SILICON CONCENTRATOR CELLS IN A TWO-STAGE PHOTOVOLTAIC SYSTEM WITH A CONCENTRATION FACTOR OF 300x

Transcription

1 SILICON CONCENTRATOR CELLS IN A TWO-STAGE PHOTOVOLTAIC SYSTEM WITH A CONCENTRATION FACTOR OF 300x Dissertation zur Erlangung des Doktorgrades der Fakultät für Angewandte Wissenschaften der Albert-Ludwigs-Universität Freiburg im Breisgau vorgelegt von Andreas Mohr aus Stegen Juni 2005 Freiburg im Breisgau

2 Dekan: Prof. Dr. Jan G. Korvink Datum der Promotion: Erstgutachter: PD Dr. V. Wittwer Zweitgutachter: Prof. Dr. O. Paul Vorsitzender der Prüfungskommission: Prof. Dr. H. Zappe Beisitzer: Prof. Dr. P. Woias

3 Photography: Silicon concentrator cell together with a loupe.

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5 Abstract Photovoltaic concentrators have a high potential to achieve cost reduction for solargenerated electricity. Different studies indicate that energy production cost of photovoltaic concentrators can occur at a fraction of the energy production costs of flat module plants in countries having high direct solar irradiation over the year. This cost reduction is achieved by a reduction of the area of highly-efficient and quite expensive solar cells using cheap optical elements concentrating the light. The fraction of the solar cell cost of the concentrator system decreases with increasing concentration factor, while the fraction of the costs of the optical elements and the tracking system increase. A tracking system is needed in order to collect the direct sun light using optical elements. The concentrator system cost goes up to a higher cost level if two-axis tracking systems instead of one-axis tracking systems are used in order to allow a high concentration factor. Two-axis tracking means usually a more complex mechanical setup and is thus more complicated to control. One-axis tracking systems are usual designed for relatively low concentration levels up to 50x. Fraunhofer ISE developed an one-axis tracking PV concentrator system enabling a high geometrical concentration of around 300x. This concentrator system was optimised and realised in this work. The system uses a parabolic trough mirror and a three-dimensional second stage consisting of compound parabolic concentrators (CPCs). Due to the two stages of this system it is named BICON (BI Two stages; CON Concentrator) system. The used CPCs are optimised for concentrating the sunlight by total internal reflection up to a geometrical concentration ratio of 7.7x. Together with the first stage concentration of 40.4x, a geometric concentration of around 300 suns can be achieved. The three-dimensional second stage consisting of the dielectric CPCs has an acceptance angle of +/ allowing one-axis tracking from summer to winter solstice. The twostage system is tracked around a polar-oriented axis. For this linear concentrator system and particularly for an easy mounting on the CPCs a rear-line-contacted (RLCC) silicon concentrator solar cell was developed in this work. In order to find an optimum cell structure, a set of masks was designed for processing 85 different RLCC cells on one single four-inch wafer. On this set of masks all the important cell parameters are varied. Extensive data from these solar cells is presented in order to display major trends in cell design and performance. As a result of this optimisation study, a 25% efficient RLCC cell at around 100 suns at 25 C was fabricated. An analysis of the recombination effects in the RLCC cell shows that the carrier recombination is dominated by the emitter recombination due to the highly doped cell regions, which are still in low-level injection, while the base of the cell is in high-level injection. At even higher injection the influence of the Auger recombination in the base

6 Abstract becomes more and more important. In investigating the RLCC cell, a shunting effect at the rear side of the cell and a nonideal diode characteristics become apparent. Both effects could be successfully implemented in an one-dimensional numerical simulation model. Utilising measurements from several runs of the solar cell, a two-dimensional model was implemented in order to investigate physical effects due to the lateral current flow in the RLCC cell. For this the complete cell is divided into three regions: The peripheral bus bars and a inner cell element. Using network simulations of all three elements, the measured cell performance could be represented precisely. The optical performance of the compound parabolic concentrators (CPCs), which are used as second stage in the concentrator, were analysed by using angular-dependent measurements and a laser mapping system. With the laser mapping system twodimensional plots of all optical losses of the CPCs mounted on a RLCC cell are possible including reflection losses at the front surface, absorption losses in the material, internal reflectance losses and coupling out losses at the exit aperture. An averaged high optical efficiency of over 81% of PMMA CPCs mounted on rear-contacted concentrator cells could be realised. Additionally, the CPCs reach the acceptance angle condition (± 23.5 ) of the BICON system. As the final part of this work, a complete cell receiver consisting of six rear-linecontacted concentrator cells and six CPCs was successfully integrated into the one-axis tracking BICON system. A high system efficiency of 16.2% could be realised at around 800 W/m 2 direct normal irradiance under realistic outdoor conditions (not temperature corrected). This is around 4% absolute higher than system efficiencies of standard commercial available flat-plate modules under PVUSA Testing Conditions (AM 1.5g, 1000 W/m 2, 20 C ambient temperature, 1 m/sec wind speed). A detailed analysis of the BICON component efficiencies indicates that the BICON system efficiency should stay nearly stable all year through and efficiencies of over 17.0% may be realised in the near future.

