17th European Photovoltaic Solar Energy Conference, Munich, Germany, 22-26 October 2001

                              CHARACTERISTICS OF SOLAR CELLS

                                                  Mark J. Kerr and Andres Cuevas
                                                Centre for Sustainable Energy Systems
                     Department of Engineering, The Australian National University, Canberra ACT 0200, Australia,
                       Email:, Phone: (+61) 2 6125 0078, Fax: (+61) 2 6125 0506

          ABSTRACT: The current-voltage characteristics of solar cells and photodiodes can be determined by measuring the
          open-circuit voltage as a function of a slowly varying light intensity. This paper presents a detailed theoretical
          analysis and interpretation of such quasi-steady-state Voc measurements (QssVoc). The ability of this analysis to
          accurately obtain the true steady-state device characteristics even in the case of high lifetime, high resistivity silicon
          devices is demonstrated experimentally. The QssVoc technique can be used to determine the minority carrier lifetime,
          and the new generalised analysis is required to do this accurately. An important outcome is that solar cell and diode
          device characteristics can be obtained from measurements of either the photoconductance or the open-circuit voltage,
          even using transient decay techniques.
          Keywords: Quasi-steady-state - 1: I-V Characteristics - 2: Lifetime - 3

      1. INTRODUCTION                                                      widespread. As shown in this paper, even if the light
                                                                           intensity and the electric current are zero, it is still possible
          The recombination properties of semiconductor devices            to obtain the characteristic curves of the device from PCD
      strongly affect their electrical performance. Experimentally         and OCVD measurements. The vertical axis of these plots
      measuring these recombination properties in a simple and             is the rate of change of the carrier density, which represents
      easy way has been significantly facilitated by the                   the natural recombination rate in these situations. This
      introduction of quasi-steady-state (Qss) techniques [1, 2].          generalisation of the device characteristics is conceptually
      In the QssPC technique, a simple flashlamp is used to                very important, and provides a broader understanding of
      produce a slowly varying illumination and the resulting              the different physical mechanisms present in the
      time dependence of the excess photoconductance (PC) of               semiconductor.
      the sample is measured. Analysis of the data gives the
      minority carrier effective lifetime as a function of injection
      level, over a wide range of injection levels. It is possible to      2. GENERALIZATION   OF   EFFECTIVE
      analyse the data in a way that accommodates the possible                LIFETIME MEASUREMENTS
      different time dependencies of the light excitation and the
      excess carrier density in the semiconductor. Such a                  2.1 Previous Work
      generalised analysis has been applied to improve the                     Three operating regimes for the light excitation can be
      accuracy of QssPC measurements [3].                                  identified. The first involves an abrupt cessation of the
                                                                           illumination. This is the traditional transient photo-
          The Qss approach has also been applied to measure the            conductance technique. The second regime is the steady-
      open-circuit voltage of solar cells as a function of the             state illumination, which has also been historically used. In
      incident light intensity [4]. Again, a monotonically varying         the third regime the intensity of the illumination varies
      illumination from a flashlamp is used to produce a voltage           monotonically with time, which is the basis of the quasi-
      vs. illumination curve. This quasi-steady state open-circuit         steady-state method. Proper analysis of all the different
      voltage method (QssVoc) has important practical                      possibilities requires use of the continuity equation for the
      advantages over the classic Isc-Voc technique. Besides an            excess minority carriers:
      almost direct measurement of the I-V characteristics, the
                                                                                 ¶D n                                 1 dJ n
      QssVoc technique provides an insight into carrier                               = G b (t , x ) - U b (t , x ) +                   (1 )
      recombination and an alternative path to determine the                      ¶t                                  q dx
      minority carrier lifetime. A generalised analysis of QssVoc                    For regime 1: Gb(t,x) = 0, while for regime 2:
      measurements is presented in this paper. This corrected              ¶Dn/¶t = 0. Further, the transport term reduces to surface
      analysis is particularly important for solar cells made on           recombination terms when equation (1) is integrated over
      high resistivity, high lifetime silicon, and we have used            the sample width. Nagel et al. [3] generalised the analysis
      such devices to demonstrate the method experimentally.               procedure to define an effective minority carrier lifetime
                                                                           regardless of the flashlamp characteristics by combining
          The extended analysis of time-dependent measurements             the bulk and surface recombination rates into an effective
      of the excess carrier density under illumination by either           recombination rate (Ueff) and not dropping any of the terms
      conductance or voltage techniques goes beyond pure                   in equation (1):
      mathematical accuracy. Applying it to the special case
                                                                                                        D n av ( t )
      where the light excitation is abruptly terminated and the                       t eff =
                                                                                                             ¶ D n av ( t )
                                                                                                                                 (2 )
      transient decay of the photoconductance or the voltage is                                 G av ( t ) -
      recorded with time is very important, since the PCD and                                                       ¶t
      OCVD methods have historically been, and still are,

