants should not be high, as the phonon

frequencies along [100] and [110] planes are directly related to the elastic stiffness

constants. Figure 4.31 shows the presently calculated quasi harmonic p-dos at 0 K and

at 300 K temperature. Figure 4.32 shows the temperature dependent Gibbs free energy

function for ZrC. Figure 4.33 shows the calculated temperature dependent isothermal

and adiabatic bulk modulus. Figure 4.34 shows the calculated Debye temperature of

ZrC along with the experimental results [26]. As, we are using the classical approach

for inclusion of ionic motion at finite temperatures, we do not observe any quantum

mechanical effect present at low temperature. On the other hand, the experimental

values show deep in low temperature region and then increases. However, the

Chapter-4_________________________________________________________


137

experimental values are available up to limited temperature range, beyond which

Debye temperature should decrease. Figures 4.35 and 4.36 show respectively, the

specific heats and thermodynamic Grüneisen parameter at different temperatures.

Figure 4.36 also shows the experimental values [26] of thermodynamic Grüneisen

parameter of ZrC. At low temperatures, the difference between presently computed

values and the experimental values is observed. However, it is observed that at high

temperatures ( DT θ> ), presently calculated values and experimental results are nearly

same. This validates the present approach at high temperatures in case of ZrC. All the

properties calculated here, indicates that B1-ZrC is a mechanically hard material.

Figure 4.28. Calculated equilibrium lattice constant of ZrC along with the

experimental results [24].

Figure 4.29. Thermal eos of ZrC at various temperatures. At 0 K (full line), 500

K (dashed line) and at 1000 K (dotted line).

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138

Figure 4.30. Quasi harmonic pdc of ZrC at 0 K (full line) and at 300 K (dashed

line).

Figure 4.31. Quasi harmonic p-dos of ZrC at various temperatures. 0 K (full line) and

at 300 K (dashed line).

Chapter-4_________________________________________________________


139

Figure 4.32. Temperature dependent Gibbs function of ZrC.

Figure 4.33. Isothermal (full line) and adiabatic (dashed line) bulk modulus of ZrC at

various temperatures.

Chapter-4_________________________________________________________


140

Figure 4.34. Temperature dependent Debye temperature of ZrC along with

experimental results (dots) [26].

Figure 4.35. Specific heat of ZrC at constant volume (full line) and at constant


Chapter-4_________________________________________________________


141

Figure 4.36. Thermodynamic Grüneisen parameter of ZrC as a function of

temperature along with experimental results (dots) [26].

Fluorides

(1) Lithium Fluoride

Among all alkali halides and ionic crystals, LiF have found much attraction

due to its interesting physical properties. It is a material with low absorption, low

scattering factor for sample measurement through Diamond Anvil Cell (DAC). As a

result, it has found application as pressure transmitting medium as well as pressure

sensor. Structural stability of LiF up to about mega bar pressure also makes it an

important material [27]. Previously, experimental and theoretical work on LiF is

performed mostly at ambient conditions. Studies on thermophysical properties of LiF

are reported in limited range of temperature and pressure. Liu et al [27] have studied

thermophysical properties up to 37 GPa pressure and ~1000 K temperature. Most

recently, Smirnov [28] has studied various ground state properties of LiF using full

potential linear muffin tin orbital method as implemented in LMTART code with few

modifications. Further, he has studied pressure dependence of elastic constants and

structural stability under pressure. In his work, thermal contribution at finite

temperature was taken into account using modified Debye model. In the present work,

Chapter-4_________________________________________________________


142

we describe ground state and finite temperature/pressure thermophysical properties of

rock salt structured (B1) LiF using plane wave pseudopotential density functional

theory as implemented in Quantum ESPRESSO package [1] in conjunction with quasi

harmonic Debye model [2]. The value of Poisson’s ratio of LiF is taken from Ref. [8].