7 Contents 1 Overview of silicon concentrator cells and concentrator systems Concepts for silicon concentrator solar cells Concepts for concentrating sunlight onto the cells Basics of solar cells and concentrator systems Characteristic parameters of solar cells Operation mode of solar cells The output current The output voltage Recombination processes in solar cells Recombination in highly doped regions Radiative recombination in the base Defect recombination in the base Auger recombination in the base Surface recombination Current voltage characteristics of solar cells Series resistance of the RLCC solar cell Concentrator systems Optics of concentrators Tracking modes Cells at high concentration levels Recombination Mobility and conductivity Band gap narrowing Current crowding i

8 Contents 3.5 Edge losses Temperature coefficient Design and technology of the rear-contacted silicon concentrator cell Design of the concentrator cell Set of masks Fabrication process Process flow Technology Mounting of the cells Cell testing Standard measurement systems Dark and light IV curves Spectral response and external quantum efficiency Spectral mismatch correction Measurement errors at one-sun Determination of the series resistance versus the concentration Measurement method Analysis of the measurement error Needle array measurement setup Cell testing under concentration Measurement setup (KoSim) Measurement method Simulation Simulation process Two-dimensional simulation of the three-dimensional RLCC cells Simulation parameters...69 ii

9 Contents Doping profiles Reflection losses and generation profile Surface recombination velocity Simulated trends of the RLCC cell Variation of the cell thickness on a 1 Ω cm substrate Variation of the RLCC cell thickness on a 100 Ω cm substrate Variation of the contact window width Experimental trends Position of the bus bars Grid geometry Finger distance and contact windows Cell thickness Base doping concentration Thermal performance of the RLCC cells Determination of the series resistance Analytical calculation Experimental determination Summary of the parameter study % efficient RLCC cell Modelling of rear-line-contacted concentrator cells Determination of the injection level Analysis of the recombination mechanism for different concentration levels Measurement method Analysis One-dimensional model for implementing the V oc characteristics of the RLCC cell Influence of the bus bars on the RLCC cell performance iii

10 Contents 8.5 Summary of the chapter The BICON system Assembly of the BICON concentrator system The construction of the parabolic mirror and of the dielectric secondaries Fabrication and characterisation of the CPCs Lateral homogeneity of the CPCs Absolute optical performance as a function of incidence angle Determination of the surface roughness Indoor characterisation of the system under concentration Outdoor measurements of the BICON system Summary of the chapter Conclusions Summary Outlook Appendix: Detailed fabrication flow of the RLCC cell Shortcuts Variables Constants Publications Danksagung Bibliography iv

11 1 Overview of silicon concentrator cells and concentrator systems Many basic concentrator cell and concentrator system concepts came off during the seventies due to the 1973 oil crisis. At this time a lot of government efforts were funded for concentrators in the United States of America. In Europe and Japan, concentrator activities were viewed less favourably because of the low direct solar irradiation all year through. During the eighties the oil crisis was overcome, the oil price and the urgency of the energy crisis passed. So, the government efforts and the concentrator activities were strongly scaled back and unfortunately, there was no commercial success in concentrator PV. Since the middle of the nineties the photovoltaic activities are reinforced world-wide. This is again due the eventual spark of reduced availability of fossil fuel in the near future and in contrast to 1973 this time not only the governments but also big oil companies as e.g. Shell or BP are interested in renewable energies. Different governments, especially in Germany and in Spain, push and fund photovoltaic energy and thus the solar industry grew up very fast over the last years. Since the demand for silicon in the PV market increased faster than expected, a lack of the feedstock of the silicon base material between the producer and the PV industry came off and is now present. Using the concentrator technology, less silicon material is needed and this could be the reason why there is so much interest in innovative concentrator concepts world-wide today. 1.1 Concepts for silicon concentrator solar cells The concentrator cell concepts, which are developed over the last 20 years, are based on four main cell designs briefly discussed in this chapter. All these cell concepts are optimised in respect to at least one of the following demands. Minimised shadowing losses at the front side. Low series resistance losses in the grid structure. Small resistance losses due to the lateral current flow in the diffused layer between the grid lines. Low lateral current in the base. Low contact resistance and recombination current underneath the contacts. Good light trapping for the optimal use of the incoming light. One basic cell concept is the V-groove cell (Figure 1.1) [1]. This cell is optimised for reducing the front reflection while keeping the series resistance losses low. The idea of 1