     To understand the physical meaning underlying this                                                               102                                                                              xx

                                                                  Equivalent Steady-State Illumination (Inet -suns)
generalised definition of lifetime, it should be realised that                                                                           uncorrected data - transient decay                          xx
                                                                                                                                         corrected data - transient decay                          xx
the recombination rate occurring within an illuminated                                                                                                                                         xx
                                                                                                                                   x     uncorrected data - 2ms flash                        xx
sample can be affected by both the photogeneration rate at                                                            101                                                                  xx
                                                                                                                                         corrected data - 2ms flash                    xx
that instance, as well as the history of the sample (through                                                                                                                         xx
the time-dependent carrier density - ¶Dn/¶t). What is not                                                             100                                                 x xxx
obvious from the generalised lifetime definition of Nagel et                                                                                                            xx
al. is how to determine other device parameters, like the                                                                                                       x xxx
                                                                                                                                                                 x xx
true steady-state open-circuit voltage or the current-voltage                                                         10-1                                      x
                                                                                                                                                           x xx
                                                                                                                                                           x x
                                                                                                                                                           x xx
characteristics of a solar cell, from QssPC or transient                                                                                                   x
                                                                                                                                                          x xx
decay measurements.                                                                                                   10-2
                                                                                                                                                    x x
                                                                                                                                                   x xx
                                                                                                                                                x xx x x
                                                                                                                                                x x
                                                                                                                                             x x x
                                                                                                                                             x x x
                                                                                                                                             xx x
                                                                                                                                     x      xx x
                                                                                                                                     x      xx
2.2 Determination of implied Voc vs. illumination from                                                                           x x
                                                                                                                                 x x
                                                                                                                                     x    x
                                                                                                                                 x x
                                                                                                                                x xx
    lifetime measurements                                                                                             10-3     xx xx
    Effectively, the correction of Nagel et al. is equivalent
to defining the denominator of equation (2) as a net                                                                    0
generation rate (Gnet) that incorporates the actual
                                                                                                                            0.50         0.55     0.60       0.65        0.70       0.75       0.80
photogeneration (Gav) and the carrier density history:
                                                                                                                                                         Implied Voc (V)
        Gnet = Gav -                                     (3)      Figure 1: Corrected and uncorrected implied open-circuit
    In the context of characterizing the recombination            voltage vs. light intensity curves for a solar cell precursor
parameters of solar cells and/or their precursors under real      from photoconductance measurements. A flashlamp with a
conditions (ie under true steady-state conditions), the           decay time of ~2ms has been used as well as the transient
concept of net generation can be greatly expanded. The            mode where there is no actual illumination of the sample
simplest extension of the net generation concept is for           during the measurement. The vertical axis is determined
obtaining implied open-circuit voltage vs. light intensity        from Gav for the uncorrected data and Gnet for the corrected
curves      from      photoconductance     based     lifetime     data using equation (5).
measurements [2]. Under the quasi-steady state
assumption, the actual illumination level (Ilight - in suns)      seen that the uncorrected data is actually quite
can be plotted against an implied open-circuit voltage            discontinuous, particularly at lower light levels, while the
(Voc,imp) using the relationship:                                 corrected data produces a smooth, continuous curve which
                                                                  is physically what would be expected. Further, by
                                               qV oc              comparing the uncorrected and corrected quasi-steady-state
 np » D n ( 0 )[ N A + D n ( 0 )] = n i2 exp              (4 )    data it can be seen that the correction starts to become
                                                                  significant at light intensities below »1 sun for this sample.
      We know however that the actual illumination level
                                                                  The effect of the correction is to provide a more realistic
(Ilight), and therefore the actual photogeneration rate (Gav),
                                                                  determination of the electronic properties of the solar cell
does not necessarily account for all the recombination
                                                                  substrate. That is, the uncorrected data over-estimates the
events occurring within the sample. The net generation rate
                                                                  quality of the solar cell substrate by over-predicting the
(Gnet), as defined in equation (3), needs to be used to
determine the equivalent, steady-state illumination level
                                                                       An interesting, not so obvious consequence of using
(Inet) that would need to be used to obtain the same total
                                                                  the net generation concept is that transient
recombination rate. This can be expressed in unit of suns
                                                                  photoconductance lifetime measurements can be used to
(1sun=1kW/m2) via:
                                                                  determine implied Voc vs. Inet curves even though the actual
                          WG net                                  light intensity is zero. Included in Figure 1 are corrected
           I net =                                (5 )
                     f abs N ph |1 sun                            and uncorrected implied open-circuit voltage data when the
                                                                  lifetime is measured by the transient method. It can be seen
    Where W is the sample width, ¦abs is the optical              that the two corrected curves agree very well,
absorption fraction of the sample to take into account            demonstrating the validity of the approach.
reflection and absorption losses and Nph½1sun is the density           The agreement between PCD and QssPC data can be
of photons in solar light with an irradiance of 1sun              exploited to determine, or verify, the optical properties of
(1kW/m2). Alternatively, ¦absNph½1sun can be replaced with        the wafer, that is the factor fabs in eq. (5). Further, the
an estimated or measured short-circuit current density (Jsc)      standard diode analysis and interpretation can be made on
under one sun standard illumination.                              the Ilight-Voc,imp curves, thus relating the injection level
    An example of an implied open-circuit voltage vs. light       dependent effective lifetime of the sample to ideality factor
intensity curve both before the correction (Ilight-Voc,imp) and   at a given open-circuit voltage. For example, the corrected
after the correction (Inet-Voc,imp) is given in Figure 1. The     Inet-Voc,imp curve in Figure 1 shows non-ideal diode
flashlamp used showed an approximately exponential                behaviour at lower voltages (n¹1) due to an injection level
reduction in light intensity over time with a decay time of       dependent surface recombination velocity and bulk
»2ms. The sample being measured is a high lifetime solar          lifetime.
cell precursor consisting of a phosphorus diffused, high
resistivity silicon substrate passivated with high quality
TCA based silicon oxide on the front and rear. It can be