Figure 4.37 shows the presently calculated relative equilibrium lattice constant as a

function of temperature along with the experimental results [29]. A good agreement is

observed at low temperatures. Further, the trend of variation of lattice constants in

presently calculated values and experimental values is also same. Experimental values

are available only up to room temperature. The maximum deviation of presently

calculated values is of the order of 0.3% from the experimental results. Figure 4.38

shows the presently calculated thermal eos of LiF at various temperature along with

the room temperature experimental results [27]. A good agreement with the presently

calculated value and the corresponding experimental results is found. The high

temperature thermal eos also shows the proper inclusion of anharmonic effects in the

present calculation. Figure 4.39 shows the presently calculated quasi harmonic pdc of

LiF at 0 K and at 300 K temperature. It is observed that at higher temperatures, the

effect is to reduce magnitude of phonon frequencies along all symmetry directions.

Figure 4.40 shows the quasi harmonic p-dos at 0 K and at 300 K temperature. Again,

at high temperatures, the peak of p-dos shifts towards lower frequency side, showing

weak correlation among atoms at finite temperature. Figure 4.41 shows the presently

calculated Gibbs free energy function at various temperatures. Figure 4.42 shows the

calculated isothermal and adiabatic bulk modulus along with the experimental values

of isothermal bulk modulus [30]. The trend of variation of isothermal bulk modulus is

almost same. The presently calculated values of temperature coefficient of isothermal

bulk modulus is -0.033 GPa/K, while, the experimental value is -0.0218 GPa/K.

Figure 4.43 shows the presently calculated Debye temperature at various temperatures

along with the experimental data [30]. Again, trend of variation of Debye temperature

in the presently calculated values and the experimental one is found to be same. The

temperature coefficient of Debye temperature in the present work is -0.13, while the

experimental value is -0.15. Figure 4.44 shows the presently calculated specific heats

along with the experimental data of room temperature specific heat at constant

pressure [31]. A good agreement is found. The constant pressure specific heat

increases beyond the Dulong-Peltit limit at high temperature, showing the role of

anharmonicity at high temperatures. Figure 4.45 shows the presently calculated values

Chapter-4_________________________________________________________


143

of thermodynamic Grüneisen parameter at various temperatures along with the

experimental results [32]. Unlike experimental results, presently computed

thermodynamic Grüneisen parameter increases with temperature. On the other hand,

experimental values show first increase, then decease and then increasing values.

Such difference requires further study in this field. All the finite temperature

thermodynamic properties calculated here, shows good agreement with the

corresponding experimental data available in literature.

Figure 4.37. Calculated equilibrium lattice constants of LiF at various

temperatures.

Figure 4.38. Thermal eos of B1-LiF. At 300 K (full line), 500 K (dashed line) and at

1000 K (dotted line). Symbols are the experimental results at room temperature [27].

Chapter-4_________________________________________________________


144

Figure 4.39. Quasi harmonic pdc in B1-LiF at various temperatures. At 0 K (full line)

and at 300 K (dashed line).

Figure 4.40. Quasi harmonic p-dos at various temperatures. At 0 K (full line) and at

300 K(dashed line).

Chapter-4_________________________________________________________


145

Figure 4.41. Temperature dependent Gibbs free energy function of LiF.

Figure 4.42. Isothermal (full line) and adiabatic (dashed line) bulk modulus of B1-

LiF at various temperatures. Symbols are the experimental values of isothermal bulk

modulus [30].

Chapter-4_________________________________________________________


146

Figure 4.43. Temperature dependent Debye temperature of B1-LiF along with the

experimental results (dots with line) [30].

Figure 4.44. Specific heats of LiF at constant volume (full line) and at constant

pressure (dashed line) along with experimental constant pressure specific heat (dot)

[31].

Chapter-4_________________________________________________________


147

Figure 4.45. Calculated thermodynamic Grüneisen parameter of LiF at various

temperatures along with experimental results (dots) [32].

(2) Sodium Fluoride

Sodium fluoride is also a technologically important material having found

application in the field of high pressure physics [27]. It can be used as a pressure

transmitting medium in high pressure experiments. The study of thermophysical

properties of NaF at very low pressure is reported by Yagi et al [33] and Sato-

Sorensen [34]. Most recently Liu et al [27] have reported thermal expansion of NaF.