12 1 Overview of silicon concentrator cells and concentrator systems this concept is to use highly reflective metal on one side of the grooves. The coverage of this metal is unimportant since all incoming light is reflected by the metal to the opposite side of the V-grooves. Due to the structured surface the light trapping of this cell is increased. The fabrication can be easily realised by using metal evaporation at an defined angle to the cell surface. Unfortunately, due to the high metal coverage of the surface without any deep diffusion underneath the contact, the recombination at the contacts and the contact resistance are high. A high resistance limits the cell performance under high concentration levels. This disadvantage is hardly avoidable in a simple process of the cell. Different other groups used similar concepts in order to reduce shadowing losses. Due to the contact problems, the best application field for the V-groove cell concept seems to be low-cost concentrator systems with a geometrical concentration ratio in a range from 5x to 40x. metal light diffusion metal Figure 1.1: A V-groove solar cell which allows all light, reflected from the top contact metal, to impinge the other side of the grooved surface [1]. Another approach to design a cell for the use under concentration is the vertical multijunction cell [2]. The top contact is formed by grooving the surface with a laser or scriber and plating metal into the grooves. light plating p+-diffusion n+-diffusion plating Figure 1.2: The plated vertical junction solar cell [2]. The principle of this cell is presented in Figure 1.2. Using the vertical junctions, the 2

13 1 Overview of silicon concentrator cells and concentrator systems junction area is increased and all the carriers are generated next to the junctions independent of their generation position. This leads to a high probability of collecting the generated carriers. One disadvantage of this cell design is that the recombination at the large metal semiconductor contact area is high, leading to low open-circuit voltages. The cell may be used for a concentration of up to 40x. The BP Saturn cells are based on this concept and are applied in the EUCLIDES power plant, which will be described in the next chapter. A useful one-sun solar cell design, which can be optimised for high concentration levels, is the p ++ -n-n ++ cell from the Sandia National Laboratories from 1982, which is shown in Figure 1.3. Efficiencies of around 20% were reached from 40 to 200 suns [3]. The cell performance under concentration is limited due to series resistance losses in the front grid, because the geometric dimensions of the front grid must be optimised for two contrary effects. On the one hand the grid fingers must be small for low reflection losses and on the other hand the front fingers must be broad for low series resistance losses. A reasonable application of this cell concept seems to be possible up to 150x. p ++ Ag n-doped substrate n ++ Ag Figure 1.3: A conventional solar cell optimised for high concentration and developed at the Sandia National Laboratories [3]. The interdigitated back-contact cell (see Figure 1.4) has both electrical contacts on the rear side of the cell [4]. So, there are no shadowing losses at the front side. This design uses alternating n ++ - and p ++ -diffusion lines on the rear side so that 50% of the back is covered by diffusions. The lateral series resistance losses due to the current flow in the diffused areas can be neglected. The metal semiconductor contact resistance losses are very small due to the high doping concentration of the diffusion lines. One critical point of this design is that most carriers are generated at the front side and have to diffuse to the rear side of the cell. So, recombination losses in the bulk must be reduced in order to collect most of the generated electron/hole pairs at the rear contacts. If 50% of the rear side is covered by high doping diffusions the recombination losses are high at the rear side. For high voltages the dopant coverage of the rear side must be decreased. 3

14 1 Overview of silicon concentrator cells and concentrator systems SiO 2 SiO 2 n++ p++ n++ p++ n++ p++ n++ metal Figure 1.4: The interdigitated back contacted cell (IBC) has no metal on the front side in order to reduce the reflection losses at the front side [4]. The back-junction point-contact silicon solar cell, developed at the University of Stanford (see Figure 1.5) [5], has also both contacts on the rear of the cell side in the same way as the interdigitated back-contact cell. The major difference between these cell types is, that instead of broad line diffusion underneath the contacts, only small local diffusion points underneath the contacts are used. This leads to high voltages and low contact resistances. The back-junction point-contact cell scheme can be applied in systems with a geometrical concentration of over 200 suns. n ++ p ++ n ++ p ++ n p n p Figure 1.5: A cross section of a textured point-contact solar cell of SUNPOWER TM, which has point contacts at the rear side [5]. 1.2 Concepts for concentrating sunlight onto the cells Over the last 20 years the developed concentrator systems use either reflecting or refracting cheap optical elements in order to concentrate light onto the solar cells. Middle scaled concentrator power plants in the range of some 100 kilowatts peak were built up in order to demonstrate the long time stability and reliability of concentrator systems. A detailed overview of the concentrator activities all over the world is given in [6]. In this work only the most important basic concentrator concepts are summarised and examples are presented. Concentrators with reflecting optical element work either with a parabolic mirror having a focus line (Figure 1.6) or a parabolic dish mirror (Figure 1.8) focusing the light onto a closed packed PV element. 4

