Lifetime change

Some impurities, such as iron or oxygen, form a kind of molecule with silicon or dopant atoms within the lattice. In this case, their recombination properties differ from their properties as single atoms. If the impurity additionally permits to toggle between the single and the molecule state, the difference in lifetime can be used to calculate the concentration of this impurity. This method is mainly used for the interstitial iron content in boron-doped silicon, because iron boron pairs can be easily dissociated by illumination. As a second advantage, the dissociation is reversible. The method has also been used to study light degradation of boron-doped silicon by Schmidt and Bothe [SB04]. In this section, we will derive the theory for this measurement principle following the approach of Macdonald et al. [MGA04] for interstitial iron in boron-doped silicon.

In boron-doped silicon interstitial iron Feiforms FeB pairs with the boron atoms. As there is much more dopant than impurity, due to the low solubility of iron in silicon, all interstitial iron can be assumed to be paired in the dark. Illumination separates these pairs and at first we assume all pairs to be dissociated. Later the model is extended to incomplete association and dissociation. As the inverse lifetimes add according to (2.7), the effective lifetime can be split into the contributions of interstitial iron Fe_i, iron-boron pairs FeB and other recombination mechanisms, which are not influenced by illumination. In the dark the effective lifetime is given as

3.3 Impurities

Calculation of the difference of the inverse effective lifetimes eliminates the effect of the other recombination mechanisms, provided that they are not influenced by illumination. Therefore, this method is relatively insensitive against other recombination effects.

To calculate the iron concentration, we use the Shockley-Read-Hall lifetime model (2.11) for Fe_i as well as for FeB-pairs. For reasons of clarity, we rearrange this equation

τSRH= 1 and define the latter factor as excess carrier density dependant inverse capture cross section

1 inserted into the difference above, we obtain

1

τ_illu − 1

τ_dark = 1

τ_Fei − 1

τ_FeB =N v_th(σ_eff,Fei(∆n)−σ_eff,FeB(∆n)), (3.20) where N is the density of interstitial iron, which must be the same in the associated as well as the dissociated case. Solving this equation forN leads to the relation

N =C Now we extend the model to incomplete association and dissociation. Letαbe the fraction of the interstitial iron to be paired with boron before dissociation and β the fraction to be paired after the dissociation process. αcan be smaller than unity because of thermal excitation and also depends on the doping level, but is usually above 99 % according to Macdonaldet al.

[MGA04, p. 1023]. Then the effective lifetime in the dark has to be substituted as follows 1

For the effective lifetime after illumination we can use the same equation, but β instead ofα.

Then we obtain for the difference 1

Defect σ_p in cm² σ_n in cm² E in eV.

Fe_i 7·10⁻¹⁷ 5·10⁻¹⁴ E_V + 0.38 [MGA04]

7·10⁻¹⁷ 4·10⁻¹⁴ EV + 0.38 [IHW99]

∗ 3.6·10⁻¹⁵ E_V + 0.39 [RG05]

FeB 2·10⁻¹⁵ 3·10⁻¹⁴ E_C−0.23 [MGA04]

3·10⁻¹⁵ 5·10⁻¹⁵ E_C−0.26 [MRD⁺06]

5.5·10⁻¹⁵ 2.5·10⁻¹⁵ EC−0.26 [RG05]

3·10⁻¹⁴ 2.5·10⁻¹⁵ E_C−0.26 [IHW99]

∗ this study used the value from [IHW99] and calculated from it σ_nwith the symmetry factor.

Table 3.1: Overview over often used values for the iron defects.

and hence for the iron concentration N = 1

β−αC 1

τ_illu − 1 τ_dark

(3.23) with the same C as above.

Beside the measurement of lifetime vs. excess carrier density curves, this method requires knowledge about the SRH parameters of the recombination centers. Some recent or often used values for the capture cross sections and energy levels are shown in Table 3.1. From this table one can see, that there is little doubt about the energy levels, but the capture cross sections vary strongly. In Figure 3.6 on the left some of the resulting conversion factors C (as defined above) are shown for different combinations. As the variation is within a factor of three ([MGA04] vs. [IHW99] & [RG05]), the absolute iron concentration may match the real iron concentration only in the order of magnitude, but if all measurements are done using the same model, relative comparisons are quite reliable. Macdonald et al. [MRD⁺06] recommend the use of the values from [IHW99] for Fe_i and the values from [MRD⁺06] for FeB. If not otherwise mentioned, we will use these values.

As one can see in Figure3.6on the right, the conversion factor depends heavily on the excess carrier density. For low excess carrier densities, the lifetime decreases after illumination and the conversion factor is almost constant. Therefore this region would be ideal to measure the iron content. But this condition can only be fulfilled by surface photovoltage (carried out by Zoth and Bergholz [ZB90]) and photoluminescence measurements (carried out by Macdonald et al. [MRD⁺06]). In our laboratory we use photoconductance methods, such as QSSPC and MWPCD. Typical excess carrier densities are between 10¹³ and 10¹⁶cm⁻³. As the calibration factor has a singularity in this region, the excess carrier density where the lifetime is reported must be chosen carefully. For high excess carrier densities the conversion factor seems to become constant, but on the other hand, other recombination processes like radiative and Auger recombination become more important. The resort is to choose an excess carrier density near the local maximum, which implies little variation. It is common to use the excess carrier density of 1·10¹⁵cm⁻³with high injection measurements [Sin11, p. 14]. At this point, lifetime increases after dissociation.

