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Figure 5. Comparison of the optical absorption coefficient of a-Si:H, nc-Si:H, and c-Si. The EMA results are given by solid circles. Reprinted with permission from [11], K. Shimakawa, J. Non-Cryst. Solids 266-269, 223 (2000). © 2000, Elsevier Science.

calculation. The calculated results agree very well with the experimental data, except at energy of around 1.7 eV. This suggests that a mean field constructed by mixture of amorphous and crystalline states dominates the optical absorption in nc-Si:H. The multiple light scattering seems to be not so important in this energy range.

### 4.3. Transport Mechanism

We understand in the preceding section that the EMA is useful to understand the macroscopic electronic and optical properties of composite materials, in particular the dependence of crystalline volume fraction of physical properties of nc-Si:H. In this section, we discuss the microscopic electronic transport mechanisms of nc-Si:H. For example, the mechanism of dc conductivity is very unclear.

Data shown in Figure 6 exhibit the temperature dependence of dc conductivity for n-type nc-Si:H films [6]. These cannot be fitted to the conventional relation adc = a0 exp(- AE/kT) with a single activation energy at higher

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Figure 6. Band diagram near the conduction band. A percolation threshold for electronic transport by thermionic emission process is represented by the dashed line. The Fermi level is given by the dash-dotted line.

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Figure 6. Band diagram near the conduction band. A percolation threshold for electronic transport by thermionic emission process is represented by the dashed line. The Fermi level is given by the dash-dotted line.

temperatures. Above 20 K, Finger and co-workers [6] have suggested that conduction occurs across potential barrier $ (=AE) between different crystalline regions and a barrier energy distribution leads to a curvature of ln o"dc versus 1/T. They further suggested that the transport takes place across higher barriers at higher temperatures and lower barriers at lower temperatures [8].

More quantitative discussion of the transport mechanism is given as follows by Shimakawa [14]. At relatively high temperatures, the dc conductivity due to thermionic emission of electrons over a potential barrier between crystalline grain clusters is expected to occur and can be given as adc = AT2 eW exp(—$/kT)/kT

where A is the Richardson constant, W the barrier width, and $ the barrier height measured from the Fermi level [30]. Figure 7 shows the band diagram representing this situation. The barrier height may be randomly distributed and hence the electronic transport can be dominated by the percolation level Epc (dashed line). $ defined here should therefore correspond to Epc — EF. Note that Eq. (2) is based on one-dimensional array of potential barriers and therefore, strictly speaking, Eq. (2) cannot be applied to the present materials, since we should consider three-dimensionally distributed potential barriers. This point will be discussed in a future publication.

Here the nonactivated behavior suggests that $ (=Epc — Ef) decreases with decreasing temperature, which can be attributed to the temperature-dependent ionization of donors (upward shift of the Fermi level). From this temperature-dependent conductivity, temperature variation of Epc — EF can be deduced. The temperature dependence of Epc — EF should include the temperature dependence of bandgap in the discussed temperature range, that is, yT, where y is the coefficient of temperature variation in bandgap and is taken to be 4 x 10—4 eV/K [31]. The net temperature variation of Epc — EF (=$(T) — yT) is given in Figure 8. From the pre-exponential term in Eq. (2), the barrier width W is also estimated. It is of interest that W = (2-4) x10—6 cm is estimated for all the samples. This value seems to be a reasonable value.

A linear temperature dependence of the value of Epc — Ef, at relatively high temperatures, suggests that the donors are essentially all ionized ("exhaustion" or "saturation" regime) [31]. In this regime, Epc — EF is given as follows:

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