Absorption coefficient modeling of microcrystalline silicon thin film using Maxwell-Garnett effective medium theory

Sheng-Hui Chen; Hsuan-Wen Wang; Ting-Wei Chang

doi:10.1364/OE.20.00A197

1. Introduction

The absorption coefficient (α) of thin-film material is a very important factor when it is applied to thin-film solar cells. One of the main reasons is that many material properties such as bandgap, structure disorder and defect density can be obtained quantitatively from the absorption spectrum [1–3]. In the case of silicon thin-film solar cell, not only can the spectral response be determined by the absorption feature of its intrinsic layer, the maximum short-circuit current density can be deduced via the integration of solar spectrum into the spectral response of the cell as well.

Microcrystalline silicon (μc-Si) thin films have been widely employed in silicon thin-film solar cells since the first announcement by Vetterl et al. in 2000 [4]. The absorption coefficient, however, varies with its crystalline volume fraction (X_C) and is sensitive to the film structure, causing surface or internal scattering [5]. The prediction of α for μc-Si thin film is, therefore, a key to the application of solar cells.

Effective medium theory (EMT) is useful to statistically explain the optical behavior of material. Originally, the Maxwell-Garnett EMT was proposed to simulate the effective refractive index and extinction coefficient of metal grains embedded in a dielectric material [6]. Recently, regardless of the optical matrix element, the EMT has been applied to successfully explain the optical constants of composite thin film with structures such as dielectric-dielectric mixture [7], and quantum dots thin film [8]. The low absorption of μc-Si thin film was understood by mixing the dielectric constant of amorphous silicon and crystalline silicon using effective medium approximation [9]. Krnakenhagen et al. [10] has also simulated the absorption coefficient of μc-Si thin film using a linear superposition of absorption curve of single crystal silicon (c-Si) and amorphous silicon (a-Si) with constant defect density. However, the linear superposition approach did not meet the experiment curve very well.

In this study, we model the absorption coefficient of microcrystalline silicon using the Maxwell-Garnett EMT. With the employment of a genetic algorithm and the introduction of scattering correction, our simulated profile of α, constructed from the Mott-Davis density of state (DOS) model of a-Si, fits well to the experiment results that are measured by the constant photocurrent method (CPM).

2. Model and theoretical basis

In this section, we briefly describe our approach and address its theoretical basis. Shown in Fig. 1 is the flow chart of the process employed in our model. The main feature of our approach is that the extinction coefficient (κ) of a-Si is constructed using the Mott-Davis DOS model, which allows for important parameters such as mobility gap, the slope of Urbach tail and defect distribution function to be retained. It deserves mentioning that the refractive index (n) and extinction coefficient (κ) of a-Si and c-Si must be acquired first in order to use the equation of EMT. Once they are available, the value of X_C is then substituted into the EMT formula to model the α curve of μc-Si. A genetic algorithm is subsequently applied to fit the modeled profiles to the experimental profiles through iterations. This step allows two key parameters, the pre-factor (N_d) of defect distribution function and scattering factor (M), that construct the α to be verified.

Fig. 1 Flow chart of the process.

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2.1 Absorption of amorphous silicon constructed by the DOS model

Vanecek et al. [11] proposed that the absorption profile of a-Si in different energy (E) can be obtained numerically by the following convolution

α (h υ) = \frac{K}{h υ} \int N_{i} (E) f (E) N_{f} (E + h υ) [1 - f (E + h υ)] \cdot d (E),

where N_i(E)f(E) represents the carrier density of which the excitation originates, N_f(E + hυ)[1-f(E + hυ)] represents the carrier density of the excitation ends, and hυ is the absorbed photon energy. If the carriers are transited from the valance band (E_V) to the conduction band (E_c), then N_i(E)f(E) is the valance band carrier density, and N_f(E + hυ)[1-f(E + hυ)] is the conduction band carrier density, where N_i, N_f and f(E) are the valance band DOS, conduction band DOS, and the Fermi-Dirac distribution function, respectively.

