Determination of the diffusion length of solar cells
Mäckel & Cuevas
Determination of the diffusion length of solar cells from
the spectral response of the open-circuit voltage
H. Mäckel and A. Cuevas
Department of Engineering, Faculty of Engineering and IT
Australian National University
Acton 0200, ACT, AUSTRALIA
E-mail: helmut.maeckel@anu.edu.au
The diffusion length of a solar cell can be determined from a large signal measurement of the spectral
response of the open-circuit voltage. This new method is demonstrated with a solar cell fabricated on
aluminium-doped p-silicon. The relatively low carrier lifetime of this particular material ensures that the
diffusion length can be unambiguously extracted by a straightforward linear fit of the inverse internal
quantum efficiency versus the absorption length. The method is compared to the conventional
determination of the diffusion length from the spectral response of the short-circuit current. An
excellent agreement between both techniques has been found.
INTRODUCTION
The diffusion length is a fundamental parameter that determines the quality of the semiconductor
material used for solar cells. Its value directly affects the performance of the solar cell, both the opencircuit voltage and the short-circuit current. A sufficiently high diffusion length is needed to assure that
generated carriers reach the contact areas of the solar cells in order to be collected before they can
recombine. The measurement of this parameter is therefore an elementary part of solar cell
characterization and the aim of several different measurement methods.
Two methods that have been particularly well established in the past, draw the determination of the
diffusion length from the fact that fundamental magnitudes of the solar cell, such as the open-circuit
voltage and the short-circuit current, depend on the wavelength of the light. These are the steady-state
1
surface photo voltage (SPV) method and the spectral response of the short-circuit current. Both
methods operate with a very similar set-up and use a small light excitation at different wavelengths.
The small signal ensures a linear relationship between incoming light intensity and output signal, but
the data signals have to be amplified with the lock-in technique, which proves to be very costintensive. The small signals are in contrast to the high intensities solar cells face in normal operation,
which is circumvented by the use of a bias light in the case of the short-circuit method. This, however,
2
leads to the measurements of differential parameters, which have to be integrated in a subsequent
step.
We present in this investigation a direct method employing large signals, avoiding a bias light and
therefore giving the true steady-state, rather than the differential parameters: the spectral response of
the steady-state open-circuit voltage. The method uses bandpass filters instead of a monocromator
and has no lock-in amplification. The set-up is therefore very simple and cost-effective. In a recent
investigation, we demonstrated with this method that the external quantum efficiency of the open3
circuit voltage depends strongly on the wavelength. The open-circuit external quantum efficiency
showed a similar insight into loss mechanism of solar cells than the standard external quantum
efficiency. The technique uses a photographic flash, which declines over a broad range of light
intensities and thus creates a wide range of voltages in the sample. Hence, the external quantum
efficiency and thus the diffusion length can be additionally investigated as a function of intensity or
voltage in one single measurement.
In this paper we give first an introduction into the theory behind the measurements. We shortly sketch
the measurement set-up involved and finally describe the results of the experiments.
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Determination of the diffusion length of solar cells
Mäckel & Cuevas
THEORY
A straightforward analysis of the spectral res ponse of the short-circuit current and the SPV technique
consists in a graphical solution, as the incident light intensity, I 0 is a linear function of the reciprocal
-1
1
absorption coefficient, α , defined as the absorption length:
-1 -1
I 0 = constant x (L α + 1),
(11)
-1
Where L is the diffusion length of the base region. The plot of I0 against α has an extrapolated
intercept of I 0 at α−1 = L. In the short-circuit method, it is the inverse internal quantum efficiency,
-1 -1
4
1/IQE(Jsc), which is directly related to (L α + 1).