In the present work, we have reported the finite temperature thermophysical

properties of NaF. The Poisson’s ratio of NaF is taken from Ref. [9]. Figure 4.46

shows the relative lattice constants of NaF at various temperatures. Figure 4.47 shows

the presently calculated thermal eos of NaF at 300 K, 500 K and 1000 K along with

the room temperature experimental results [27]. A very nice agreement of the

presently computed room temperature thermal eos is found with the experimental

results. The calculated thermal eos of NaF shows that the cold pressure entirely

controls the high temperature thermal eos and is a right indication of inclusion of

anharmonic effects. Figure 4.48 shows the quasi harmonic pdc of NaF. It is observed

that the effect of the temperature is to reduce the phonon frequency along all principal

symmetry directions. Figure 4.49 shows the presently calculated quasi harmonic p-dos

Chapter-4_________________________________________________________


148

at various temperatures. The calculated p-dos at finite temperature show the effect of

weak correlation among atoms. Figure 4.50 shows the presently calculated Gibbs free

energy function at various temperatures. Figure 4.51 shows the calculated isothermal

and adiabatic bulk modulus at various temperatures. Figure 4.52 shows the presently

calculated temperature dependent Debye temperature. No experimental results are

found for comparison. Figures 4.53 and 4.54 shows the computed specific heats and

Grüneisen parameter at various temperatures along with experimental results [31],

respectively. In case of specific heats, data for only room temperature specific heat is

available. Experimental result is slightly higher. Interestingly, thermodynamic

Grüneisen parameter remains constant in the temperature range studied here. The

presently calculated value at room temperature is in close agreement with

experimental and other theoretical values. Present findings are useful set of data from

many points of view. Firstly, no such study of full set of thermodynamic properties of

NaF is reported. On the other hand, looking to the applications of NaF in the extreme

environments, presently computed results may be useful to study structure, elasticity

and vibrational response of NaF at finite temperatures and/or pressures.

Figure 4.46. Calculated equilibrium lattice constants of B1-NaF at various

temperatures.

Chapter-4_________________________________________________________


149

Figure 4.47. Thermal eos of B1-LiF. 0 K (full line), 500 K (dashed line) and at 1000

K (dotted line). Dots are the experimental results at 0 K [27].

Figure 4.48. Quasi harmonic pdc in B1-NaF at various temperatures. At 0 K (full


Chapter-4_________________________________________________________


150

Figure 4.49. Quasi harmonic p-dos of NaF at various temperatures. At 0 K (full line)

and at 300 K(dashed line).

Figure 4.50. Temperature dependent Gibbs free energy function of NaF.

Chapter-4_________________________________________________________


151


NaF at various temperatures.

Figure 4.52. Temperature dependent Debye temperature of B1-NaF.

Chapter-4_________________________________________________________


152

Figure 4.53. Specific heats of NaF at constant volume (full line) and at constant

pressure (dashed line) along with experimental results [31].

Figure 4.54. Calculated thermodynamic Grüneisen parameter of NaF at various

temperatures along with experimental (filled circle) and theoretical

results (open circle) [31].

Chapter-4_________________________________________________________


153

Oxides

(1) Calcium Oxide (CaO)

Alkaline earth oxide forms a technologically important class of materials.

They have found large number of applications. CaO is a major constituent of lower

mantle and its thermo-elasticity is important for the understanding of processes

including brittle failure, flexure and propagation of elastic waves [35]. Very few

theoretical studies of finite temperature thermophysical properties of CaO are

reported. Most of these studies include the evaluation of few thermophysical

properties using one of the classical models [36, 37]. In the present work, we have

calculated thermodynamic properties of CaO using density functional theory [1] in

conjunction with the quasi harmonic Debye model [2]. The value of Poisson’s ratio of