In this study, we use the Mott-Davis DOS model to derive the absorption spectrum of a-Si. An example of the DOS distribution, which utilizes the parameters [12] listed in Table 1 , is depicted in Fig. 2 . In Table 1, E_g stands for the mobility gap of a-Si. E_0v and E_0c are the slope of band tails. N_V and N_C are the effective DOS in the valance band and conduction band, respectively. E_d, E_a, and ΔE_cor are the position of the donor-like and accepter-like defect states distribution and their separation. The distribution function $N_{D} (E)$ of those two defect states are both the Gaussian type, which is expressed as:

N_{D} (E) = N_{d} \exp (\frac{{| E |}^{2}}{2 W^{2}}),

where

N_{d}

and

W

represent the pre-factor and full-with-half-maximum (FWHM) of the distribution function, respectively. The total defect density

N_{D}

can be therefore, acquired by the integration of

N_{D} (E)

within the mobility gap as written in Eq. (3) [13].

Table 1. Parameters of DOS Model

View Table

Fig. 2 Example of constructed DOS distributions of the a-Si with N_d 4 × 10¹⁷cm⁻³eV⁻¹.

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N_{D} = \int_{0}^{E_{g}} N_{D} (E) d E

From the DOS distribution displayed in Fig. 2 and Eq. (1), we are able to create an absorption spectrum of a-Si shown in Fig. 3 using the parameters listed in Table 1. Since the constructed absorption comprises the defect density in a-Si, this method provides us with flexibility for varying the $N_{d}$ to obtain a total defect density $N_{D}$ , instead of using a constant, during the iterations of fitting the α values of the μc-Si absorption curve in the genetic algorithm.

Fig. 3 Absorption spectrum of a-Si with mobility gap 1.8eV and N_d 4 × 10¹⁷ cm⁻³eV⁻¹.

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2.2 Modeling of absorption coefficient

Generally speaking, the silicon crystals of μc-Si thin films are randomly distributed in the amorphous tissue. The crystal sizes are usually smaller than 20nm. In the modeling of absorption coefficient we assume, macroscopically, the optical property of μc-Si thin film is equivalent to a composite material where small crystals are embedded in amorphous medium, and can be simulated by the Maxwell-Garnett EMT. Based on the Clausius-Mossotti relation the Maxwell-Garnett EMT was proposed to simulate the effective refractive index and extinction coefficient of metal grains embedded in a dielectric material [6]. Normally, when an electromagnetic (EM) wave propagates in an inhomogeneous material, the EM wave will be scattered. However if the size of the inhomogeneous material is less than the wavelength of the incident light, the EM wave cannot identify the each scattering center. To meet the requirement the inhomogeneous size and the wavelength have to under the condition shown as follows [14]:

\frac{4 π^{2} D^{2}}{15 λ^{2}} \frac{(ε + 4) (ε + 2)}{(2 ε + 3)} ≪ 1

In our case the size D, which refers to the grain size of silicon crystals, is about 10 nm. Therefore, the Maxwell-Garnett EMT can be applied in between visible to IR range.

The effective dielectric constant of μc-Si ${\tilde{ε}}_{μ c}^{M G}$ is expressed as:

{\tilde{ε}}_{μ c}^{M G} (E) = {\tilde{ε}}_{a} (E) \frac{{\tilde{ε}}_{C} (E) + 2 \cdot {\tilde{ε}}_{a} (E) + 2 \cdot X c \cdot [{\tilde{ε}}_{C} (E) - {\tilde{ε}}_{a} (E)]}{{\tilde{ε}}_{C} (E) + 2 \cdot {\tilde{ε}}_{a} (E) - X c \cdot [{\tilde{ε}}_{C} (E) - {\tilde{ε}}_{a} (E)]},

where

{\tilde{ε}}_{a}

and

{\tilde{ε}}_{C}

are the complex dielectric constant of a-Si and c-Si, respectively, and X_C is the crystalline volume fraction of the μc-Si thin film. The complex dielectric constant of a-Si and c-Si can be obtained from Eq. (6) and Eq. (7), where

n_{a - S i}

,

κ_{a - S i}

,

n_{C - S i}

,

κ_{C - S i}

are the refractive index (n) and extinction coefficient (κ) of a-Si and c-Si, respectively.

