International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 1
ISSN 2229-5518
On the formation energy of Schottky defects in
MgO
Dr. Dologlou Elizabeth
Abstract- The formation entropy for Schotkky vacancies in magnesium oxide has been estimated for low and high temperatures, by using a recent value of the formation enthalpy and with the aid of a thermodynamic model which interrelates point defect parameters with bulk properties. Our calculated values obtained from macroscopic data, are comparable with recent theoretical results derived from detailed, microscopic calculations.
Index Terms- Schottky formation entropy, formation enthalpy, MgO , bulk modulus, thermal expansion coefficient, defect, thermodynamic model.
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TUDIES on mass transport mechanisms in magnesium oxide have been extensively
deployed and a sizeable portion of literature on
experimental and theoretical results of diffusion processes and defect parameters is available [1], [2], [3], [4]. Among various theoretical approaches, studies of absolute diffusion rates showed in many cases a good agreement with experiments [5] although a scatter of orders of magnitude in between different experimental results has been observed [3], [6]. Recently, formation energies and entropies of Schottky defects in MgO by means of total energy and phonon calculations in supercell configurations have been estimated by Runevall and Sandberg [7].
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Solid State Section, Department of Physics, University of Athens,
Here we propose an alternative theormodynamic
model for the calculation of formation entropy of Schottky defects in MgO which interrelates bulk properties with defect parameters and we compare our findings to recent theoretical results.
Defect parameters can be directly estimated by means of a thermodynamic model, termed as cBΩ model, which interrelates the Gibbs energy gi with the isothermal bulk modulus B according to the formula [8], [9], [10], [11]
gi = ci BΩ (1)
where the superscript ‘i’ refers to the different process mechanism, (formation, migration and activation), Ω is the mean atomic volume per atom and ci is a dimensionless constant which can be considered as independent of temperature and pressure. This model has many successful applications such as in alkali and silver halides [12], in diamond [13], in fluorine superionic semiconductors [14], as well as in the case of seismic electric signals (SES) detected prior to large earthquakes [15], [16], [17], [18], [19], [20], [21].
The formation entropy sf is obtained by inserting
(1) into the relation sf = - (dgf /dT)P which leads to :
Panepistimiopolis, Zografou 157 84 Athens, Greece. E-mail:
sf = -cf Ω βB + B
(2)
T P
edologl@phys.uoa.gr
where β stands for the volume thermal expansion coefficient and T for the temperature. From the thermodynamic formula hf = gf - T (dgf /dT)P and (1) the formation enthalpy hf is given as:
IJSER © 2012
International Journal of Scientific & Engineering Research, Volume 3, Issue 11, November-2012 2
ISSN 2229-5518
hf = cf Ω B - TβB - T B
(3)
(9.85 0.05)k, which is more or less comparable with the
T P
Combining now (2) and (3) the formation entropy can be calculated through the formula:
calculated one 7.29k by Runevall and Sandberg [7]. Both
these calculated values of the formation entropy are high compared to that in non ionic materials, as expected [11].
Next, we proceed to the estimation of the
βB + B
formation entropy s
f2000
at T=2000K. Applying (2) for T
sf = - T P hf
B - TβB - T B
(4)
=300K and 2000K respectively, we get the ratio R as:
T P
Ω ( β B
+ dB )
sf 2000 2000 2000
2000 P
R = 2000 = dT
(5)
sf Ω
( β B
+ dB )
300
300 300 300
dT 300 P
Here, we test the validity of the above thermodynamic model in the case of MgO by means of recent thermodynamic data derived from another theoretical model.
We estimate the formation entropy for Schottky
vacancies first, at ambient conditions and then at temperature T =2000K and pressure P=0 GPa. We consider for the formation enthalpy hf the value 5.79 eV found by Runevall and Sandberg [7] for Schottky defects by means of total energy and phonon calculations in supercell configurations. The isothermal bulk modulus and its temperature derivative dB/dT at ambient conditions are taken as B300=162 GPa and (dB/dT)300= -0.03 GPa/K while at T=2000K and P=0 GPa as B2000=111 GPa and (dB/dT)2000= -
0.304 GPa/K respectively, (Table 6 of Sushil [22]). As far for the volume thermal expansion coefficient at T=300K and T=2000K we consider the values β300=3.17x10-5 K-1 and β2000=5.48x10-5 K-1 (Table 5 of of Sushil [22] ). Inserting all the appropriate values in (4) we find for the formation entropy for Schottky defects, at room conditions, sf300 =
where the value Ω2000/Ω300= V2000/V300=1.0801 at P=0 GPa is
taken from Table 2 of Sushil [22]. The derived formation
entropy has a somewhat of higher value, i.e., sf2000
=(10.41 0.05)k. Such a value is not unreasonable if we
consider that in typical ionic crystals like NaCl, KCl, the sf
values for a Schottky defect is close to 10k (see [11], pp
288).
The formation entropy for Schottky defects in MgO has been estimated, for low and high temperature, by using a recent value of the formation enthalpy and with the aid of a thermodynamic model which interrelates bulk properties with defect parameters. Although our results are derived following a simple macroscopic model, they are comparable with recent theoretical values obtained through more sophisticated microscopic calculations.
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