BONDING CHARACTERISTICS
CRYSTAL DEFECTS
2. General thermodynamic relations
2.1. Formation energy, enthalpy, entropy and volume
The formation energy, E^f, of a defect is the energy difference between a crystal with a defect, and a perfect crystal, containing the same num-ber of atoms. The formation volume Vdef is the difference in volume
18
General thermodynamic relations 19
Table 2.1
Some characteristic energies of defects in elements, in units of kg Tfus
Vacancy formation energy 8-13 Activation energy of self-diffusion 15-19
Heatoffusiona 0.9-1.3
Surface energyb 5-10
Grain boundary energyb 3-6
a Per atom.
Per area of one atom (in a monolayer).
between the two crystals. Similarly, one can define the thermodynamic quantities H^u ^def, GW and Sdef• The usual thermodynamic relations for the enthalpy H, the Helmholtz energy F etc., are also valid for quantities like //def and Fdef, if they involve derivatives with respect to intensive variables, such as p and T. For instance, the formation volume can be written as
/ 3Gdef \
v —Ur) r - <21)
One has to be careful in the use of relations involving derivatives with respect to extensive variables, such as the volume V (Howard and Lidiard 1964, Levinson and Nabarro 1967).
2.2. Defect concentration in thermodynamic equilibrium
The defects present in a state of complete thermodynamic equilibrium must have low formation energies Z?def• This limits us to point defects and small aggregates of them. Let N be the number of lattice sites. The number of defects in equilibrium is
TVdef = N exp(Sdef,tot/*s) e x p ( - / /de f / ^ ) . (2.2) This expression assumes that iVdef <^C N and that interactions between
defects can be neglected. At ambient pressure, we can put p = 0 so that Hdef — Edef. The total entropy 5def,tot has two parts:
,tot — ^geom + Sdef- (2.3)
0.90
1000/T (K'1)
1.00 1.10
P (GPa)
Fig. 2.1. The quenched-in electrical resistance in gold plotted as \og(AR/Ro) ver-sus l/T (upper scale) and p (lower scale) yields straight lines with slopes from which the vacancy formation energy £Vac and formation volume VVac can be obtained.
Data from Huebener and Homan (1963).
ln(Zdef) *s a temperature-independent geometrical term, zdef being the number of configurations of a defect associated with a partic-ular site in the lattice, and with proper consideration of double counting.
We normally encounter it as a term wp{Sgeom/ kB) = £def- If there are several atoms per primitive cell, one may have to assign different values i/def, Sgeom a nd Sdef to different configurations or lattice sites.
2.3. Defect parameters from an Arrhenius plot
A physical quantity showing an exponential temperature dependence such as in eq. (2.2) is said to obey an Arrhenius law. Then //def (or £def) can be determined from a plot of ln(Nfef/N) versus X/ksT which yields a straight line with the slope — H^f (cf. fig. 2.1). One has
d(\/kBT) \dT ) \ dT )
+ //def = fldef(r). (2.4)
General thermodynamic relations 21
Here we have used the thermodynamic relation T(dS/dT)p = (dH/dT)p. Thus, a meaningful formation enthalpy Hdef(T), or forma-tion energy Edef(T), can be obtained even when these quantities are temperature dependent, so that the Arrhenius plot is curved.
From eqs. (2.1) and (2.4) we obtain the formation volume
v _ /9Gdef\ _ / 9 ( # d e f - TSfof)
\ dp ) T V dP
= _^p») r . (25)
A plot of ln(7Vdef/A0 versus p thus gives the formation volume Vdef (Levinson and Nabarro 1967).
Example: temperature-dependent Hdef(T) for vacancies. We will study the variation of Hdef(T) with T and make an expansion in T near the melting temperature Tfus:
O Hdef Hdef(T) = //def (Tfus) + (T — Jfus)
(2.6)
d T 7 P;T=T{US
(
9 5def \For harmonic lattice vibrations, and in the high temperature limit, we have (cf. eq. (9.21))
Sde{ = 3NkBln[6(0)/9de{(0)l (2.7)
where #def(0) and 6(0) are "entropy Debye temperatures" for the crys-tal with and without a vacancy (cf. Chapter 9, §4.5). From eq. (2.7), (dSdQf/dT)p = 0. To obtain a temperature-dependent Z/def, anharmonic effects must be included. If this is done within the quasiharmonic model, and with equal Griineisen parameters for the perfect and the defect state, we still obtain (dS^f/dT)p = 0. One has to go beyond such a simple description to get a temperature-dependent //def (Girifalco 1967, Levinson and Nabarro 1967).