{\tilde{ε}}_{a} (E) = [n_{a - S i} {(E)}^{2} - κ_{a - S i} {(E)}^{2}] + i [2 n_{a - S i} (E) κ_{a - S i} (E)]

{\tilde{ε}}_{C} (E) = [n_{C - S i} {(E)}^{2} - κ_{C - S i} {(E)}^{2}] + i [2 n_{C - S i} (E) κ_{C - S i} (E)]

The extinction coefficient κ of a-Si is obtained through $α = \frac{4 π κ E}{h c}$ , where α is obtained by the DOS model, as mentioned previously, and h is the plank constant and c is the speed of light. The modeled α of μc-Si is derived as:

α_{μ c}^{M G} (E) = \frac{4 π E}{h c} {\frac{{[ε_{μ c}^{Re} {(E)}^{2} + ε_{μ c}^{Im} {(E)}^{2}]}^{\frac{1}{2}} - ε_{μ c}^{Re} (E)}{2}}^{\frac{1}{2}},

where

ε_{μ c}^{Re} (E)

and

ε_{μ c}^{Im} (E)

are the real part and imaginary part of the effective dielectric constant

{\tilde{ε}}_{μ c}^{M G} (E)

, respectively. Readers are referred to [15,16] for n, κ of c-Si and n of a-Si.

2.3 Scattering correction

Several research groups have discovered the contribution of α due to scattering while measuring the absorption of μc-Si film by the constant photocurrent method (CPM) or by photo-thermal deflection spectroscopy (PDS) [5,10,17]. The scattering comprises surface and bulk scattering due to surface roughness and internal grain boundaries of the film. Although the effect of surface scattering can be avoided by backside (i.e. substrate/air interface) illumination, the bulk scattering is inevitable. Poruba et al. indicated that the contribution of α from bulk scattering was a form of Rayleigh scattering, which was expressed as $α_{s c} = Μ \cdot E^{4}$ [17]. In our model, the scattering correction variable, M, has also been considered during the iterations in the genetic algorithm.

3. Fabrication and characterization of microcrystalline silicon thin films

Hydrogenated microcrystalline silicon (μc-Si:H) thin films were deposited on Corning Eagle 2000 glass substrate by plasma-enhanced chemical deposition (PECVD). Different silane dilution ratios (gas flow of H₂:SiH₄ = 10:1 to 15:1) were employed to control the crystalline volume fraction X_C of the μc-Si:H films to lie between 40% and 60%. The choice of this range was based on the findings reported by Vetterl et al., which indicated that the highest conversion efficiency of μc-Si:H solar cells was around 50% [4]. The thickness of all samples was about 1μm, measured by the stylus profiler, and Xc was calculated using the deconvolution of three peaks located at 480cm⁻¹,510cm⁻¹,520 cm⁻¹ in the Raman spectra. Absorption of the films was measured by the transmittance type CPM (T-CPM) with back side illumination to reduce the surface scattering effect, and the α was calculated from the measured results using Ritter-Weiser formula [18]. Eight μc-Si:H samples were fabricated. The X_C values of these samples were 41.2%, 42.5%, 53.4%, 56.7%, 57.4%, 58.5%, 60.8% and 63.4%. Since the X_C of samples were between 40~65%, and the grain sizes were around 10nm, the increase of the optical band gap of due to the quantum size effect [19] is negligible.

4. Results

Figure 4 shows the absorption spectra measured by T-CPM and the modeling curves fitted by the genetic algorithm for the eight samples mentioned in the previous section. As seen from this figure, the fitted results are in good agreement with the measured results for the values of X_C ranging between 41.2% and 63.4%. One can also find a discrepancy in α between the fitted and measured results as the X_C increase when the photon energy is below 0.9eV. Although the optical matrix elements are complicated for μc-Si:H, the grain boundary defects, which are also detrimental, can be recognized from the absorption coefficient below 0.9eV. Vanecek et al. proposed the grain boundary defects reflect on the α(0.8eV) [20], while Klein et al. showed the α(0.7eV) is possibly attributed to grain boundary defects [21]. Grain boundary defects are proportional to the absorption coefficient at 0.8eV and 0.7eV as proposed by Vanecek et al. and Klein et al., respectively. The grain boundary defect is not considered in our assumption and is not easy to be described perfectly by mathematical formula. It is possible that the volume ratio of grain boundaries is not “dense” enough to be observed by CPM for samples of low X_C, while the effect is more intense in high X_C. A better model that includes the contribution of grain boundary defects to the absorption profile is currently under investigation. Interestingly, we observed, during the fitting process, that the fitted defect density ( $N_{D}$ ) in a-Si and the scattering correction factor M, are correlated with the X_C. This observation is described below.

