Lower resolutions

1060 1080

1100 1120

1140 wavenumber in cm⁻¹ 0.25

(a) Comparison of spectra with resolution 1 cm⁻¹, 8 cm⁻¹ and 1 cm⁻¹ with moving average with a win-dow size of 20 data points, corresponding to 4.8 cm⁻¹

0 1000 2000 3000 4000 5000 6000 7000

index

(b) Fast Fourier transformation of the spectra. The 8 cm⁻¹spectrum is stretched inxdirection, to be com-parable to the 1 cm⁻¹ spectrum

Figure 5.5: Mathematical treatments of the spectrum.

The thicknesses are below the limit of 366µm, obtained in the previous section for interference free measurements with a resolution of 4 cm⁻¹. Therefore we can expect to see interferences for a single wafer and hope to make them vanish for the stack. The spectrum of the two stacked wafers obtained with a resolution of 4 cm⁻¹ is shown in Figure 5.6.

As can bee seen, the result is not convincing. In addition to the interferences from the thickness, we see additional interference fringes, which correspond to an air gap of around 18µm. Hence, we tried to remove the air by evacuation of the samples in the sample holder to about 600 mbar below atmospheric pressure. After that treatment, the sample was placed into the FTIR spectroscope under atmospheric conditions and a spectrum was registered, which is also shown in Figure 5.6. One can see, that the period of the interference fringes increases, which corresponds to an air gap of 10µm after step one and 8µm after step two. Further evacuations even with lower pressures did not improve the spectra, so this gap must be due to mechanical roughness of the sample. Additionally, the interference fringe due to the sample thickness did not vanish either. We conclude, that this method cannot resolve our problem.

The other physical option, according to Eq. (5.10), is to reduce the reflectivity by means of an anti reflection coating, for example by a layer of silicon nitride. In this case, one has to keep in mind, that this may introduce additional absorption bands, which can overlay the desired peaks. This approach has not been investigated in this work.

5.3 Lower resolutions

As seen in the previous section, it is difficult to eliminate the interferences mathematically as well as physically. As they rise, if the thickness falls below a resolution dependent minimal thickness, the other option is, to decrease resolution. With a resolution of 8 cm⁻¹, wafers down to 183µm could be measured without interference fringes. However, this resolution does not agree with the protocols.

500 1000

1500 2000

2500 3000

3500

4000 wavenumber in cm

⁻¹

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

absorbance

without treatment 1st vacuum 2nd vacuum

Figure 5.6: Spectrum of two wafers, stacked together

Therefore, we carried out an experiment with eight 2000µm thick two-side polished p-type wafers of multicrystalline silicon. The thickness specifications of the oxygen protocol are not met, hence we cannot rely on these values. The other drawback is, that the resistivity is with 1.0 to 2.8 Ω cm below the specifications and the transmission at 1600 cm⁻¹ relative to the reference is only matched within 10 %. In the introduction of [sem04] is stated, that these resistivities cause the available energy to fall below satisfactory operation. However, the spectra obtained from our instrument show little noise in the regions of interest and even for the carbon- and oxygen-only spectra the signal to noise ratio is satisfactory, because we used samples with high carbon content. We recorded a spectrum at a resolution of 2 cm⁻¹ and a number of scans of 64 and a spectrum at 8 cm⁻¹ with the same number of scans for each of our eight samples.

The carbon concentration has been calculated using the baseline criterion. The results are shown in Figure 5.7a. A linear relationship between the concentrations obtained for different resolutions is obtained. The proportionality constant is 0.657. As lower resolutions tend to lower the peak height, a constant below 1 is to be expected. The maximum deviation from the data to the fit line is 9 %. The conversion coefficients for a resolution of 8 cm⁻¹ are then

CC,ppma= 2.50 ppma mm (5.12)

or C_C,dens = 1.25·10¹⁸atoms cm⁻². (5.13)

We also calculated the oxygen content from the same spectra according to the protocol, which is shown in Figure 5.7b. Here we also obtain a linear relationship, with a proportionality constant of 0.984. The maximum deviation from this fit is 2 %. Hence, the resolution change from 2 cm⁻¹ to 8 cm⁻¹changes the peak height a little bit. From this we obtain the conversion coefficients for a resolution of 8 cm⁻¹

C_O,ppm = 6.38 ppm cm (5.14)

and C_O,dens = 3.19·10¹⁷atoms cm⁻². (5.15)

Despite the variation of sample thickness, resistivity, carbon and oxygen concentration, the relation between concentrations at the two resolutions is very well described by a linear

5.4 Conclusions

Figure 5.7: Comparison of carbon and oxygen concentrations measured with a resolution of 2 cm⁻¹ and 8 cm⁻¹. Each point represents a sample. The fit function is linear.

relation. Hence we conclude, that the change in resolution only affects the peak height. This enables us to measure with lower resolution, to avoid interferences. However, as the absorption amplitude is lowered, the lower detection limit may have to be increased. This experiment cannot state anything about the absolute concentration values. The values might be biased due to increased free carrier absorption or in the oxygen case insufficient elimination of the silicon bands because of different thicknesses. To estimate or correct for these errors, a calibration with another technique would be required.