15 1 Overview of silicon concentrator cells and concentrator systems cell receiver parabolic mirror Figure 1.6: Parabolic mirror reflecting the incoming sunlight onto a focus line. Figure 1.7: EUCLIDES TM concentrator plant in Tenerife. Concentrator systems using parabolic mirrors reach a geometrical concentration from 2x up to 50x. The EUCLIDES TM concentrator plant (Figure 1.7) in the south of Tenerife is a project of different solar research groups. It is one of the largest parabolic mirror concentrator power plant world wide [7]. The plant is composed of 14 arrays each 84 meters long and its nominal output power is 480 kwp. The concentration of the system is 38.2x, the modules are cooled with a passive heat sink and the system is one-axis tracked. The used cells are Saturn cells from BP Solar, whose cell concept is based on the vertical junction cell as already described before. solar cell array parabolic dish mirror light Figure 1.8: Dish concentrator concept. The light is reflected by a parabolic dish mirror to a PV array at the focus. Figure 1.9: Dish concentrator system of the company SOLAR SYSTEMS TM in Australia. The company SOLAR SYSTEMS TM in Australia and SUNPOWER TM in the US are developing two-axes tracked reflective dish concentrators and water-cooled close- 5

16 1 Overview of silicon concentrator cells and concentrator systems packed PV arrays for use in the focus (Figure 1.9). The parabolic reflective dishes have a geometrical concentration of around 340x. The receiver consists of a array of 16 PV modules (each 6 cm x 6 cm) and a power plant of 14 parabolic concentrators was outdoor tested reaching a high electrical system efficiency of around 20% under PVUSA Testing Conditions, i.e. 850 W/m 2 direct irradiation, 20 C ambient temperature and 1 m/sec wind speed. Concentrator systems with refractive optical elements work either with Fresnel lenses (Figure 1.10) concentrating the sunlight onto one point or linear Fresnel lenses (Figure 1.12) having a focus line. light Fresnel lens v-trough secondary solar cell Figure 1.10: The incoming light is concentrated by using thin Fresnel lenses. Figure 1.11: Five Mega Modules of AMONIX TM assembled on a 20 kwp generating system. linear Fresnel lens cell receiver Figure 1.12: Linear Fresnel lens concentrator concept. Figure 1.13: ENTECH TM 100 kw PV power plant Two-axes tracked point-fresnel lens arrays are being developed by AMONIX TM, USA. 20 kwp power plants were built up and a system efficiency of 18% at a geometrical 6

17 1 Overview of silicon concentrator cells and concentrator systems concentration of around 250x was reached. This system uses secondary optical elements called V-trough secondaries in the centre of the lens in order to increase the acceptance angle of the system and to homogenise the illumination on the cell level. V-trough secondaries are hollow pieces with reflective surfaces using multi reflection. The company ENTECH TM fabricates line-focus Fresnel concentrators operating at 20x. 100 kwp power plants are being under development in the US. Module efficiencies of around 15% at 20x under PVUSA were reached. 7

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19 2 Basics of solar cells and concentrator systems In this chapter the theoretical basics of solar cells and concentrator systems are summarised. The main focus is set on recombination effects and series resistance losses, which are the limiting parameters in the concentrator solar cell under high concentration levels. At the end of this chapter, the optical basics of concentrator systems are introduced. 2.1 Characteristic parameters of solar cells Short-circuit current If V = 0 V, the short-circuit current I sc is determined by the division of the short-circuit current density (J sc J ph ) and the active cell area A cell. The active cell area is the region of the silicon, where the solar cell process is applied. I sc = J A J A Equation 2.1 sc cell ph cell Open-circuit voltage If J out = 0 and the surface and SRH recombination are neglected, the open-circuit voltage V oc can be approximated by kt J sc kt J sc V ln + 1 ln oc, Equation 2.2 q J 0 q J 0 where k is the Boltzman s constant, T is the absolute temperature, q is the elementary charge and J 0 is dark diffusion saturation current density. Efficiency and the maximum power point The efficiency η is the maximum output power divided by the incoming irradiance G [W/m 2 ]. J mpp Vmpp Acell η = Equation 2.3 G V mpp and J mpp are the voltage and the current at the maximum power point of the IVcurve. The efficiency can also be expressed as 9

20 2 Basics of solar cells and concentrator systems J sc Voc FF Acell η =. Equation 2.4 G The fill factor FF is given by FF J mpp mpp =. Equation 2.5 J sc V V oc Cell parameters under concentration In order to deliver a simple insight of the cell performance under concentration C, the solar cell can be described by the illuminated one-diode current characteristics. J out V JRs, total = J 0 exp CJ V 1 T ph, Equation 2.6 where V T = kt/q and R s,total is the area weighted total series resistance of the cell. Equation 2.6 is only valid if the base is in low-level injection, which means that N a >> p in a p-doped base or N d >> n in a n-doped base. If the base is in high-level injection the illumination-dependent changes of some parameters, which are discussed in detail in Chapter 3, must be considered. For low-level injection the empirical cell parameters in dependence of the concentration are listed in Table 2.1 [8]. J V sc ( C) CJ ph, one sun oc ( C) V FF( C) η( C) = J T C J ln J ph, one sun 0 ph, one sun s ( FF + ) one sun lnC 1 Voc ( C) sc ( C) V oc C G ( C) FF( C) one sun C J R C max ( η max ) R s J V T ph, one sun 0.66Ωcm R s 2 Table 2.1: Solar cell parameters in dependence of the concentration. The series resistance R s is weighted by the solar cell area and the one-sun parameters are the values of P, J ph and J sc at one-sun under Standard Test Conditions. It can be seen that the efficiency of the solar cell increases with increasing concentration due to the increase of the open-circuit voltage V oc with the natural logarithm. The concentration level at which the cell efficiency peaks depends strongly on the series 10

