## Electronic And Optical Properties

In this section, we concentrate on the electronic and optical properties of nanocrystalline Si prepared by the PECVD method. Since particular interest has been paid to the electronic and optical properties of these films, the thermal and mechanical properties have not been discussed so far. As the

20 nm

Disorder (Region B)

deposition temperature is around 200 °C, the physical properties should be stable below this temperature. Si is classified into a "hard" material and thus nanocrystalline Si should also be expected as a mechanically tough material.

### 4.1. Effective Medium Approximation

We briefly introduce the effective medium approximation (EMA) before proceeding with the discussion. The EMA predicts the total network conductance om for composite materials in D dimensions:

where a is a random variable of conductivity. Assuming that a random mixture of particles of two different conductivities, for example, a volume fraction, C, has conductivity of a0, and the remainder has conductivity of ar, substantially less than a0, simple analytical expressions of dc conductivity and Hall mobility as a function of C have been derived (see the pioneering works by Kirkpatrik [10] and Cohen and Jort-ner [22]). EMA has been also extended to calculate the ac conductivity in which case a in Eq. (1) becomes a complex admittance (a* = a1 + ia2) [23]. As the dielectric constant, e* = e1 — ie2 = a2/m — iat/w, is closely related to a*, optical absorption coefficient a(a) can be calculated using a(a) = 4wa(w)/cn, where c is the light speed and n the refractive index [11].

4.2. Application of EMA

to the Experimental Data

Solid circles in Figure 2 show the dc conductivity at room temperature as a function of crystalline volume fraction, Xc, for a series of undoped ^c-Si:H films [3]. Note that theoretical prediction of such a dependence for complicated systems is a very hard task. Shimakawa [11] has applied the

EMA to the experimental data and the result of the EMA is shown by the solid line, where o1 = 3 x 10-9 (Xc = 0) and o0 = 3 x 10-2 S cm-1 (Xc = 1) were used in D = 3. The experimental data do not fit the solid line very closely, which may be attributed to the varying grain sizes. In other words, the system we treated cannot be an ideal case for EMA. Following the solid line, however, the percolation threshold appears at Xc = 0.33, which agrees very well with the computer simulation (Xc = 0.32) [9].

Solid circles in Figure 3 show the room temperature Hall mobility in undoped ^c-Si:H as a function of volume fraction of crystalline Si [24]. Shimakawa [11] has tried to apply the Cohen and Jortner approach (EMA) [22] and the solid line shows the calculated result with /x1 = 0.2 (Xc = 0) and /x0 = 2 cm2 V-1 s-1 (Xc = 1). Note, however, that /x1 = 0.2 cm2 V-1 s-1 used here is not the true Hall mobility of the amorphous state (Xc = 0), because the well-known anomaly of the Hall effect observed in hydrogenated amorphous silicon (a-Si:H) interferes with obtaining a proper value of Hall coefficient [25]. The fitting of the calculation to the experimental data is reasonably good and the percolation threshold is shown to exist at Xc = 0.33 [11]. The small Hall mobility (~2 cm2 V-1 s-1) even at Xc = 1.0 is about three orders of magnitude smaller than that for single-crystalline Si [26]. This can be attributed to the complicated structure of ^c-Si:H. As already shown in Figure 1, the grain boundaries (between region A) dominate the transport properties and hence the mobility of ^c-Si:H films is reduced significantly. This means that crystalline Si wears "grain boundary" and hence the mobility for a crystalline itself is still small.

Next, we argue the size effect of crystallite on the Hall mobility. In earlier works [3, 5, 26], the Hall mobility with the same Xc has been suggested to increase linearly with crystallite size 8 and has been explained by the grain-boundary trapping model (electronic transport is dominated by thermionic emission over a potential barrier). In commercial viewpoint, this may be very hopeful, since higher-mobility materials can be easily realized by bigger size of m

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