The last experiment in this section investigates the absorption coefficient as function of the resolution. Here, one sample of the previous and the reference were measured at resolutions from 2 cm⁻¹ to 16 cm⁻¹. As the protocol’s requirements are not met, we only consider the normalized concentration, defined as concentration per concentration at the highest resolution (1 cm⁻¹). In Figure 5.8a can be seen that the carbon absorbance varies heavily with the resolution and the oxygen absorbance varies much less. This may be due to the FWHM of the peaks, which is 6 cm⁻¹ for carbon and 32 cm⁻¹ for oxygen following the protocols. To correct for this effect, the normalized concentration is also plotted against the normalized resolution, which is defined in this case as resolution divided by FWHM. From this graphic, we may obtain, that for a normalized resolution below 0.2, there is no effect on the absorbance.

However, this is only one single measurement and therefore waits for further investigation.

5.4 Conclusions

The protocols for oxygen and carbon content are not well suited for common wafers, because they require large wafer thicknesses, compared to the currently used wafers. We thinned wafers, to see how the spectrum and the concentration changed. The main problem for thin wafers is, that there appear interference fringes if the thickness falls below a resolution dependent thickness. Then we tried to remove the interference fringes both mathematically as well as physically. The applied methods did not solve the problem. The anti-reflection coating may be investigated in the future. On the other hand, measurements can be carried out with lower

0 2 4 6 8 10 12 14 16 18 resolution in cm⁻¹

0.30.4 0.50.6 0.70.8 0.91.01.1

Normalized concentration

O_i C_s

(a)Concentration relative to the concentration at the best resolution

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized resolution 0.30.4

0.50.6 0.70.8 0.91.01.1

Normalized concentration

O_i C_s

(b)Concentration relative to the concentration at the best resolution plotted against the resolution divided through the FWHM of the respective peak

Figure 5.8: Concentrations obtained for different resolutions

resolutions. We found a linear relation between the absorption coefficient at resolution 2 cm⁻¹ and 8 cm⁻¹. This enables us to correct the conversion factor for the lower resolution and makes carbon and oxygen measurements possible even with thin wafers. Further investigations should be carried out, to confirm these relationships for lower carbon and oxygen concentrations, as well as higher resistivities.

6 Lifetime measurements on silicon ingots and discs

Sawing bricks into wafers leads to a loss of about 49 % of the brick [SE02, Fig. 7]. Additionally there is waste of energy and abrasion of the tools. Therefore it is interesting to determine in advance, if the ingot is worth cutting or if this material has to be recycled. Criteria can be lifetime or impurity concentrations. We have a brand new bulk lifetime tester and examine the application and its limits on Czochralski grown ingots.

6.1 How the BLS works

Lifetime measurements on ingots and discs are performed using the lifetime tester BLS-I from Sinton Instruments [Ins], which being a commercial product is a black box for us: insert an ingot and obtain a lifetime. As the first results of Czochralski Si-ingots provided by Centro de Tecnolog´ıa del Silicio Solar (CENTESIL, Madrid) were quite far from the expected lifetimes, it is important to know in depth, how the BLS-I tester works. Therefore, we will first of all try to develop a deeper understanding of the measurement method, used by the BLS-I device.

The photoconductance lifetime tester consists of two main parts, a conductance sensor and a flash unit with a reference cell. The main measurement principle is described in section 3.2.

However, some points, which could not be found in literature, like the resistivity measurement of thick ingots, are treated theoretically and experimentally in this section.

6.1.1 Resistivity theory

The BLS-I instrument uses an eddy current sensor as described in section3.1.2. However, the theory there is only applicable for thin wafers. In this section we will extend this theory to thick wafers and ingots.