Fig. 4 Absorption spectra measured by CPM and the modeling curves fitted by the genetic algorithm for the eight samples. (a)X_C = 41.2%, (B) X_C = 42.5%, (c) X_C = 53.4%, (d) X_C = 56.7%, (e) X_C = 57.4%, (f) X_C = 58.5%, (g) X_C = 60.8%, (h) X_C = 63.4%.

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4.1 Correlation between amorphous defect density and crystalline volume fraction

Figure 5 shows the correlation of the fitted defect density ( $N_{D}$ ) in amorphous tissue with respect to X_C of the samples. It is clear that the defect density in the amorphous portion reduces as X_C increases. For the two samples with Xc = 41.2% and 42.5%, the defect density in the amorphous portion is over 10¹⁸ cm⁻³, and is about two orders of magnitude larger than a “device-quality” a-Si [13]. The defect density is about 10¹⁷ cm⁻³ or less as the X_C is over 50%. It is well known that a high hydrogen flow ratio is necessary to fabricate μc-Si thin film. In addition to the chemical annealing or selective etching effect brought about by the hydrogen [22], hydrogen atoms also play a role in dangling bonds passivation. Care should be taken that fitted defect density is obtained based on the Mott-Davis model in the amorphous tissue. Although the grain boundary defect cannot be quantified by the method proposed in this research, the discrepancy in α for the samples with X_C higher than 56.7% is obvious. Therefore, the total defect density, i.e. amorphous defects plus grain boundary defects could be possibly growth as X_C increase.

Fig. 5 Correlation between amorphous defects and crystalline volume fraction.

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4.2 Correlation between scattering factor and crystalline volume fraction

In Fig. 6 , we show the scattering correction factor as a function of X_C. Generally speaking, the M value increases with X_C, indicating an enhancement of internal bulk scattering. The increase in M is especially noticeable, which is more than a double when X_C is over 57.5%. Since bulk scattering can be expressed as a form of Rayleigh scattering, the increase in M reflects a pronounced absorption in the short wavelength region due to bulk scattering. Jun et al. has observed that the grain boundary scattering can enhance the absorption of μc-Si film, especially when the photon energy is larger than 1.7eV, and the enhancement is proportional to the volume ratio of grain boundary [23]. Since our μc-Si samples, with X_C around 40~60%, do not have wide variations in grain size, about 10~15nm, and all the films are controlled to be about 1μm thick, samples with higher X_C possess a larger grain boundary volume ratio and, therefore, a strong internal bulk scattering could be expected.

Fig. 6 Correlation between scattering factor and crystalline volume fraction.

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5. Conclusions

For developing a reliable prediction of the efficiency of a microcrystalline silicon thin film solar cell, construction of a model that is able to precisely describe the variation of absorption is of crucial importance. In this paper, we have proposed a model that utilizes the Maxwell-Garnett EMT to fit the absorption coefficient profile of microcrystalline silicon thin films measured by the constant photocurrent method. Satisfactory fitting results have been obtained from our studies when the crystalline volume fraction varies from 41.2% to 63.4%. Our results show that the consideration of bulk scattering and adjustable defect density with respect to the crystalline volume fraction is crucial to have a good fitting. Furthermore, our findings indicate that, as the crystal volume fraction increases, not only do the defects in amorphous silicon reduce, but the bulk scattering effect is gradually enhanced as well.

Acknowledgments

The authors would like to thank the National Science Council of Taiwan for the financial supports of this research under Contract Nos. 100-2120-M-008-002, 100-2627-E-008-001 and 100-2221-E-008-111.

References and links

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Absorption coefficient modeling of microcrystalline silicon thin film using Maxwell-Garnett effective medium theory

Abstract

1. Introduction

2. Model and theoretical basis

2.1 Absorption of amorphous silicon constructed by the DOS model

2.2 Modeling of absorption coefficient

2.3 Scattering correction

3. Fabrication and characterization of microcrystalline silicon thin films

4. Results

4.1 Correlation between amorphous defect density and crystalline volume fraction

4.2 Correlation between scattering factor and crystalline volume fraction

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (8)

Optics Express