CaO is taken from Ref. [10]. Figure 4.55 shows the presently calculated values of

relative equilibrium lattice constants as a function of temperature along with the

experimental results [38]. The authors of Ref. [38] have reported the relative lattice

constants with respect to room temperature. We have reported the relative lattice

constants with respect to ambient temperature. However, good agreement is observed

between the presently calculated values and experimental one. Figure 4.56 shows the

presently calculated thermal eos of CaO at various temperatures. Figure 4.57 shows

the quasi harmonic pdc of CaO. It is observed that at finite temperatures, the phonon

frequencies along all principal symmetry directions decreases. Figure 4.58 shows the

calculated quasi harmonic p-dos of CaO. In the calculated p-dos of CaO, the peak of

p-dos shifts towards lower frequency side. The height of p-dos is not found to be

affected much at finite temperatures. Figure 4.59 shows the presently calculated

Gibbs free energy function at various temperatures. It decreases with temperature as

expected. Figure 4.60 shows the calculated temperature dependent isothermal and

adiabatic bulk modulus along with the experimental results [38] of adiabatic bulk

modulus. The trend of variation of bulk is almost same. Both of them decrease with

temperature. The temperature coefficient of presently computed adiabatic bulk

modulus is -0.0163 GPa/K and for experimental results it is -0.02 GPa/K. Figure 4.61

shows the calculated temperature dependent Debye temperature. The Debye

temperature decrease with temperature and as we are using the classical approach, we

do not observe any quantum mechanical effect at low temperature. Figure 4.62 shows

the presently calculated specific heats at different temperatures. The specific heat at

constant pressure increases beyond the Dulong-Peltit limit, which shows the dominant

Chapter-4_________________________________________________________


154

role of anharmonic effect at high temperatures. Figure 4.63 shows the temperature

dependent thermodynamic Grüneisen parameter.

No experimental data for the sake of comparison is found for the finite

temperature thermophysical properties except equilibrium lattice constants and bulk

modulus. Quasi harmonic pdc and p-dos of CaO are calculated for the first time. The

properties calculated here form a complete set of thermodynamic properties of rock

salt structured CaO, which will be helpful to understand the dynamics behind the

finite temperature thermophysical properties of B1-CaO.

Figure 4.55. Calculated equilibrium lattice constants of CaO at various

temperatures along with the experimental results [38].

Figure 4.56. Thermal eos of B1-CaO. At 0 K (full line), 500 K (dashed line) and 1000

K (dotted line).

Chapter-4_________________________________________________________


155

Figure 4.57. Quasi harmonic pdc in B1-CaO at various temperatures. At 0 K (full


Figure 4.58. Quasi harmonic p-dos at various temperatures. At 0 K (full line) and at

300 K(dashed line).

Chapter-4_________________________________________________________


156

Figure 4.59. Temperature dependent Gibbs free energy function of CaO.

Figure 4.60. Isothermal (dashed line) and adiabatic (full line) bulk modulus of B1

CaO at various temperatures along with experimental results [38].

Chapter-4_________________________________________________________


157

Figure 4.61. Temperature dependent Debye temperature of B1-CaO.

Figure 4.62. Specific heats of CaO at constant volume (full line) and at constant


Chapter-4_________________________________________________________


158

Figure 4.63. Calculated thermodynamic Grüneisen parameter of CaO at various

temperatures.

(2) Strontium Oxide (SrO)

Like Calcium oxide, Strontium oxide is also a alkaline earth based oxide,

having found many applications. Till date only the study of finite temperature specific

heat of SrO is studied experimentally [39]. Some theoretical studies are also reported

[40-42]. In the present work, we have adopted the Poisson’s ratio from the Ref. [11].

Figure 4.64 shows the calculated relative equilibrium lattice constants at various

temperatures. Figure 4.65 shows the thermal eos of SrO at various temperatures.

