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The idea of equilibrium

The nature of defects in amorphous silicon has been a topic of much discussion since the first report on light induced changes in the photoconductivity and their reversal by annealing which was published by Staebler and Wronski [Staebler-1977apl]. We have come to understand a few things but as I write these lines there is still dispute on some the the underlying processes.

In the paper of Staebler, it was speculated that the loss in photoconductivty could be attributed to changes in the bonding environment. The actual breaking of bonds was somewhat later proposed by Pankove in 1980 [Pankove-1980apl]; these authors proposed that illumination would break weak Si-Si bonds, thus creating two dangling bonds. This simple model was contradicted by the observation that degradation is less severe in heavily reverse biased solar cells, and forward biased solar cells have been observed to degrade even in dark, leading to the conclusion that actually the recombination of charge carriers is responsible for the degradation [Staebler-1981].

A common aspect to the processes mentioned above is the creation of new defects. Smith and Wagner contributed the idea of defect formation as equilibrium process that takes place already during the deposition [Smith-1985prb]; taking up the idea that recombination processes create defects, the authors proposed that defects are formed already during growth by the recombination of thermally activted charge carriers. During the cool-down from the growth temperature, the equilibrium defect density N_D is frozen in at certain temperature like the position of molecules when glass is cooled from the melt.

Stutzmann related the equilibrium idea with a closer look at into the breaking of weak bonds [Stutzmann-1987pm]. Imagine two silicon dangling bonds that are not too far apart. The hybridized sp³ orbitals can go into a weak bonding state which splits the energy levels into a bonding and an anti-bonding state. Stutzmann identified this situation with the disordered bonds of the amorphous network. The valence band tail represents the low energy state of the weak covalent bond filled with two electrons, the conduction band tail corresponds to empty anti-bonding states at higher energy. The broken weak bond corresponds to states at intermediate energy in the band gap. This concept turned out to be so successful that during its further development the distinction between inherent defects and light induced defects got somewhat neglected. After outlinig some basic aspects of the equilibrium models in the following sections, we will come back to a discussion of different defect types.

The dangling bond can take up three charge states; breaking a bond means the low lying covalent bonding states are lifted back to a non-bonding sp³ orbital which is neutral when occupied with a sinlge electron. This defect is therefore called D⁰. We can of couse put a second electron into this state, but the energy level of the resulting D^- state is slightly higher because of repulsive forces. Likewise, stripping the electron out of this level yields a D⁺ defect. In the figure below they are denoted by distributions that extend over a certain range of energy.

The weak bond theory relates bonding and anti-bonding orbitals with the valence band and the conduction band, respectively. Weak bonds are thus identified with the tail states, broken bonds correspond to levels at mid-gap.

Let us stay with the neutral defect for the time being. Smith and Wagner argued that bond breaking at the cost of a certain amount of energy can be treated statistically. They calculated the number of possibilities we can choose N_D broken bonds out of a total of N₀ bonds [Smith-1987prl]. The number W of such arrangements is obviously given by the binomial. We can approximate the factorials by the Stirling formula and plug the result into a calculation of the entropy which is given by S=klnW. Next, we have to minimize the free energy F = N_DU - TS. Note that U is the energy required to break the weak bond. The minimum condition yields the equilibrium value for N_D: $N_D = N_0 e^{-\frac{U}{kT}}$ The model can nicely explain why the effects of light induced degradation can be recovered by annealing because upon cool-down, the frozen-in equilibrium defect density depends only on the cooling rate, but not on the sample history. Smith and Wagner argue that not all silicon bonds take part in the equilibrium process but only those of the valence band tail. Therefore, they use a variable defect creation energy U whose value represents the separation from all possible energies in the valence band to the given defect state at fixed energy in the gap. The defect density is obtained by integration over all possible values of U.

Equilibrium defect densites above the freeze-in temperature [Street-1989prb].

The diagram above shows data on on the equilibrium defect density at various temperatures by Street and Winer [Street-1989prb]; starting from a high annealing temperature, they cooled their samples slowly in order to maintain the thermal equilibrium between weak bonds and defects. At a desired freeze-in temperature they quenched the defect density of this particular temperature by rapid cooling. The diagram shows that the equilibrium defect densities indeed show the behaviour predicted earlier. The defect formation energy U is around 0.2 eV and does not change much for different samples.

In the treatment above both, the weak bond and the defect are assumed to occur at fixed energy. This is clearly at odds with the idea of a disordered material. Go ahead to see how we can fix this issue.

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Last updated August 27th, 2010