This approach is only valid, if several assumptions are made: 1) Only absorption in the base region is
considered, which implies that the absorption length has to be much larger than the thickness of the
emitter region. 2) The diffusion length has to be much smaller than the thickness of the solar cell. 3)
Any reflection at the back of the solar cell is neglected, i.e. the penetration depth has to be
considerably smaller than the wafer thickness. 4) Low-injection conditions prevail throughout the base
region. Assumptions 1) and 3) essentially limit the wavelength range to λ > 700 nm and λ = 950 nm,
depending on the thickness of the emitter region and the device. The second assumption has a much
wider implication: it restricts the validity of the method to thick solar cells (= 500 µm) or solar cells
fabricated on low quality silicon. For all other cases, it has been shown that the intercept of equation 1
5
represents an effective diffusion length, Leff, which can vary considerably from the actual
6
L.
Recently, other models have been proposed, which allow the determination of the actual rather than
the effective diffusion length. Spiegel et al. suggested a model, which fits both L and Leff to the inverse
7
IQE(Jsc). While this model leads to better fits for higher absorption lengths, where the linearity of
equation 1 is not assured anymore, it is restricted by the same assumptions made for equation 1.
Isenberg et al. went one step further and proved that a full analytical model of IQE(Jsc), which includes
8
internal light reflection, can be equally successful than the model of Spiegel. The inclusion of the back
reflection makes their model even applicable for larger penetration depth or thinner devices.
In this paper we are going to demonstrate that the spectral response of the open-circuit voltage is
equally able to extract the diffusion lengths of solar cells as the standard techniques. Instead of using
the advanced models of Spiegel and Isenberg, which involve complicated fitting procedures and are
prone to errors common for non-linear fits, we restrict our investigation to the model of equation 1,
using a solar cell, which fulfils all necessary requirements. This ensures the unambiguous extraction of
the diffusion length L and renders the comparison between the short-circuit and open-circuit method
more valuable.
In order to explain the extraction of the diffusion length from a spectral measurement of the opencircuit voltage, we begin with the standard relationship between the current density and the voltage of
an illuminated p -n junction in low injection:
 qV
 qD p n i2
qD n n 2i
J = (IQE n + IQE p ) qN ph (1 − R ) = 
Ξ( d ) +
Ξ( H ) (e kT − 1) ,

 Lp N D
Ln N A


(2)
where we expressed the current density in terms of the internal quantum efficiency of the emitter
region, IQEp, and base region, IQEn. Nph is the photon flux, R the reflectance of the cell. Dn, Ln, Dp,
and Lp denote the diffusion constants and diffusion lengths of the electrons and holes, respectively. ND
and NA are the doping density of the emitter and base region, while Ξ is a geometrical factor, which
reflects the fact that the cell dimensions are finite and that the diffusion length is affected by surface
recombination. In this sense, we can relate the effective diffusion length, Leff, of the base region to the
actual diffusion length, Ln, by Leff = Ln/ Ξ (H). d and H are the emitter and base thickness, respectively.
The wavelength dependent parameters are IQE, Nph, R and V.
Assumption 1) implies that the internal quantum efficiency of the emitter, IQEp ˜ 0, and leads to the
approximation of Ξ (d) ˜ SpLp/Dp. Assumption 2) sets Leff equal to Ln. Low -injection conditions are
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Determination of the diffusion length of solar cells
Mäckel & Cuevas
achieved by choosing low light intensities between 0.1 – 1 suns. As in the SPV technique, there exist
1
two possible implementations for the extraction of Ln: The measurement at constant voltage or at
constant photon flux. In the first case, we obtain:
 Sp ni2
D n2
N ph (1 − R ) = 
+ n i
Leff NA
 ND
 qV
 ( e kT − 1) ⋅ IQE n-1 = c1 ⋅ IQE n-1 ,


(3)
and in the second:
 S n2
D n2
= p i + n i
 ND
Leff N A
− 1) 
(1 − R )
(e
qV
kT

 N ph ⋅ IQE n-1 = c 2 ⋅ IQE n-1 ,


(4)
In the following we use the prefixes ‘sc’ and ‘oc’ to distinguish between short- and open-circuit IQE. In
open-circuit conditions, IQEnoc is mathematically the same as IQEnsc and can therefore be
approximated by the same models discussed above. c1 and c2 are approximately constant under lowinjection conditions in terms of their wavelength dependence. They are also obtained from the linear
fit, which opens the way for further analysis:
o
o
oc
The total open-circuit IQE can be determined indirectly from equation 3 and 4 by using c1
oc
and c2, respectively, for all wavelengths. This expression strictly represents IQE only for the
wavelength used in the linear fit, i.e. for 700 nm < λ < 1000nm, since the emitter contribution
oc
and back reflection have been neglected. Still, the so-deduced IQE can show valid
information about efficiency losses, as we will see later in section 4.
