Chapter 3 Theory
3.5 Annealing
In the context of this Thesis, the term annealing refers to the permanent alteration of a defect as a result of thermal treatment. Such alterations occur in samples
where the defect concentrations differ from the equilibrium value at the annealing temperature. Defect concentrations can be reduced as a result of:
1. Defects becoming mobile and migrating towards a sink, i.e. a surface, dislo- cation, vacancy cluster or other defect, with which it then combines.
2. The defect-complex may dissociate to form two separate defects.
3. Defects may combine to form a new complex, e.g. V +V →V2.
Defects can also change from being EPR-‘observable’ (or EPR-‘active’) to
EPR-‘silent’ in different charge states. The effect of a charge transfer reaction on defect concentrations can be described in the same way as other annealing pro-
cesses. However, this process is reversible and will usually occur at temperatures lower than those where the defects involved will break-up. To avoid complication,
in this Thesis if the heat treatment being applied has a temporary effect caused by charge transfer it will be referred to as ‘heat or thermal treatment’. If the treatment induces irreversible changes, i.e. the formation or break-up of defects it will be referred to as‘annealing’. From now on, this chapter will discuss annealing behaviour, but identical descriptions are also suitable for charge transfer.
It is important to state that if the reaction requires two stages e.g. a break-up
of one defect and formation of another, then the activation energy of the reaction is controlled by the stage that requires the most energy.
Chapter 3. Theory
3.5.1
First order reaction
A first order reaction is the simplest, but there are assumptions concerning the
reaction that must hold in order to assume first order kinetics. These assumptions will be discussed by way of an example where the defect of interest is labelled X
in Equation 3-28.
[X] + [Y] → [XY] (3-28)
1. X must be involved in the reaction only once and not involved subsequently.
2. X and Y must be homogeneously distributed throughout the sample and separated by a distance that is smaller than the mean free path of each of
the interacting defects4.
3. The concentration of defect Y must be significantly larger than the con- centration of defect X, so that the concentration of Y remains effectively
constant.
If these conditions are fulfilled then the solution to a first order reaction is:
[X] = [X0]exp(−Kt) (3-29)
where K is the decay rate and can be expressed as:
K=υ0exp[ −
E kBT]
(3-30)
where υ0 and E are the characteristic attempt frequency and activation energy
respectively. kB is the Boltzman constant andT is the treatment temperature. K
has units of inverse time.
3.5.2
Second order reaction
If the initial concentration of bothX and Y defects (Equation 3-28) is equal then
the reaction can be described by second order kinetics. The solution to second
4In the case of charge transfer this could involve the conduction or valence band, in which
case the distance between defects being smaller than the mean free path is not a requirement. However, if the charge transfer involves only states within the band-gap then the impurities must be sufficiency close for direct electron transfer or quantum mechanical tunnelling between defects (defect band).
Chapter 3. Theory
order kinetics is given by:
[X] = 1
[X0]−1+Kt
(3-31)
If the concentrations ofX andY differ, but are of similar magnitudes the problem is more complex. Any solutions relevant to specific, more complex, reactions will
be discussed in the appropriate chapter of this Thesis.
3.5.3
Experimental methods
Experimentally annealing can be carried out in two ways.
1. Isochronal annealing - Each treatment is performed at increasing tempera- tures, each treatment lasting for the same period of time. From isochronal annealing data, if the decays are assumed to be exponential, then an esti-
mate of K and E can be obtained. The errors involved are typically large but the advantage of such experiments is that the ‘characteristic annealing temperature’, where reactions occur, can easily be determined.
2. Isothermal annealing - The temperature of each heat treatment is kept con- stant and the treatment is carried out over varying lengths of time producing
a decay curve at the treatment temperature. If the decay curves are expo- nential then at each temperature a decay rate, K can be obtained from
Equation 3-29 and from this an Arrhenius plot (lnK verses 1/T) can be constructed. From an Arrhenius plot a more accurate activation energy
and attempt frequency can be obtained from the gradient and y-intercept respectively.
References
References
[1] O. Stern and W. Gerlach, Zeitschrift fur Physik9, 353 (1922). [2] E. Zavoisky, J. Phys. USSR 9, 245 (1945).
[3] F. Bloch, Phys. Rev.70, 460 (1946). [4] A. M. Portis, Phys. Rev. 100, 1219 (1955).
[5] W. V. Smith, I. L. Gelles, and P. P. Sorokin, Phys. Rev. Lett. 2, 39 (1959). [6] W. V. Smith, P. P. Sorokin, I. L. Gelles, and G. J. Lasher, Phys. Rev. 115, 1546
(1959).
[7] C. A. J. Ammerlaan, Paramagnetic centers in diamond (Springer, 2001).
[8] C. D. Clark, R. W. Ditchburn, and H. B. Dyer, Proc. R. Soc. Lond. A. 237, 75 (1956).
[9] J. A. Weil, J. R. Bolton, and J. E. Wertz,Electron paramagnetic resonance(Wiley- Interscience, New York, 1994), 2nd ed.
[10] A. M. Stoneham, Theory of defects in solids (Clarendon Press, Oxford, 1975), Chapter 13.
[11] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Oxford University Press, 1970).
[12] S. S. M. Wong,Introductory nuclear physics (Wiley, 1998). [13] W. Jevons and A. G. Shenstone, Rep. Prog. Phys.5, 210 (1938).
[14] C. P. Slichter, Principles of magnetic resonance (Springer-Verlag, 1989).
[15] J. A. van Wyk, E. C. Reynhardt, G. L. High, and I. Kiflawi, J. Phys. D: Appl. Phys. 30, 1790 (1997).
[16] M. H. Levitt,Spin dynamics: Basics of nuclear magnetic resonance (Wiley, 2001). [17] M. Weger, Bell Syst. Tech. J. 39, 1013 (1960).
[18] J. W. Stoner, D. Szymanski, S. S. Eaton, R. W. Quine, G. A. Rinard, and G. R. Eaton, J. Magn. Reson. 170, 127 (2004).
[19] M. J. Shaw, P. R. Briddon, J. P. Goss, M. J. Rayson, A. Kerridge, A. H. Harker, and A. M. Stoneham, Phys. Rev. Lett.95, 105502 (2005).
[20] A. M. Edmonds, PhD. Thesis, University of Warwick (2008). [21] A. Cox, D. Phil. Thesis, University of Oxford (1993).