Correction of the minority carrier lifetime measurements

The different techniques for the determination of the carrier lifetime work on the premise that the carrier mobility is known. In the following, the influence of the carrier mobility on the lifetime measurements is first shown for two exemplary techniques, which are very important in today’s material characterization: the quasi-steady state photoconduc-tance (QSSPC) [80] and photoluminescence imaging (PLI) [81].

A) Quasi-steady state photoconductance

During the QSSPC measurements, excess carriers are injected by illuminating the sample with a photo flash providing intensities between 0-500 suns during a controlled intensity decay. The time-dependent light intensity S_av(t) is detected with a reference solar cell; at

the same time, the change in the conductivity Δσ(t) is recorded with the help of an induc-tion-coil.

The analysis of the minority carrier lifetime then consists in the calculation of the aver-aged excess carrier density Δnav(t) in the sample of thickness d via

( ) ( )

which corresponds to the definition of the conductivity in equation (3-32). At the same time, the generation G_av(t) of carriers is computed to

( ) ( )

with N_ph^1sun the photon flux at an intensity of 1 sun and f_abs a correction factor which ac-counts for the fact that not all incident photons are absorbed.

In the mathematically simplest case, the quasi-static analysis of the carrier lifetime reads as

( )

( )

eff Δ

τ = (4-16)

where the dependency on the carrier mobility is introduced via equation (4-14)¹⁶. The QSSPC measurement tool implements the mobility model by Dannhäuser and Krausse [83, 84], which describes the mobility as a function of the (uncompensated) doping con-centration and the excess carrier density Δn_av. During the QSSPC-measurements, the determination of Δn_av is repeated, until the excess carrier density and the mobility are consistent in equation (4-14).

As a rule, if during the QSSPC-measurements of compensated silicon samples the lower minority and majority carrier mobilities are not accounted for, the resulting measured lifetimes underestimate the real carrier lifetime.

Therefore, the evaluation tool of the QSSPC setup has been adjusted to the use of com-pensated silicon samples in the following way¹⁷: Instead of giving the base resistivity as input parameter from which the mobility is calculated, the software asks directly for the sum of the conductivity mobilities (µ_C,e+µ_C,h). Lacking alternatives, these mobilities are derived from Klaassen’s model for the measured dopant densities N_A and N_D, giving a good description at least for low compensation. In the following, the same iterative pro-cedure as described above determines the excess carrier density.

A comparison of measurements of the uncorrected and corrected effective carrier lifetime is shown in Figure 4.19. The uncorrected lifetime values are based on the measured re-sistivities. The samples were taken from different positions over ingot height of the weakly compensated ingot “Comp Cz 1”, hence the net doping concentrations and the conductivity mobilities vary strongly, which is reflected in the lifetime values. The high net doping concentration in the seed end of the crystal leads to very low lifetime due to

16 In the transient [80] and generalized [82] mode of the QSSPC technique, the influence of the mobility on the carrier lifetime measurements is similar.

17 Adjustment done by J. Geilker.

the predominant boron-oxygen defect which will be discussed in section 4.3.3. With de-creasing p0 towards the tail end of the ingot, the carrier lifetime increases.

The difference between the mobility models can significantly influence the measure-ments: At the seed end, the correct lifetime exceeds the uncorrected value by about 10%. As the enhanced dopant densities towards the tail end result in lowered conductiv-ity mobilities, the difference in this ingot part even amounts to more than 30%.

In the following sections, therefore, the lifetime measured via QSSPC on compensated samples is always corrected for the dopant concentration-dependent conductivity mobil-ity.

Figure 4.19: Comparison between uncorrected QSSPC-measurements and for mobility corrected values exemplified by samples from ingot “Comp Cz 1” in the degraded state versus ingot height (=varying mobilities). The effective lifetime was evaluated at Δn=0.1xp0. After ref. [54].

B) Photoluminescence imaging

Photoluminescence imaging makes use of radiative band-to-band recombination de-scribed in section 3.2.2. Excess carriers are generated by illumination with a laser. The intensity of the resulting radiative recombination, which is detected by a silicon charge coupled device camera, is proportional to the electron and hole densities. For p-type compensated silicon, the signal depends therefore on the net doping density:

n n≈ Δ

n p

p= ₀ +Δ (4-17).

From equations (3-17) and (4-17), it follows that the detected intensity per unit volume SPL,dV equates

(

)

,

, A Bp n n

S_PLdV = _idV Δ +Δ (4-18)

with Ai,dV describing a factor depending on the geometry of the measurement setup. The excess carrier density Δn, which is then used for the calculation of the carrier lifetime according to eq. (4-16), depends on the net doping concentration as

B

Hence, the measurement of the excess carrier density per se does not depend on the conductivity mobility and can be determined when both N_A and N_D are known.

However, a calibration of the camera signal – corresponding to the determination of the parameter A_i,dV – is needed, which may involve the carrier mobility. For example, in ref-erence [85], three approaches are presented:

i. Self-consistent lifetime calibration

The generation of excess carriers is approximated by measuring the incident photon flux and calculating the absorption via eq. (3-13). In this method, the carrier mobility is not involved.

ii. Calibration by means of comparison with alternative lifetime measurement techniques This calibration is probably most commonly used. As reference lifetime measurement, the QSSPC method is often employed. Therefore, the PLI necessitates a correct analysis of the QSSPC lifetime as described in the previous subsection.

iii. Calibration by use of selective filters

In this method, only selected wavelengths of the entire spectrum of radiative recom-bination are measured with the help of band-pass filters. By combining two or more images at different wavelengths, the carrier diffusion length can be estimated, which is proportional to the effective carrier lifetime. However, the diffusion length L, eq.

(3-25), involves the conductivity mobility, see the following section. Care has hence to be taken when using this calibration.

4.3.2 Impact of reduced conductivity mobility on the effective