For lowly doped base material, the approximation Sp/ND « D n/(NA Leff) is usually made. This
allows either a second evaluation of Leff. Alternatively, by using Leff obtained from the fit, a
determination of either Dn, ni or NA is in principle possible.
EXPERIMENTAL
The measurements of the open-circuit spectral response are based on the Illumination-Voc technique
9
introduced by Sinton et al. The experimental set-up uses a photographic flash as light source. Two
concave lenses collimate the scattered light of the flash towards the filter. 16 bandpass filters with a
bandwidth of 10 nm covering the range of 400 - 1150 nm in an interval of 50 nm produce the required
monochromatic light. The samples are placed on a copper stage whose temperature is kept at 25 °C.
In a first measurement, the photon flux is determined by a calibrated reference cell. Subsequently, the
quasi steady-state voltage of the sample is measured. The data is read out by means of a data
acquisition card and analysed in the computer. Additionally, the internal quantum efficiency of the
short-circuit current has been measured at the Fraunhofer Institute for Solar Energy, Germany. The
reflectance of the cell has been measured with a spectrophotometer.
We demonstrate the new technique with a solar cell based on aluminium doped 0.15 Ωcm Czochralski
silicon wafer. This material is known to have a very low diffusion length due to an aluminium related
defect. The PERC-type cell with random pyramids has been fabricated at the Fraunhofer Institute for
2
Solar Energy. Its output parameters are: η = 11.8 %, V oc = 585.1 mV, Jsc = 25.24 mA/cm and FF =
0.801. The low V oc and Jsc reflect the losses in the bulk due to the aluminium related defects.
RESULTS AND DISCUSSION
A typical measurement of the photon flux-voltage curve at different wavelengths, λ, is pictured in
16
Figure 1. We adjusted the photon flux with grey filters so that it covers at least the range from 2x10
17 -1
-2
to 4x10 s cm for each λ. This is equivalent, in terms of carrier density, to a white light intensity of
0.04 - 1.4 suns. A clear wavelength dependence of the Nph-Voc-curve can be seen in Figure 1.
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Determination of the diffusion length of solar cells
Mäckel & Cuevas
Wavelength [nm]:
400
450
500
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1E17
-1
-2
Photon flux [s cm ]
N ph, max
N ph, min
1E16
Voc, min
300
400
500
V oc, max
600
Voltage [mV]
Figure 1: Photon flux – voltage curve of the solar cell at monochromatic light in the range 400 1150 nm. The minimum and maximum voltage and photon flux show the range in which the
diffusion length has been fitted
Especially for wavelength greater than 900 nm, the shift of the Nph-Voc-curve to lower voltages at a
given photon flux is noticeable. In addition, for higher wavelengths and low photon flux a deviation
from the ideal diode behaviour becomes increasingly visible as a shunt-like shape of the N ph-Voc-curve
for IR light. The exact nature of this phenomenon is still under investigation, presumably it is simply
18
1/IQE
16
sc
Fit
oc
1/IQE | V
1/IQEoc| N
oc
ph
14
Fit
Fit
=const
=const
1/IQE
12
10
8
6
4
2
0
50
100
150
1/α [µm]
Figure 2: Inverse internal quantum efficiency, 1/IQE, as a function of the absorption length 1/α .
sc
1/IQE has been obtained directly from a spectral response measurement of the short-circuit
oc
current. 1/IQE has been determined indirectly by the linear fit to equation 3 and 4. The lines
denote the linear fit to 1/IQE.