3.3 Impurities

Figure 3.6: The conversion factorCfor a dopant densityNA= 1.5·10¹⁶cm⁻³ to convert the inverse lifetime difference into iron concentration for the different values from Table 3.1 in the typical range of excess carriers for photoconductance decay methods. As C has a singularity in the region between 10¹⁴ and 10¹⁵cm⁻³, we plot also 1/C over a more extended region.

Although the singularity of the conversion factor limits the accuracy of this method with high injection measurements, it can also be considered as an advantage. The singularity corresponds to the excess carrier density where the lifetime does not change. Below, the lifetime decreases and above the lifetime increases after illumination. This crossover point does not depend on the quantity of iron, and shows only slight dependence on the acceptor concentrationN_A. This makes it a robust fingerprint for iron. If you have a lifetime change after illumination, but no crossover point, the change cannot be attributed to the dissociation of iron boron pairs, which prohibits the use of this method. You might look for another explanation, such as boron oxygen compounds.

To solve the full equation (3.22) for the cross over point, one must find the root of a third or-der polynomial as carried out by Birkholzet al. [BBMS05]. The solution can be approximated by

∆nCOP= σ⁻¹_n,FeB−σ_n,Fe⁻¹

i

σ_p,FeB⁻¹ NA+ σp,Fei

σ_p,FeBn1,FeB(T), (3.24) if the dopant concentrationN_Ais above 10¹⁴cm⁻³. For the meaning of the symbols see section 2.2. The left hand term depends only on the doping, the right hand term only on the temper-ature under the assumption, that the capture cross sections are tempertemper-ature independent.

In Figure3.7athe conversion factor (3.22) is shown as function of the excess carrier density and dopant concentration for a constant temperature. The approximation (3.24) is shown for T = 300 K as white line. One can see, that the approximation is very good, because it coincides with the color change from red to blue, which is due to the sign change at the singularity. We can also notice, that the conversion factor below the cross over point is fairly constant, and there is a local maximum above the cross over point for every dopant concentration. For better resolution of the low excess carrier density region, the same values are given as contour map in Figure 3.7b. Hence we see, that the change in this region is below 10 %, if we are not close to the singularity. Above the cross over point, one has to take care of both, the excess carrier

1.0E+09 4.0E+10 1.6E+12 6.4E+13 2.5E+15 excess carrier density in cm⁻³ 1.0E+14

4.0E+14 1.6E+15 6.3E+15 2.5E+16

dopant concentration NA in cm−3

9.0 7.5 6.0 4.5 3.0 1.5 0.0

iron conversion factor in cm−3µs

1e13

(a) Color coded map. The white line corresponds to the approximation of Birkholzet al.

10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶

excess carrier density in cm⁻³

10¹⁴ 10¹⁵ 10¹⁶

dopant concentration NA in cm−3

2.5E+12 2.7E+12 3.0E+12 4.0E+126.0E+12

-2.7E+13 -3.0E+13 -3.5E+13 -4.0E+13 -5.0E+13 -6.0E+13 -8.0E+13

(b) Contour map with selected values, the unla-beled line corresponds to the singularity

Figure 3.7: Conversion factor for iron calculation as function of excess carrier density (abscissa) and dopant concentration (ordinate). The temperature is fixed at 300 K. The color scale is the same in both diagrams.

density and the dopant concentration. In our case, all measurements are carried out at room temperature, so there is little influence of temperature variations. In general one can notice, that changes are more pronounced, when the dopant concentration is low.

4 Interstitial iron mapping

Interstitial iron is an important recombination center in boron doped silicon. Therefore it is subject to defect engineering processes. For evaluation purposes, we are interested in the lateral distribution of iron, especially in multicrystalline wafers. Lauer et al. [LLb⁺08] have developed an iron mapping method using a microwave photoconductance decay (MWPCD) instrument. According to this article, we try to develop this system for our purposes. We use the Semilab WT2000 [Sem] as MWPCD device.

4.1 Measurement principle

The concentration of interstitial iron in boron doped silicon is usually measured by comparing minority carrier lifetimes before and after dissociation of boron-iron-pairs through illumination.

The method is described in detail in section 3.3.2. If lifetime is measured in high injection, as done with the MWPCD method, it is necessary to know the excess carrier density correspond-ing to the lifetime. The MWPCD instrument however usually does not evaluate the excess carrier density depending lifetime, as explained in 3.2.4.

The method of Lauer et al. uses the evaluation of the whole transient signal. According to Eq. (3.12), the lifetime can be calculated as function of the MWPCD reflectance voltage.

As the relations between excess carrier density, microwave reflectance and MWPCD voltage are assumed to be linear, a calibration of the MWPCD voltage can be done using a single known point. This can either be done by comparison to an QSSPC measurement or using the crossover point of the lifetime curves of paired and interstitial iron. The first suffers from the different measurement spot sizes (QSSPC ∼1 cm², MWPCD∼1 mm², [LLb⁺08, p. 104503-6]), which make it useless for multicrystalline wafers. The latter method can be applied to every single point and therefore also correct for shifts in the reflectance amplitude.

Once the calibration is known, the iron content can be calculated according to Eq. (3.21) for every excess carrier density. Since the iron concentration must not change with excess carrier density, this can be used to check the validity of the method. Lauer et al. concluded from simulations, that this method requires thin . 200µm and surface passivated silicon wafers with a bulk lifetimeτ_b with 1µs< τ_b<100µs in order to obtain a homogeneous excess carrier profile in the wafer.