The power loss in a thin conductor is given by Eq. (3.4). We will now consider the ingot to be made up of many infinite thin slices with the thickness dx. Then the power loss dP in the slice at the distance x from surface is

dP(x) =V²(x) σ

8πn²dx. (6.1)

Let us consider an electromagnetic wave impinging on a conductor. Inside, the intensity will decay exponentially. As the intensity is proportional to the square of the electric field or its root mean square, we assume that the square of the voltageV in the previous equation varies also exponentially with distance in the conductor, thus

V²(x) =V²(0) exp

−x δ

, (6.2)

0 2 4 6 8 10 12

Figure 6.1: Conductivity sensor voltage vs. stack thickness. The different thicknesses were obtained by stacking 500±10µm wafers (new ones to the bottom).

where δ is the penetration depth of the electromagnetic wave. Putting this together, one obtains the differential equation

The conductivityσcan be a function of the depth, but to simplify the calculations, we consider it to be homogeneous. Integrating both sides from surface (x = 0) to the ingot thickness t leads to the total power absorption

P = σ

in the case of infinite thickness and the original equation for tδ. The same considerations as in section 3.1.2lead to the relation

I−I0 ∝σδ

To verify this relationship an experiment was performed. Starting with one wafer with a thickness of about (500±10)µm and a resistivity of (4±1) Ω cm, up to 24 wafers with the same characteristics were stacked one by one. Each time the corresponding voltage was recorded. The current measurement from section 3.1 is converted to a voltage measurement for technical reasons - the proportionality remains the same. The stack was built, adding the following wafers to the bottom of the stack, to avoid conductivity changes due to slightly different conductivities of the wafers.

The voltages corrected for the zero voltage are shown in Figure 6.1. As can be seen by the least squares fit, the relation (6.6) describes the data well. But the penetration depth obtained

6.1 How the BLS works

A B C

ρ 3.521 287 16·10⁻⁴ SV⁻²cm⁻¹ 2.933 505 73·10⁻² SV⁻¹cm⁻¹ 8.968 506 45·10⁻² V R_s 1.266 352 38·10⁻³ SV⁻² 1.375 693 42·10⁻¹ SV⁻¹ 4.193 428 97·10⁻² V Table 6.1: Calibration coefficients of BLS-I for resistivityρand sheet resistanceRs

by the fit as δ_fit = 2.4 mm is much lower than the theoretical skin depth δ_theo = 27.3 mm using Eq. (2.5). For the calculation we assumed a resistivity of 4 Ωcm corresponding to the average resistivity of the wafers and a rf circuit frequency of f = 13.56 MHz (date given from the manufacturer). The reason for this low penetration depth is unclear. The manufacturer states, this is due to the geometry of the setting. Fortunately, the measurements done in the next sections and the statement in the User Manual [Sin11, p. 4] give similar results, so we can consider this depth assense depth of the apparatus.

6.1.2 Sense depth

In the previous section we made the ad hoc assumption that the decay constant of the fit describes the depth sensitivity of the instrument. In this section we will carry out further experiments to fortify this hypothesis. The method described by Swirhun et al. [SSFM10]

is based on the comparison of sheet resistance and resistivity calibration of the instrument.

Sheet resistanceRsand resistivity ρ differ in a multiplication by the sheet thickness, thus the apparent sheet thicknesstapp can be obtained by dividing resistivity through sheet resistance

tapp = Rs

ρ . (6.7)

These calculations can be done for every voltage in the calibration range, hence one obtains an apparent sheet thickness - resistivity curve, which is shown in Figure 6.2 on the left. For conductivity and sheet conductance the calibration function

Y =A((V −V0)−C)²+B((V −V0)−C) (6.8) is used. Y has to be replaced by the quantity to be calibrated and A, B and C are obtained by a least squares fit. The parameters shipped with our instrument are shown in Table 6.1.

Although it is a polynomial of degree 2, the linear term is dominant under normal measurement conditions.

Compared to the curve of Swirhun et al. [SSFM10, Fig. 2], the variation of the apparent thickness is very low in our case and additionally, we find an increase for low resistivities, whereas they find a significant decrease of apparent thickness. The absolute values in the constant regions are 2.6 mm for Swirhun et al. and 2.2 mm in our case. In every case these values are much smaller than the theoretical skin depth, which is shown for different frequencies in Figure 6.5 on the right, and they are comparable to the value obtained in the previous section.

However, there arise some questions. On which material has the machine been calibrated, wafers or ingots? How will sheet resistance be defined for ingots? Since resistivity does not change with thickness, sheet resistance will vanish with thicker samples. As we have no

0 2 4 6 8 10 Resistivity in

Ω·

cm

1.6 1.8 2.0 2.2 2.4 2.6 2.8

t

app

in m m

10

^-1

10

⁰

10

¹

Resistivity in

^Ω·

cm 10

^-1

10

⁰

10

¹

10

²

10

³

Sense Depth in mm

t_app 1.000 MHz 13.56 MHz 1000 MHz

Figure 6.2:left: the apparent sheet thickness - resistivity curve, obtained from the calibration values.

right: the maximum skin depths for different frequencies

information about the manufacturer’s calibration method, there remains the doubt, if the apparent thickness is only the thickness of the wafers used for calibration.