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Determination of the diffusion length of solar cells
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due to the fact that shunting effects become dominant as the voltage across the diode decreases, the
latter being a consequence of the spectral dependence at long wavelenghts. It is clear that the
increased ideality factor at higher wavelengths and lower photon flux would alter the determination of
the diffusion length. We therefore limited the analysis to photon fluxes where the ideal behaviour is still
preserved. As the discussion of Figure 2 will show, the analysis of Ln by the voltage technique has
been restricted to wavelength = 950 nm, which in turn increases the range of possible photon fluxes.
For the two implementations of the technique, evaluation at constant voltage and constant photon flux,
a minimum and maximum value has been chosen. Within this window, the diffusion length has been
determined for each constant value of voltage and photon flux, respectively. This results in a diffusion
length as a function of voltage and photon flux (Figure 4). The mean value of the diffusion length of
sc
both methods have been calculated and compared to the fit of the inverse IQE (Table 1). Due to the
preset photon flux, the spectral response at a constant voltage can only be analyzed within a relatively
small window of injection levels. Still, it is in general possible to consider the spectral response at the
actual open-circuit voltage and at the maximum power point of the cell. The voltage range of 480 –
12
-3
560 mV corresponds to a carrier density of 2.5 – 4.2x10 cm . Compared to the doping density of
17
-3
1.37x10 cm , this ensures low i njection conditions.
sc
The inverse IQE has been plotted as a function of the absorption length, 1/α, for all wavelengths
(Figure 2). For 1/α up to 160 µm (λ = 1000 nm), the measured points lie on a straight line, as the
linear fit proves. Higher wavelengths deviate from the linear fit, since back reflection cannot anymore
oc
neglected. For a better comparison, the internal quantum efficiency of the open-circuit voltage, IQE ,
has been added rather than the actually measured parameters of the left-hand side of equations 3 and
oc
4. From the linear fit of these equations, the constant c1 and c2 have been dete rmined and IQE has
oc
been calculated for all λ. As an example, Figure 2 shows the results of the inverse IQE at a constant
17 -1
-2
voltage of 510 mV and at a constant photon flux of 1.41x10 s cm as a function of 1/α. The photon
flux corresponds to a white light intensity in terms of carrier density of 0.5 suns. The data form a
straight line for 1/α > 70 µm (λ > 950 nm). It is unclear why the divergence from the linear fit appears
already at λ > 950 nm for the voltage measurements. We chose to exclude them from the fit.
Table 1. Extraction of the diffusion length.
Parameter
Method
Ln
IQE(Jsc)
Cell
Thickness [µm]
[µm]
RPAl44
335
9.3
Ln
oc
IQE
V oc=const Nph=const
[µm]
[µm ]
9.5
8.3
Table 1 shows the diffusion length determined by the short-circuit and open-circuit spectral response.
An excellent agreement is found between the different techniques. It seems that the extraction of Ln at
a constant open-circuit voltage results in slightly more consistent results. Ln is significantly lower than
the cell thickness, which validates the linear model. The internal quantum efficiency measured directly
by the standard spectral response technique and determined indirectly by the new method are
oc
depictured in Figure 4. IQE |Voc=const is the average of the internal quantum efficiency of each constant
oc
voltage, the same holds for IQE |Nph=const. Both implementations are in good agreement, but
sc
considerably lower than IQE for λ in the range 500 – 900 nm. First, differences may occur due to the
oc
oc
fact that IQE is determined indirectly using a fit, which is prone to errors. Second, IQE is strictly only
valid for the wavelength range 700 – 950 nm, as discussed in section 2. But apart from the errors due
oc
to the calculation of IQE , there exists a fundamental difference between the short- and open-circuit
measurem ents. Since the spectral response of the short-circuit current uses a small signal, which is
sc
2
overlapped by a large bias light, IQE is a differential measure. In order to obtain the actual values,
sc
IQE has to be measured at various intensities and subsequently integrated over these intensities.
The cell investigated in this study has a MOS structure at the rear surface, which is characterized by
sc
an injection dependent recombination. The IQE is therefore expected to show an increasing
response with increasing light intensity. This leads to an overestimation of the actual internal quantum
sc
efficiency, what can be seen in Figure 3. The intensity dependence of IQE is undermined by the
measurement of the relative external quantum efficiency, EQE(V oc)|rel at Nph=const (Figure 4).