21 2 Basics of solar cells and concentrator systems resistance R s. For J ph,one-sun = 40 ma/cm 2 and V T = 26 mv at room temperature, a maximum efficiency of around 100 suns can only be reached for R s values smaller than 6.6x10-3 Ω cm Operation mode of solar cells The output current An often shown way to analyse the current behaviour of silicon solar cells is to apply the current density, the drift plus diffusion and the continuity equation. This is a good approach to understand the transport processes in the pn-junction of a device. However, the operation of high-efficiency solar cells is not mainly controlled by current transport processes but by generation and recombination processes. So, Swanson and Sinton [9] use another approach to describe highly-efficient solar cells. For this an integral formulation of the continuity equation is applied describing the relationship between generation and recombination of carriers and brings out the output current at the contacts. Using this approach, high-injection effects, which are needed for describing concentrator cells under high concentration, can be simply introduced. The integral method of Swanson and Sinton is summarised in this chapter. The applied basics of semiconductors and silicon solar cells can be found in [10],[11],[12]. For modelling the steady-state carrier transport in silicon, the standard equations are: The current transport equations. At a pn-junction an electric field is present in addition to a concentration gradient leading to drift current and diffusion current flow. v v v J e = qµ ene + qde n Equation 2.7 v v v J = qµ pe qd p Equation 2.8 h h h where E v is the electric field, µ e,h are the mobilities of the carriers, D e,h are the diffusion coefficients and n, p are the hole and electron densities. The total current density is the sum of Equation 2.7 and Equation 2.8. v v v J = J + J Equation 2.9 The continuity equations. cond e h The number of carriers flowing into a volume minus the carriers which recombine and plus the carriers which are generated in this volume equals the number of carriers flowing out of the volume. 11

22 2 Basics of solar cells and concentrator systems v J v e = q( Rrec Gl ) Equation 2.10 v J v = q R G ) Equation 2.11 h ( rec l where R rec and G l are the recombination and generation rates. The Poissson equation. The constant Fermi level required at thermal equilibrum results in an unique space charge distribution at the pn-junction. The unique space charge distribution and the electrostatic potential are given by the Poisson equation. v q 2 + ψ = ( p + N D n N A ) Equation 2.12 ε where N d and N a are ionized doping densities and Ψ is the potential referenced to the intrinsic level. The carrier density equations. The electron and hole densities are in terms of the intrinsic carrier concentration n i and the intrinsic Fermi level E i n = n i p = n i EF, n EF, i exp Equation 2.13 kt EF, i EF, p exp Equation 2.14 kt where E F,i is the intrinsic Fermi level and E F,n and E F,p are the electron and hole quasi-fermi levels. For determining the output current of a solar cell, the continuity Equation 2.10 and Equation 2.11 are integrated over the device volume. v J v dv = q ( R G )dv Equation 2.15 V e V rec ( R G ) hdv = q rec V V l v J v dv Equation 2.16 With the Gauss divergence theorem the left-hand sides of Equation 2.15 and Equation 2.16 are converted to surface integrals over the complete device surface S. v J nˆ ds = q R G dv Equation 2.17 S S v J h e ( ) rec l V nˆ ds = q ( R ) rec Gl V where nˆ is the normal vector directed outward of the device (see Figure 2.1). l dv Equation

23 2 Basics of solar cells and concentrator systems The complete device surface S can be separated into three single surfaces. The region around the p-contact should be S 1, the region around the n-contact should be S 2 and the rest should be S 3 as shown in Figure 2.1. Iout1 n^ Jp p-contact Jn S1 S3 Iout2 n^ S2 n-contact Figure 2.1: Separation of the complete device surface into three single surfaces. v v J e nˆ ds = J e nˆ ds + J e nˆ ds + J e S S1 S2 S3 v v J h nˆ ds = J h nˆ ds + J h nˆ ds + J h S S1 S2 S3 v v v v nˆ ds nˆ ds Equation 2.19 Equation 2.20 At the p contact the current is I out1 = S1 v J e nˆ ds Replacing J v v ˆ h nds by Equation 2.20 and J h nˆ ds S1 this into Equation 2.21 it follows I out1 = q v Gl dv + q Rrec dv + J h nˆ ds + J h nˆ ds J e V V S3 S2 S1 S1 v J h nˆ ds. Equation 2.21 Since the total current is I out = I out1 = -I out2 and if taking into account that I = q G dv (photogenerated current), ph V l I, = q R dv (recombination in the base), I I rec base rec, surface rec, contact V S3 c v = J nˆ ds (recombination at the surface), = S2 h v J h nˆ ds S1 v J e nˆ ds (recombination at the contact), S v by Equation 2.18 and inserting v nˆ ds. Equation