Figure 4.66 shows the quasi harmonic pdc of SrO at ambient temperature and at 500

K. Like, all other materials studied here, in SrO also, the phonon frequencies

decreases at high temperatures. This feature indicates that SrO at finite temperature

becomes mechanically soft. Figure 4.67 shows the calculated quasi harmonic p-dos of

SrO. The peak of p-dos shifts towards lower frequency side. The height of p-dos is

not found to be affected much at finite temperatures. Figure 4.68 shows the presently

calculated Gibbs free energy function at various temperatures. It decreases with

temperature. Figure 4.69 shows the calculated temperature dependent isothermal and

adiabatic bulk modulus of SrO. Figure 4.70 shows the calculated temperature

dependent Debye temperature. The Debye temperature decrease with temperature and

Chapter-4_________________________________________________________


159

as we are using the classical approach, we do not observe any quantum mechanical

effect at low temperature. The calculated Debye temperature decrease with

temperature, which is expected as the phonon frequencies also decreases and the p-

dos at high temperature moves towards lower frequency side. Figure 4.71 shows the

presently calculated specific heats at different temperatures along with the

experimental results [39]. A good agreement is observed. The specific heat at constant

pressure increases beyond the Dulong-Peltit limit, which shows the dominant role of

anharmonic effect at high temperatures. Also, it overestimates the experimental

results at high temperatures. Figure 4.63 shows the temperature dependent

thermodynamic Grüneisen parameter.

Figure 4.64. Calculated equilibrium lattice constants of SrO at various

temperatures.

Chapter-4_________________________________________________________


160

Figure 4.65. Thermal eos of B1-SrO. At 0 K (full line), 500 K (dashed line) and 1000

K (dotted line).

Figure 4.66. Quasi harmonic pdc in B1-SrO at various temperatures. At 0 K (full


Chapter-4_________________________________________________________


161

Figure 4.67. Quasi harmonic p-dos of SrO at various temperatures. At 0 K (full line)

and at 500 K(dashed line).

Figure 4.68. Temperature dependent Gibbs free energy function of SrO.

Chapter-4_________________________________________________________


162


SrO at various temperatures.

Figure 4.70. Temperature dependent Debye temperature of B1-SrO.

Chapter-4_________________________________________________________


163

Figure 4.71. Specific heats of SrO at constant volume (full line) and at constant

pressure (dashed line) along with the experimental results [39].

Figure 4.72. Calculated thermodynamic Grüneisen parameter of SrO at various

temperatures.

Chapter-4_________________________________________________________


164

(3) Barium Oxide (BaO)

JANAF table [42] have noted that the data of thermodynamic properties of

alkaline earth metal oxide like BaO are suspect since the results of the calorimetric

studies, on which their evaluation is based, are ambiguous [39]. To the best of our

knowledge, no theoretical study of thermodynamic properties of BaO is performed so

far. So, we in the present work intend to report the thermodynamic properties of B1-

BaO. The value of Poisson’s ratio is taken from Ref. [12]. Figure 4.73 shows the

calculated value of equilibrium relative lattice constants at various temperatures. No

experimental data of thermal expansion in BaO is available. Figure 4.74 shows the

calculated thermal eos of BaO at various temperatures. p-V graph clearly suggest that

anharmonic effects are properly included in the present work. Figure 4.75 shows the

Gibbs free energy function as a function of temperature. It decreases with

temperature. Figure 4.76 shows the isothermal and adiabatic bulk modulus of BaO at

various temperatures. It is observed that bulk modulus decrease almost linearly with

temperature. In Figure 4.77, we have shown the temperature dependent Debye

temperature. It decreases with temperature. As we are using the classical approach to

include the ionic motion at finite temperatures, we do not observe any deep in the

temperature dependent Debye temperature. Figure 4.78 shows the temperature

dependent constant volume and constant pressure heat capacities along with the

experimental values of constant pressure specific heat [39]. It is observed that the

constant pressure specific heat in the present study is slightly overestimated compared

to the experimental data. Figure 4.79 shows the presently computed thermodynamic

Grüneisen parameter as a function of temperature. Except specific heat at constant

pressure, no experimental data are available in literature for comparison. The reported

thermal expansion, thermal eos, variation of Gibbs function, bulk modulli, Debye

temperature show proper inclusion of thermal properties in the present calculation.

Confirmation to the present work is observed when comparison is made of presently

computed specific heat with the experimental results. Thus, it is to be observed that

presently computed values form a reliable set of data finite temperature

thermophysical properties of BaO, which shall work as an important guideline for

further research in this field.

Chapter-4_________________________________________________________


165

Figure 4.73. Calculated equilibrium lattice constants of BaO at various

temperatures.