      3. GENERALIZATION OF OPEN-CIRCUIT                                                                                                        102
      3.1 Introduction

                                                                                    Illumination Intensity (suns)
                                                                                                                                                                                                                   x xx  x
          The objective of a QssVoc measurement is to obtain the                                                                                                                                                    xx x
      diode characteristics of a finished solar cell [4]. These                                                                                100                                                                x
      characteristics come about by simply plotting the actual                                                                                                                                                xxx
                                                                                                                                                                                             x             xxx
      open-circuit voltage of the device as a function of the light                                                                                                                           x
                                                                                                                                                                                              x           xx
                                                                                                                                               10   -1                                       x          xx
      intensity. We are seeking the dc, or steady-state                                                                                                                                      x        xx
                                                                                                                                                                                           xx       xx
                                                                                                                                                                                           x       xx
      characteristics of the device, that is, the dc value of Voc that                                                                                                 xxx
      corresponds to a given value of steady-state light intensity.                                                                                                      x            xx
                                                                                                                                               10   -2                    x         xx
                                                                                                                                                                         x         xx
      Analogous to the discussion in section 2.1, the actual                                                                                                            x
                                                                                                                                                                               x xx
                                                                                                                                                                       x       x
      illumination level at the time (and therefore the                                                                                                              x
      photogeneration rate) does not necessarily account for all                                                                               10-3
      the carriers in the sample. In particular, it does not account
      for all the carriers at the boundary of the space charge                                                                                   0

                                                                                       Equivalent Steady-State Illumination Intensity (suns)
      region, which in-turn, determine the open-circuit voltage.                                                                                               2(b)
      Again, we need to consider the photogeneration history of
      the sample by using the net generation rate of equation (3).                                                                             101
      This can be achieved by solving the quadratic expression of                                                                                                                                                       xx
      equation (4) to determine Dn (only the positive root has a                                                                                                                                                      x
      physical meaning) and its derivative:                                                                                                                                                                      xx
                                                                                                                                               100                                                           xxxx
                                               qV oc                                                                                                                                                 xx
                 N    2
                          + 4 n i2 exp                 - N                                                                                                                                        xx
                      A                                      A                                                                                                                              xx
                                                                                                                                                                                            xx xxxx
                                                kT                                                                                                                                         xx
      Dn =                                                            (6 )   and                                                               10-1                                      xxx
                                       2                                                                                                                                           xxx
                                                                                                                                                                               x                     true steady-state        data
                                           qV oc
                           qn i2 exp                                                                                                                                     xx
                                                                                                                                                                        xxx                            4ms flash
       ¶Dn                                                   ¶ V oc                                                                            10-2                       x
                                            kT                               (7 )
           =                                                                                                                                                         xx
                                                                                                                                                                      x                                2ms flash
        ¶t                    2                    qV oc       ¶t                                                                                                xx
                                                                                                                                                                   x                         x         0.35ms flash
                 kT       N   A   +   4 n i2   exp                                                                                                                                                     transient decay
                                                    kT                                                                                         10   -3