Ch. 2. Crystal defects
2.4. Constant pressure and constant volume
Sometimes it is essential to distinguish between quantities considered at constant volume (subscript V) and at constant pressure (subscript p).
Experiments are usually performed at constant pressure (p « 0) but model calculations may be more conveniently carried out at constant volume. If (Vdef)/? is the increase in the specimen volume when a certain defect is introduced at constant pressure, and (/?def)v is the increase in pressure if a defect is added at constant volume, they are related by (KT
is the isothermal bulk modulus)
(Vdef)pKT = (/?def)y V, (2.8)
with similar relations for other thermodynamic properties of defects (Catlowetal. 1981).
3. Vacancies
Here we mainly have in mind thermally generated vacancies in elements and alloys. The formation energy £def = Ewac of a monovacancy is the energy difference between a perfect lattice with N occupied lattice sites and a similar lattice with N+l sites, one of which is void. The formation volume Vdef of a monovacancy is the difference in volume between these two crystals. The concentration of monovacancies in a monatomic solid is, in thermal equilibrium and at p = 0,
cvac = exp(SYac/kB) e x p ( - £v a c/ £Br ) , (2.9) where Svac is an entropy term related to changes in the vibrational
spec-trum of the lattice, eq. (9.28). Since there is only one configuration (orientation etc.) for a vacancy, we have Zdef = 1 and, hence, Sgeom = 0.
Under a pressure /?, EWSLC in eq. (2.9) is replaced by EWSiC+pVV3iC. The role of vacancy formation in the heat capacity of solids close to the melting point is considered in Chapter 11, §3.
The Landolt-Bornstein tables (Ullmaier 1991) give values of £Vac and SVac derived by various experimental techniques. While different sources typically agree in their values for EV3LC to within ~10%, the values for Svac a*"6 verY uncertain. It is not unusual that quoted SVSLC/kB for a certain element range from, e.g. 1 to 2. Chapter 19 (§7) deals
Vacancies 23 Table 2.2
Vacancy defect parameters for Cu and Al
Cu
with the estimate of Lvac and other defect energies from the melting temperature 7fus.
Example: vacancies in Cu and Al. Vacancies increase the length L of a specimen but do not affect the lattice parameter a, as measured by X-ray methods. Simmons and Balluffi (1960) used this fact to derive the vacancy formation energy of Al from the relation
f AL Aa 1
cVac = 3 . (2.10)
{ L a \
Here AL includes the effect of vacancies as well as ordinary thermal expansion, while Aa only includes the thermal expansion. This method has been refined by Hehenkamp et al. (1992), to get cvac, LVac and Svac
with higher accuracy than obtained in other, similar or different, exper-iments. Table 2.2 gives some results (Hehenkamp 1994). One should note that eq. (2.10) holds even when there are relaxations around the defects (Simmons and Balluffi 1960, Seeger 1973).
Example: vacancies in gold under pressure. If a specimen is quenched from a high temperature 7\ the residual resistance Ro + AR of the quenched sample has a part RQ caused by impurities and a part AR arising from the N^f quenched-in thermal defects (vacancies). Since AR is proportional to N^u a plot of ln^ef/AO can be replaced by a plot of ln(AR/Ro). Huebener and Homan (1963) measured AR for vacancies in gold at different temperatures and pressures. Figure 2.1 shows their results. From the slope of the straight lines one obtains //vac
Ch. 2. Crystal defects
= 0.98 eV and Vvac = 9.19 x KT30 m3 = 0.53£2a, where fta is the atomic volume. //Vac refers to p = 0.18 GPa. Then pVvac = 0.01 eV is negligible.
Example: non-stoichiometric carbides and nitrides. Several transition-metal carbides and nitrides crystallise in an NaCl-type lattice with a non-stoichiometric composition VCX etc., where typically x is M).8-0.9. Such vacancies on the non-metal sites do not represent thermal equilibrium, even at high temperatures. Instead, the composition of the ground state, i.e. the state of lowest energy at T = 0 K, has x < 1 as was shown in total-energy electron structure calculations for VCX (Ozolhjs and Haglund 1993).