Letter Evaluation of luminescence images of solar cells for injection-level dependent lifetimes S. Rißland n , O. Breitenstein Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany article info Article history: Received 11 October 2012 Accepted 10 December 2012 Keywords: Luminescence Series resistance Saturation current density Injection-level dependent lifetime Solar cells abstract If the minority carrier lifetime in a semiconductor is dependent on the injection level, the dark current– voltage characteristic of solar cells may show an ideality factor n 1 larger than unity. This modifies the evaluation of photoluminescence and electroluminescence imaging data for obtaining the distribution of the local series resistance and saturation current density of solar cells. In this contribution these modifications are summarized and applied to an iterative method for evaluating electroluminescence images. It is found that even a small increase of n 1 leads to substantial variations of the resulting series resistance values. & 2012 Elsevier B.V. All rights reserved. 1. Introduction Electroluminescence (EL) and photoluminescence (PL) imaging are versatile tools to image local variations of the effective diffusion length L eff in solar cells [1] as well as of the lateral series resistance R s and the dark saturation current density J 01 [2–6]. All these methods describe the dark current–voltage (I–V) characteristic of the investigated solar cell by a one-diode model regarding the diffusion current density with an ideality factor n 1 of unity. This assumption implies that the effective minority carrier lifetime in the bulk is independent from the injection level. Very often, however, this is not the case. This holds e.g. for oxidized surfaces [7] and also for multicrystalline solar cells [8], which then have to be described by an ideality factor larger than unity. The aim of this work is to include the concept of an injection-level dependent lifetime in the quantitative evaluation of EL and PL images. If the lifetime is governed by a single Shockley–Read–Hall (SRH) recombination center, and the recombination channel is saturable (low capture cross section for majority carriers), in the low-injection regime (n onet doping concentration) the lifetime t depends in the following way on the minority carrier concentra- tion n: tðnÞ¼ t 0 þ An ð1Þ here t 0 is the lifetime for low carrier concentration, and A is a constant, which depends on the electronic properties of the level. For sufficiently high carrier concentration t becomes proportional to n. Since the diode saturation current density J 01 is proportional to 1= ffiffi t p , and thus in this regime to 1/ ffiffiffi n p , with n  exp(eV/kT) the local dark current density becomes J dark V i ð Þ¼ J 01, i V i ð Þexp V i V T ¼ J # 01, i exp V i 2V T ¼ J # 01, i exp V i n 1 V T ð2Þ (V T ¼ kT/e ¼ thermal voltage, i ¼ local position index, n 1 ¼ ideality factor). J 01,i # is the voltage-independent saturation current density parameter, if the ideality factor n 1 is regarded. Hence, if the lifetime is governed by a single SRH-center, in the low voltage limit the ideality factor of the dark characteristic is unity and the lifetime is constant (t ¼ t 0 ), but with increasing voltage and current the lifetime increases and the ideality factor approaches n 1 ¼ 2. This has not to be confused with depletion region recom- bination, which for non-saturated SRH-recombination also pre- dicts an ideality factor of n 2 ¼ 2 in the whole voltage range. J 01,i (V i ) in (2) may be expressed as J 01, i V i ð Þ¼ J # 01, i exp V i V T 1 n 1 1     ð3Þ In many cases, in the injection regime of interest (usually the range between the maximum power point V mpp and the open circuit voltage V oc ), the SRH-level is only partly saturated, or there are several SRH-centers and/or other recombination channels acting in parallel [8]. Then, in this limited voltage range, the lifetime increases sub-proportional to n, and the ideality factor n 1 in (2) is lying between 1 and 2. Assuming a voltage-independent ideality factor n 1 , this can be expressed by a carrier- resp. voltage- dependent lifetime of t i ¼ t # n i N c  ð2 2=n 1 ð ÞÞ ¼ t # exp V i V T 2 2 n 1     ð4Þ Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/solmat Solar Energy Materials & Solar Cells 0927-0248/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.solmat.2012.12.024 n Corresponding author. Tel.: þ49 345 5582692; fax: þ49 345 5511223. E-mail address: rissland@mpi-halle.mpg.de (S. Rißland). Solar Energy Materials & Solar Cells 111 (2013) 112–114