          Which can be substituted into equation (3) and then into
                                                                                                                                                         0.4              0.5          0.6          0.7                              0.8
      equation (5) to obtain the equivalent, steady-state                                                                                                                    Open-circuit Voltage (V)
      illumination level (Inet) that needs to be plotted against the
      measured Voc. This procedure outlines the generalisation of                   Figure 2: Uncorrected and corrected open-circuit voltage
      Voc measurements as a function of light intensity and time.                   vs. light intensity curves for a high resistivity (90 Wcm),
                                                                                    high lifetime (1.6ms) p+nn+ BSF solar cell using four
      3.2 Experimental illustration of the correction                               different illumination conditions. An I-V curve cannot be
           The open-circuit voltage as a function of light intensity                obtained from transient OCVD using conventional
      was measured for a low recombination solar cell made at                       analysis.
      ANU on a 90 Wcm phosphorus-doped silicon wafer. The
      cell had full area boron and phosphorus diffusions at the                     the uncorrected curves are quite discontinuous. Most
      front and rear of the cell respectively, essentially making it                importantly, the predicted performance of the device would
      a 1-dimensional device. The high open-circuit voltage at                      vary greatly depending on which flashlamp was used and
      one sun, 677 mV, testifies the quality of the device                          would be significantly different from the true steady-state
      (Jsc=32.2mA/cm2, FF=0.781, h=17.1%). It also means that                       performance. Even for the 4ms flashlamp, there is a
      the excess carrier density is very high, which adds to the                    noticeable difference between the true steady-state
      problem of maintaining steady-state conditions during a                       measurements and the uncorrected Ilight-Voc data at lower
      time-dependent measurement.                                                   illumination levels.
           Four different flashlamps were used to illustrate the                        The corrected Inet-Voc curves for the device are plotted
      method. Along with the transient mode, the decay times of                     in Figure 2(b). It can be seen that applying the correction
      the flashlamps were »0.35ms, »2ms and »4ms. The 0.35ms                        again removes the discontinuities in the curves for all the
      flashlamp has been included to demonstrate the robustness                     illumination modes, but most importantly, all the curves
      of the correction. It would not typically be used for quasi-                  now overlap with each other and with the true steady-state
      steady-state measurements. Neutral density filters were                       measurements.
      used to vary the intensity of light incident on the solar cell
      to between 100suns and 0.001suns.                                             3.3 Determination of the lifetime from Voc
           Uncorrected Ilight-Voc curves for the device are plotted                     The correction is more noticeable if the measured
      in Figure 2(a) along with true steady-state measurements                      voltage is analysed to determine the effective minority
      obtained by attenuating the illumination from a solar                         carrier lifetime. The effective lifetime can be determined
      simulator with several different neutral density filters and                  from equation (2) using the minority carrier injection level
      placing the device subsequently in open and short circuit                     data derived from equation (6). Alternatively, substituting
      conditions. Consistent with the discussion in section 2.2,                    equations (6) and (7) into equation (2) gives the

generalised definition of the effective minority carrier                                     2500
                                                                                                                                        uncorrected data
lifetime from the open-circuit voltage:                                                                                                 (2ms flash)
                                                                                                                                        uncorrected data
                            2           qV oc
                        N   A   + 4 n i2 exp    - NA                                         2000                                       (4ms flash)
t eff =                                                                                                                                 corrected effective
                                        qV oc