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Determination of the diffusion length of solar cells
Mäckel & Cuevas
1.0
1.0
EQE(Voc )|rel
0.8
IQE
0.6
0.8
0.4
IQE
0.2
N ph [s cm ]
-1
16
1.4x10
16
5.5x10
16
sc
9.7x10
1.8x1017
oc
IQE |V
oc
IQE |N
oc
0.0
200
400
-2
=const
17
0.6
3.9x10
=const
ph
600
800
1000
1200
400
Wavelength [nm]
600
800
1000
1200
Wavelength [nm]
Figure 3: Internal quantum efficiency, IQE,
sc
as a function of the wavelength. IQE has
been obtained directly from a spectral
response measurement of the short-circuit
oc
current.
IQE
has
been
determined
indirectly by the linear fit to equation 3 and
4.
Figure
4:
Relative
external
quantum
efficiency,
EQE(V oc)|rel,
measured
at
different photon fluxes, as a function of the
wavelength. The reference wavelength is
700 nm.
EQE(Voc)|rel is defined as the ratio V oc(λ)/V oc(λref) at a constant photon flux. 3 The reference wavelength,
λref, has been chosen at 700 nm. EQE(Voc)|rel has been contemplated at different N ph=const
corresponding to a white light intensity of 0.05 – 1.4 suns. A clear dependence can be seen for λ >
800 nm. The response decreases strongly for decreasing photon fluxes. This consolidates the
hypothesis that IQE sc is a differential measure and thus overestimated in Figure 3.
Nevertheless, all curves show the same trend for increasing wavelengths associated with the small
diffusion length. Carriers generated by IR light are absorbed deep in the bulk and their diffusion
towards the junction is drastically limited by the low lifetime expectance in the bulk.
9.8
Diffusion length [µm]
Diffusion length [µm]
8.8
9.6
9.4
9.2
8.4
8.0
7.6
7.2
6.8
6.4
6.0
9.0
480
500
520
540
560
1x1017
2x1017
3x1017
-1
Voltage [mV]
4x10 1 7
-2
Photon flux [s cm ]
Figure 5: Voltage and photon flux dependence of the diffusion length.
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Determination of the diffusion length of solar cells
Mäckel & Cuevas
Finally, we turn our attention to the voltage and photon flux dependence of Ln (Figure 5). In both
cases, the diffusion length increases with increasing voltage and photon flux, respectively, even
though the absolute increase is very small.
CONCLUSIONS
A new technique has been introduced which determines the diffusion length of the bulk of a solar cell.
The technique is similar to the measurement of the spectral response of the short-circuit current and
the surface photovoltage technique, but uses large signals and a decaying light source. This enables
the measurement of the actual spectral response of the voltage without further integration of
differential data points. Furthermore, a dependence of the diffusion length on the voltage and photon
flux can be detected with the new technique.
This investigation shows that the extracted diffusion length coincides with the analysis of the spectral
response of the short-circuit current. The analysis enables the determination of the internal quantum
efficiency in open-circuit conditions. Both the short and open-circuit IQE show a strong decrease in
spectral response at higher wavelengths, which are related to the small diffusion length of ~9 µm.
Differences in the short and open-circuit IQE are explained by the fact that the low bulk lifetime is more
oc
strongly perceived by a voltage measurement and that the open-circuit IQE is an integrated
magnitude, hence more realistic. Within the range of constant voltages used in this investigation, the
diffusion length increases with increasing voltage. A similar dependence is found for constant photon
flux. This dependence is common for Schockley-Read-Hall recombination in p-type silicon.
ACKNOWLEDGMENTS
A special thanks goes to Stefan Glunz from the Fraunhofer Institute for Solar Energy, Germany, for
suggesting to use a solar cell produced on aluminium doped base material and for providing such a
cell. Furthermore we like to thank Elisabeth Schaeffer from the Fraunhofer Institute for Solar Energy
for the measurement of the internal quantum efficiency of the short-circuit current. This research has
been funded by the Australian Research Council.
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