Figure 4.74. Thermal eos of B1-BaO. At 0 K (full line), 500 K (dashed line) and


Chapter-4_________________________________________________________


166

Figure 4.75. Temperature dependent Gibbs free energy function of BaO.


BaO at various temperatures.

Chapter-4_________________________________________________________


167

Figure 4.77. Temperature dependent Debye temperature of B1-BaO.

Figure 4.78. Specific heats of BaO at constant volume (full line) and at constant

pressure (dashed line) with the experimental results of constants volume specific

heat [39].

Chapter-4_________________________________________________________


168

Figure 4.79. Calculated thermodynamic Grüneisen parameter of BaO at various

temperatures.

(4) Cadmium Oxide (CdO)

Cadmium oxide is a technologically important material, having found large

number of applications in the field of production of solar cells, liquid crystal displays,

electrochromic devices etc [44]. The value of Poisson’s ratio is taken from Ref. [13].

Figure 4.80 shows the calculated equilibrium lattice constants at various temperatures.

Figure 4.81 shows the calculated thermal eos of CdO at various temperatures. Again,

p-V graph clearly suggest that anharmonic effects are properly included in the present

work. Figure 4.82 shows the Gibbs free energy function as a function of temperature.

It decreases with temperature. Figure 4.83 shows the isothermal and adiabatic bulk

modulus of CdO at various temperatures. In figure 4.84, we have shown the

temperature dependent Debye temperature. It decreases with temperature. As we are

using the classical approach to include the ionic motion at finite temperatures, we do

not observe any deep in the temperature dependent Debye temperature. Figure 4.85

represents the pressure dependent Debye temperature of CdO at 0 K temperatures

along with the corresponding theoretical results of Li et al [45]. It is observed that

presently computed Debye temperature is overestimated as compared to results of Li

et al [44]. The possible reason behind such difference may be attributed to different

Chapter-4_________________________________________________________


169

theoretical approaches used. The presently computed Debye temperatures at various

pressures is fitted to 4th order polynomial and is represented by following equation,

458.09 8.4186 0.1118- 0.0016 10- )( 234-5 +++= PPPPPDθ and the results of Li et al [45]

are given by,

413.95 4.4115 0.1114- 0.0017 10- )( 234-5 +++= PPPPPDθ .

The difference between the zero pressures Debye temperature may also be attributed

to type of exchange-correlation functional used. In our work, we have used the local

spin density approximation (LSDA), while, Li et al [45] have used the generalized

gradient approximation (GGA). Figure 4.86 shows the temperature dependent

constant volume and constant pressure heat capacities. Figure 4.87 shows the

presently computed thermodynamic Grüneisen parameter as a function of

temperature. No experimental data are available in literature for comparison.

Presently calculated all thermodynamic properties show all features of proper

inclusion of thermal properties in the present calculation and will serve as a reliable

set of data of thermodynamic properties of CdO.

.

Figure 4.80. Calculated equilibrium lattice constants of CdO at various

temperatures.

Chapter-4_________________________________________________________


170

Figure 4.81. Thermal eos of B1-CdO. At 0 K (full line), 500 K (dashed line) and


Figure 4.82. Temperature dependent Gibbs free energy function of CdO.

Chapter-4_________________________________________________________


171


CdO at various temperatures.

Figure 4.84. Temperature dependent Debye temperature of B1-CdO.

Chapter-4_________________________________________________________


172

Figure 4.85. Pressure dependent Debye temperature of CdO at 0 K.

Figure 4.86. Specific heats of CdO at constant volume (full line) and at constant


Chapter-4_________________________________________________________


173

Figure 4.87. Calculated thermodynamic Grüneisen parameter of CdO at various

temperatures.

4.4 Conclusions

In conclusion, in the present chapter 4, we have reported finite temperature

thermophysical properties of technologically important materials using plane wave

pseudopotential density functional theory as implemented in Quantum ESPRESSO

package [1] in conjunction with the quasi harmonic Debye model [2]. Following

conclusions emerge out of the work presented in this chapter.