24 2 Basics of solar cells and concentrator systems it follows from Equation 2.22 I out = I b, rec + I s, rec + I cont, rec I ph = I rec I ph. Equation 2.23 Since a negative sign means a positive output power in a solar cell, the output current is the photogenerated current minus the total recombination current which consists of the recombination in the base material, the recombination at the surface and the recombination at the contacts. This equation is valid independent of whether the base is in high-level injection or in low-level injection The output voltage A typical band diagram of a high efficient silicon solar cell is shown in Figure 2.2. Such a solar cell consists of highly doped regions near the contacts (leading to an ohmic contact and a reduction of the contact recombination) and a lightly doped base. Taking E F,n and E F,p as constant through the n ++ - and p ++ -region and into the edge of the quasi-neutral base near the contacts, then the output voltage as shown in Figure 2.2 is given by out = Vi, n + Vi, p + Vb + Vc Vm. Equation 2.24 V + V c, V m and V b are the voltage losses at the contact, in the metal and in the base and they are all negative. V c and V m will be discussed in detail in Chapter 2.5. E F,n and E F,p can be taken as constant through the n ++ - and p ++ -region, respectively, because these regions are heavily doped leading to an abundance of majorities and to an independence of the quasi-fermi levels to the illumination. qv i,n and qv i,p are the differences between the quasi Fermi levels and the intrinsic Fermi level. qvb EF,n EF,i qvi,n qvi,p EF,p metal p++-doped region base n++-doped region metal Figure 2.2: Band diagram of a high efficient solar cell under illumination. Using Equation 2.13 and Equation 2.14, qv i,n and qv i,p can be expressed as 14

25 2 Basics of solar cells and concentrator systems qv i, n = E F, n E F, i = kt ln n n i Equation 2.25 p qv = = i, p EF, i E F, p kt ln. Equation 2.26 ni If the base, contact and metal voltage losses are neglected and n and p are taken as constant through the base it follows from Equation 2.24 kt pn V ln out. Equation q ni Thus, the output voltage is the separation of the quasi-fermi levels and can be determined by the pn-product in the base if transport losses are not considered. The details of the high doped p ++ - and n ++ -contacts are not of interest for calculating the output voltage. Transport losses decrease the output voltage. These transport losses can be summarised as V + V + V I R / A, Equation 2.28 b c m s, total where R s,total [Ω cm 2 ] is the area weighted series resistance. Including transport losses, the output voltage is cell kt pn V out ln I Rs, total / A 2 cell q n. Equation 2.29 i 2.3 Recombination processes in solar cells In order to determine the current voltage characteristic of the solar cell, all the recombination terms from Equation 2.23 must be defined. A detailed derivation and description of these recombination terms is given in [11],[12] Recombination in highly doped regions The base recombination in Equation 2.23 involves the n ++ - and p ++ -doped regions at the contacts, where no analytical solution for this recombination exists. Also no analytical solution exists for the recombination at the semiconductor metal contact. Del Alamo and others [13] found out that this problem can be solved by defining a new arbitrary surface around the highly doped region and the contact area. In order to get the total recombination current density into the highly-doped region and into the contact, only the current density through this arbitrary defined surface must be considered. This assumption is possible because the electrons and holes, which diffuse into the p-contact 15

26 2 Basics of solar cells and concentrator systems and n-contact region, respectively, either recombines in the doped regions or at the contacts. The complete minority carrier recombination current into a n-doped region can be written as For the p-doped region it is pn J = rec, n contact J 0n 1 Equation ni pn J = rec, p contact J 0p 1 Equation ni where J 0n and J 0p are temperature-dependent diffusion saturation currents. For the pnproduct the carrier densities at the edge of the space charge region in the neutral base can be applied Radiative recombination in the base The radiative recombination is proportional to the excess carrier densities and is given by R 2 ( pn n ) = B Equation 2.32 rec, radiative i where B is the radiative rate coefficient. Since silicon is an indirect semiconductor this recombination process is improbable and can be neglected Defect recombination in the base Defect recombination in the base can be modeled by using the approach of Shockley, Read and Hall [14],[15], so this recombination is called SRH recombination. Under low-level injection conditions, the recombination rate is proportional to the excess minority carrier density. R rec, SRH n n0 = Equation 2.33 τ SRH where τ SRH is the lifetime of the minorities. Including the effects of the majorities a more complicated expression can be obtained. Since FZ material is used for the silicon concentrator solar cells, the number of defects in the bulk and the SRH recombination is very small Auger recombination in the base The Auger recombination is a three particle process. The energy of an electron-hole recombination pair is given to a free particle either an electron or a hole. 16