6.1.3 Resistivity as function of distance to ingot

Some ingots show modulations of the radius at the beginning of the body. This makes it impossible to attach the coil properly to the ingot. There remains some air in between, which makes the resistivity value erroneous. To determine the magnitude of this error, measurements of voltage vs. distance to ingot were carried out.

Experimental setup The BLS-I device is fixed above the ingot by a tripod. Then the height is varied and the voltage is measured for every height, which is measured by a ruler. Therefore, the difference between different heights is at least 1 mm. For better resolution, i.e. smaller steps, (260±20)µm thick sheets of paper are used as spacers between ingot and BLS in a second experiment. Paper is assumed to be non magnetic and an insulator, so it would not affect the magnetic field.

Results The two voltage vs distance series are displayed in Figure 6.3, left. As the curve follows a nearly straight line in the semilogarithmic plot, we conclude an exponential relation, which is confirmed by the least squares fit. An 1/rⁿ dependency, as one could expect from multipole expansion, does not explain the data very well. For distances greater than 10 mm the exponential form is no longer valid as can be seen in Figure 6.3, right. There seems to be some kind of saturation or an additional 1/rⁿ dependency, which becomes important, when the exponential term is small. For the measurement this is irrelevant, because on one hand this makes the signal very low and on the other hand, radius variations of this magnitude are infrequent.

Additionally we notice, that there is indeed a different behaviour of paper and air with

6.1 How the BLS works

respect to absorption of magnetic fields. Although the width uncertainties overlap for air and sheet measurements, one can distinguish two slightly different decay constants of 3.0 mm for the air measurement and 2.4 mm for the sheets measurement. These values are similar to the sense depth defined in the previous section. Hence, the sense depth seems to depend more on the geometry of the device, than on the surrounding material.

6.1.4 Lateral sensitivity

To avoid boundary effects, it is useful to know the area, which is measured by the instrument.

This property can be measured by moving the instrument over the edge of a planar and homogeneously doped ingot and recording the voltage as function of the distance to the edge.

The derivation of this function is the sensitivity function of the instrument.

Short mathematical explanation: Every measurementm along an axis can be considered as convolution of the instrument sensitivity swith the material propertyr

m(x) = (s∗r)(x).

As we move over the edge of a homogeneous ingot,ris the Heaviside function, which is constant on the ingot and zero outside. According to the laws, the convolution or one of the functions in the convolution can be derived and the derivation of the Heaviside function is the Dirac distribution, which is the identity with respect to the convolution. Hence,

d which is exactly what we want to measure.

The coil has the shape of a rectangle. We define the direction parallel to the long side as axial and the perpendicular as radial. The measurements were performed on a planar multicrystalline UMG disc, which type and nominal resistivity are not known, and the distances

40 30 20 10 0 10 20 30 length in mm

0.0 0.2 0.4 0.6 0.8 1.0

normalized sensitivity

(a)axial sensitivity

15 10 5 0 5 10 15 20 25 length in mm

0.0 0.2 0.4 0.6 0.8 1.0

normalized sensitivity

(b)radial sensitivity

Figure 6.4: Lateral sensitivity. The middle of the coil is at position 0. The lines are drawn as guide to the eye.

were measured by a ruler. The derivation was obtained using a savitzky-golay filter with a window size of 3 and a linear approximation. Afterwards is was normalized to the maximum value. Those values representing the shape of the lateral sensitivity are shown in Figure 6.4 for axial and radial direction. The dotted line connects the points and is a guide to the eye.

By derivation the data gets very noisy but nevertheless the maximum measurement area can be estimated to be 50 mm in the axial and 30 mm in the radial direction. The real dimensions are 40 mm in axial and 15 mm in radial direction. In axial direction, the lateral sensitivity seems to be asymmetric, but this can also be an effect of the non-homogeneous ingot. As rule of thumb we can record, that the measurements do not suffer boundary effects, if the distance from coil to border is more than 10 mm.

6.1.5 Resistivity accuracy

Together with the measurement of the absolute values it is important, to distinguish real from random variations. We will consider two issues, the variation of the air voltage and the repeatability.

The manufacturer recommends zero calibration once a day after 30 minutes of warm-up.

Hence we recorded the air voltage from power-on for a laboratory day. The first 45 minutes a value was recorded every 5 minutes and then at least once every 90 minutes. In the meantime some measurements were performed in order to simulate a normal measurement day. For the air voltage measurement the instrument has been placed in the same location every time to avoid fluctuations due to a different environment.

In Figure6.5the resulting zero voltage vs. time curve can be seen. The value of 30 minutes

In Figure6.5the resulting zero voltage vs. time curve can be seen. The value of 30 minutes

Im Dokument Characterization of ingots and wafers for photovoltaic applications (Seite 55-67)