1) Calculated thermophysical properties of all materials agree well with the

available experimental and other theoretical findings.

2) Calculated equilibrium relative lattice constants as a function of temperature

for 3C-SiC, B1-ZrC and B1-LiF agree well with the corresponding

experimental and theoretical findings. In case of SiC, computed equilibrium

lattice constant at various temperatures show a maximum of 3% deviation

compared to molecular dynamics results. On the other hand, computed,

relative equilibrium lattice constant of ZrC show a maximum of 0.3%

deviation compared to experimental results. In a similar way, in case of LiF

also, the presently calculated relative lattice constants vary by 0.3 % only from

Chapter-4_________________________________________________________


174

experimental results. Calculation of equilibrium lattice constant is a primary

test of inclusion of proper ionic motion at finite temperature, which in our

study is properly reproduced. Thus, the good agreement validates the present

calculation of thermodynamic properties. For all other materials, no

experimental or theoretical data for thermal expansion is available. Thus, we

could not compare our results. However, as they are calculated using two

strong theoretical approaches, presently calculated values will serve as a

reliable set of data for further research.

3) Further, computed thermal eos is also found in good agreement with the

available experimental results. The computation of thermal eos in all cases

reveal that the cold pressure entirely controls the finite temperature thermal

eos. The graph of eos also indicate the proper inclusion of anharmonic effects

in the present calculations.

4) We have calculated the Gibbs free energy function at various temperatures. it

decreases with temperature. As it is observed in all cases, that equilibrium

lattice constants increases with temperature (and no negative thermal

expansion is observed in any material), for the materials to reach equilibrium

at finite temperatures, corresponding free energies should decrease.

5) Calculated values of isothermal and adiabatic bulk modulus agrees well in

case of 3C-SiC and B1-LiF. The trend of variation of bulk modulus with

temperature is same as that of experimental results. Further, the temperature

coefficient of bulk modulus is also nearly same. For other materials, no

experimental data are available.

6) Calculated Debye temperature at various temperatures, agree well in case of

B1-LiF. The temperature coefficient of Debye temperature is also nearly same.

For other materials, no experimental data are available.

7) Calculated values of specific heats (at constant pressure and at constant

volume) agree well with the corresponding available experimental results, in

case of 3C-SiC, B1-LiF, B1-SrO and B1-BaO. For other materials, no

experimental data are available for comparison.

8) The calculated values of thermodynamic Grüneisen parameter at various

temperatures (expect for NaF) indicate that it increase parabolically with

temperature. Interestingly, in case of NaF, thermodynamic Grüneisen

parameter remains constant with respect to temperature.

Chapter-4_________________________________________________________


175

9) It is observed that although Si, Ge and Sn belong to same family, the

thermodynamic properties of their carbides SiC, GeC and SnC are much

different. In particular, elastic properties of SiC is affected less by temperature

as compared to GeC and SnC. It is also observed that elastic properties of ZrC

are also affected less with temperatures. Thus, among all four carbides, SiC

and ZrC are found to retain their elastic properties at high temperatures.

10) In fluorides, LiF is found to retain is elasticity up to high temperatures. In case

of LiF, the temperature coefficient of isothermal bulk modulus is -0.073

GPa/K and in case of NaF, it is -0.0306 GPa/K. thus, it is found that bulk

modulus (which is related with elastic constants C11 and C12) decreases at

high rate with temperature in case of NaF.

11) In case of CaO, the temperature coefficient of isothermal bulk modulus is -

0.0306 GPa/K, for SrO it is -0.0203 GPa/K, for BaO it is -0.0168 GPa/K and

for CdO, it is -0.057 GPa/K. thus, among all oxides, studied here, it is

observed that elastic constants of BaO should decrease at higher rate

compared to other three oxides, which indicate that BaO is much sensitive to

temperature as compared to CaO, SrO and CdO.

12) The good agreement with the presently computed values of various

thermodynamic properties of carbides, oxides and fluorides thus confirms the

present approach (i.e. DFT + quasi harmonic Debye model) in determining

various thermodynamic properties up to their normal melting.

Chapter-4_________________________________________________________


176

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