27 R = C ( n p n p ) + C ( p n p n ) 2 Basics of solar cells and concentrator systems rec, Auger n 0 0 p 0 0 Equation 2.34 C n is the n-type Auger coefficient if the free particle is an electron and C p is the p-type Auger coefficient if the free particle is a hole Surface recombination The recombination at the surface of a solar cell can be described by using a minority recombination current into the surfaces. The recombination current density can be taken as proportional to the excess minority density. J = qs ( n n ) rec, surface 0 Equation 2.35 at a p-type surfaces. J = qs ( p p ) rec, surface 0 Equation 2.36 at a n-type surfaces. S is the surface recombination velocity which is in the range of 1 to 10 3 cm/sec for passivated surfaces. 2.4 Current voltage characteristics of solar cells For a simple analysis of the cell operation the electron and hole quasi-fermi energies are taken as constant through the base and the voltage drops along the base are ignored. Using the recombination terms from Equation 2.30 to Equation 2.36, the recombination current densities for every single recombination mechanism can be determined. For this first of all the pn-product is expressed by using Equation qv pn = ni exp. Equation 2.37 kt Under low-level injection and for a p-doped base it is then 2 qv pn = ni exp kt N A n. Equation 2.38 By replacing the pn-product in the recombination terms (Equation 2.30 to Equation 2.36) by Equation 2.38, the recombination current densities can be calculated. Taking Equation 2.30, Equation 2.31 and Equation 2.38, the recombination currents into highly doped regions are qv J rec, n contact = J 0n exp 1, Equation 2.39 kt 17

28 2 Basics of solar cells and concentrator systems qv J rec, p contact = J 0p exp 1. Equation 2.40 kt The recombination current due to the SRH recombination is what can be rewritten to I rec n n0, SRH = q dv, Equation 2.41 τ V n J rec, SRH 2 qtcni qv = exp 1, Equation 2.42 N Aτ n kt where A is the area of the device and t c is the thickness of the base. The recombination current due to the surface recombination is I rec, surface = qs ( n n0 ) da, Equation 2.43 I A surf 2 rec, surface. qasurf Sni qv = exp 1 N A kt Equation 2.44 The radiative recombination and the Auger recombination can be neglected under lowlevel injection conditions. Inserting Equation 2.39, Equation 2.40, Equation 2.42 and Equation 2.44 in Equation 2.23, the current voltage characteristic is under low-level injection J 2 2 Iout qtcn qa i surf Sni qv out = = J n J p + + A N A n A N exp 1 τ A kt n where the ideality factor n is 1. J ph, Equation 2.45 Therefore, neglecting the recombination in the depletion zone, the ideality factor n of the current voltage characteristics of a solar cell is 1 in low-level injection. The IVcharacteristics in high-level injection is considered in Chapter 3. In Chapter 3 the ideality factors for the different recombination mechanism in low- and high-level injection are summarised. 2.5 Series resistance of the RLCC solar cell Besides the recombination losses in a solar cell, the cell performance is mainly limited by the series resistance losses under concentration leading to a decrease of the fill factor (see Table 2.1). The total series resistance R s,total consists of the series resistance in the base, in the emitter and in the metal as shown in Figure 2.3. In the following the analytical expressions for the series resistance components of the RLCC cell are given. The exact derivation can be found in [16]. 18

29 2 Basics of solar cells and concentrator systems γ oxide base vertical h e floating emitter (n + ) p-silicon base lateral emitter n ++ n + oxide p++ contact metal Figure 2.3: The different components of the series resistance of a rear-contacted solar cell, which is described in detail in Chapter 4.1. γ are the incoming photons and e and h are the generated electrons and holes. Resistance in the base In the RLCC cell the generated carriers must diffuse from the front side to the rear side in order to be separated and collected at the contacts. The resulting series resistance is: R t * c s, base, vertical = ρbase, Equation 2.46 Acell where ρ base is the specific resistance of the base, t c is the thickness of the cell and A cell is the active area of the cell. In this work the series resistance R * s is given in Ω and the series resistance weighted by the active cell area R s is given in Ω cm 2. The vertical series resistance weighted by the active area of the cell is then R = ρ t s, base, vertical base c. Equation 2.47 The lateral component of the base series resistance is R 1 ρ a * base f s, base, lateral =, Equation tc l f weighted by the active cell area, it results R s, base, lateral 1 ρ = 12 t base c a 2 f, Equation 2.49 where a f is the distance between a p- and p-finger and l f is the length of a finger. 19