                                                                   Effective Lifetime (ms)
                            qn i2 exp
                                          kT           ¶ V oc                                1500
         2 ( G av -                                            )
                                               qV oc     ¶t
                     kT N A + 4 n i2 exp
For the case of low level injection, this can be simplified to                               1000
a more familiar form that is easily identifiable with the
OCVD lifetime when the generation rate is zero [5]:
          t   eff   =                                (9 )
                                   q D n ¶ V oc
                        G av     -
                                    kT     ¶t                                                   0
    The lifetime can thus be easily calculated from the                                             1012   1013       1014       1015        1016         1017
measured voltage, its time derivative and the actual                                                       Minority Carrier Injection Level (cm )-3
illumination level. Figure 3 shows the effective lifetime
                                                                        Figure 3: Minority carrier effective lifetime of a high
corresponding to a high efficiency PERL cell fabricated at
                                                                        efficiency 20Wcm n+pp+ PERL cell determined from
Fhg-ISE on a 22 Wcm gallium-doped silicon substrate
                                                                        QssVoc data before and after applying the correct
(Voc=680mV, Jsc=39.5mA/cm2, FF=0.793, h=21.2%).                         generalized analysis. The flashlamps used for illumination
Minority carrier lifetimes in excess of 1 ms have been                  the cell had a decay time of ~2ms and ~4ms.
independently     measured      for   these  wafers    by
photoconductance techniques. It can be seen from the
                                                                        5. CONCLUSION
corrected data, that the lifetime has a maximum value of
»1.1ms at an injection level of 3x1014 cm-3. From the
                                                                            The physical concept of a net photogeneration rate as
uncorrected data however, the maximum lifetime depends
                                                                        the sum of the actual photogeneration and the derivative of
on the decay time of the flashlamp, with a maximum of
                                                                        the excess carrier density with time has been introduced.
»2.2ms and »1.6ms for a 2ms and 4ms decay constant                      This leads to a generalized analysis procedure for
flashlamp respectively.                                                 illumination vs. open-circuit voltage characteristic curves
                                                                        of solar cells. In this paper we have shown that the QssVoc
4. DISCUSSION                                                           technique can be accurately applied, even to high lifetime
                                                                        devices, for a given time-dependent light source when the
    In this paper we have demonstrated the generalized                  generalized analysis is used. It also permits to obtain net
analysis procedure using high efficiency and high lifetime              illumination vs. voltage curves even for transient decay
solar cells. For the majority of industrial solar cells the             measurements of the photoconductance or the open-circuit
effective lifetime is quite small (10-50ms) and is much less            voltage, a feature previously reserved exclusively to steady-
than the decay time of the typical flashlamps used. Under               state or Qss techniques.
these conditions, the effect of the generalised analysis
presented here to determine I-V characteristics is small.               ACKNOWLEGEMENTS
     Although the experimental measurements plotted
throughout this paper have been focussed on illustrating                    This work has been funded by the Australian Research
the correction to the photogeneration rate, we have                     Council. Many thanks to S. Glunz and S. Rein of the
observed capacitive effects at very low voltages. Junction              Fraunhofer Institute for Solar Energy Systems for the use
capacitance has traditionally been one of the major                     of the solar cell used in Figure 3. Many thanks to R. Sinton
drawbacks of the OCVD method, since it dominates the                    for useful discussions.
decay of the voltage at low carrier densities [6]. The
QssVoc technique is not totally impervious to these effects,
but they only become significant at much lower voltages.
The corresponding excess carrier densities are extremely
                                                                        [1] R. A. Sinton and A. Cuevas, Appl. Phys. Lett., 69
low (of the order of the intrinsic carrier density), and both
                                                                            (1996), 2510-2512.
the recombination and generation (less than one thousand
                                                                        [2] A. Cuevas and R. A. Sinton, Prog. Photovolt., 5
of one sun) rates are very small. In such circumstances, the
                                                                            (1997), 79-90.
capacitive effect of the pn junction becomes important. We
                                                                        [3] H. Nagel, C. Berge, and A. Aberle, J. Appl. Phys., 86
have verified that the distortion produced by this
                                                                            (1999), 6218-6221.
capacitance is directly related to the light source used: it is
                                                                        [4] R. A. Sinton and A. Cuevas, 16th European PVSEC,
highest for the transient measurement and lowest for the
                                                                            Glasgow, Scotland (2000), 1152-1155.
4 ms flash. This indicates that the effect can be practically
                                                                        [5] D. K. Schroder, Semiconductor Material and Device
eliminated by using a still slower flash. Alternatively, the
                                                                            Characterization: John Wiley and Sons, 1990.
capacitance could be determined from the transient OCVD
                                                                        [6] J. E. Mahan and D. L. Barnes, Solid-State Electronics,
measurement and then its value used to correct the QssVoc
                                                                            24 (1981), 989-994.


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