Generalized Activation Energy Spectrum Theory: A New Approach for Modeling Structural Relaxation in Amorphous Solids
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bimolecular recombination [5, 8]. Finally, we assume that the reaction cross-section is independent of defect activation energy. If we use the simple theory, we can write dN(Q) dt
= =
1 (
N(Q) D(Q)-) -,expkT)
( ((1)
where g, >, and v,, are defects/flux, ion flux, respectively. va, is the reaction constant. In the case of bimolecular recombination, it would be the product of the attempt frequency, lattice constant, and the reaction cross section. The first term on the right side of Eq. (1) is the defect generation term. It ensures that the actual defect population, N(Q), saturates at its maximum value, D(Q). A similar form for generation of defects has been used successfully before [9]. The generalized activation energy spectrum theory takes into account the possibility that a defect may recombine with defects with different activation energies. This is achieved by writing
dN(Q) dt
_
gý(
1 _ NVQ) (Q) exp--Q1-)
v N(Q) [exp
0 2 ) N(Ql)dQ']
(-s) k
J
N(Q')dQ' + (2)
where v 0N(Q)exp(-Q/kT)fN(Q')dQ' is the rate at which a defect with activation energy Q annihilates defect with different activation energies, and voN(Q)fexp(-Q'/kT)N(Q')dQ' is the rate at which a defect with activation energy Q is annihilated by other defects with different activation energies. The factor of 1/2 is needed because we are counting each defect recombination event twice. In both cases, we have g and vo, as fitting parameters to be determined. The underlying assumption of the interaction of defects with different activation energies is that the defects are of similar kinds (presumably point defects), and differ only in their activation energies. The activation energies are unlikely to be that of self-diffusion, since the activation energy for self-diffusion in a-Si is between 0.13 and 0.22 eV [10]. Most likely, the activation energies are that of some rate-limiting step to defect migration, possibly that of the transition states necessary for the defects to become mobile. Both Eq. (1) and Eq. (2) were solved by computer simulations such that N(Q, t+bt) = N(Q, t)+ (dN/dt)6t, where 6R is the timestep. For simulations, the measured value of D(Q) from Ref. [3] was extended to 0.6 eV, which was the low limit used for activation energy. The results of calculations of defect dynamics in a-Si using the simple theory will be hereafter referred to as the modified simple theory, since we no longer make the step function approximation. For Eq. (1), the results of simulations were compared with analytical solution, and found to agree very well, thus confirming the accuracy of simulation. EXPERIMENTAL CONDITIONS In all experiments, a-Si resistors defined as mesas from a-Si films were used. Irradiations were performed at room temperature, with energies chosen to confine all the damage within a-Si film. All anneals were performed in a high vacuum furnace for 15 min. A more detailed description of these experiments can be found in Ref. [11]. The quantity actually measured in experiments was the electrical conductivity of a-Si. However, based on ex
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