30 2 Basics of solar cells and concentrator systems Resistance of the emitter The series resistance of the emitter (n + -diffusion), which is locally diffused underneath the n-fingers with a depth d, can be calculated by R 1 ρ w * diffusion d s, emitter =, Equation d l f where ρ diffusion is the averaged specific resistance of the diffusion and w d is the width of the n + -diffusion. The area weighted emitter series resistance is R s, emitter 1 ρ diffusion 2 = wd. Equation d Contact resistance metal diffusion (n ++ or p ++ ) contact area Figure 2.4: Cross bridge resistor for measuring the contact resistance between the n ++ - and p ++ -diffused semiconductor regions and the metal. While generating a constant current between area 2 and 3, the voltage drop between area 1 and 4 is measured for determining the contact resistance. The resistance between the semiconductor and the metal can not be calculated analytically. Therefore, cross bridge Kelvin resistor test structures [17] are integrated into the set of masks allowing to measure the contact resistance by using a four-point measurement method. While generating a constant current between area 2 and 3 in Figure 2.4, the voltage drop between area 1 and 4 is measured for determining the contact resistance. The contact resistance is then 20

31 2 Basics of solar cells and concentrator systems V test structure 14 test structure Rs, contact = Acontact, Equation 2.52 I 23 teststructure where A contact is the contact area A of the test structure between the semiconductor and the metal. For the solar cell the contact resistance is structure structure ( R test R test ) A cell R s, contact = s, contact, p+ + + s, contact, n+ +, Equation 2.53 Acontact where A cell is the active cell area and A contact is the contact area between the metal and the semiconductor. Metal resistance The ohmic losses due to the metallisation can be analytically described as R R 1 l * f s, finger = ρ metal, Equation h f w f 1 = ρ l * bus s, bus metal, Equation hbus wbus where l bus and l finger are the half of the length of the bus and the complete length of the finger, w bus and w finger are the width of the bus and the finger, h bus and h finger are the height of the bus and the finger and ρ metal is the specific resistance of the metal. For the series resistance weighted by the area it is R R s, finger = a l 2 1 f f ρ metal, Equation h f w f l A where a f is the distance between two fingers. 1 bus cell s, bus = ρ metal, Equation h f wbus In addition to the presented ohmic losses there are also so called non-generation losses, which can not be described analytically [18]. The non-generation losses are generated by the different path length of the current through the finger to the contacts. Thereby, the voltage drops along the metal structure vary. Working at the maximum power point of the complete cell, different local regions of the cell work at different maximum power points leading to current losses. The non-generation losses can be investigated by using circuit simulation. 21

32 2 Basics of solar cells and concentrator systems Influence of the total series resistance on the cell performance under concentration The total series resistance is the sum of the series resistance components. R s, total Rs, base, vertical + Rs, base, lateral + Rs, emitter + Rs, contact + Rs, bus + = R Equation 2.58 s, finger The total series resistance limits the fill factor at higher concentration levels. Some simulated IV-curves for different total series resistances of a 65 µm thick rear-contacted cell are plotted in Figure 2.5 at a concentration of 200 suns. It can be seen that with increasing series resistance the fill factor is strongly reduced. 0.0 Current [A] concentration = 200 suns; cell thickness = 65 µm; R s =0.005 Ω cm 2 R s =0.01 Ω cm 2 R s =0.02 Ω cm 2 R s =0.03 Ω cm 2 R s =0.04 Ω cm 2 FF losses Voltage [V] Figure 2.5: One-dimensional numerical simulation of a 65 µm thick rear-contacted concentrator cell using PC1D TM. 22

33 2 Basics of solar cells and concentrator systems 2.6 Concentrator systems Optics of concentrators The optical and geometrical concentration (C op, C geo ) of a system are defined as C G A η C, Equation 2.59 in in op = = op = ηop Gout Aout where η op is the optical efficiency of the system, G in and G out are the irradiances at the entry aperture A in and at the exit aperture A out. The efficiency of a concentrator system is defined as the product of the cell efficiency η cell and the efficiency of the used optics η op. η sys cell op geo = η η. Equation 2.60 In a two-dimensional space every ray entering through one point of the entry aperture can be described by using a two-dimensional phase-space volume (Etendue) consisting of x v and p v. Where x v is the ray coordinate and p v is the optical direction cosines at the entry aperture. For a homogeneous light source (e.g. sun on earth) the Etendue in the two-dimensional space can be described as dpdx = dp dx = n ε hom,2d = 4asinθ, Equation 2.61 where n is the refractive index of the surrounding medium, a is the half of the aperture and θ is the half of the aperture angle [19]. Due to the theorem of Liouville [20] the Etendue at the entry aperture of an ideal concentrator must be conserved at the exit aperture (ε in = ε out ). This means for a twodimensional homogenous source using Equation a in n sinθ = a n sinθ. Equation 2.62 in in out out Therefore, the geometrical concentration of an ideal two-dimensional concentrator system as shown in Figure 2.6 [20] is C a n sinθ out in out out geo = =. Equation 2.63 aout nin sinθ in For the three-dimensional ideal concentrator the geometrical concentration is C geo n = n out in 2 sinθ out sin. Equation 2.64 θ in Taking Equation 2.63 and Equation 2.64 into account, the conservation of the Etendue for an ideal concentrator means concentrating light by decreasing the aperture results in an increase of the divergence of the outgoing rays in contrast to the incoming rays, 23

34 2 Basics of solar cells and concentrator systems the more parallel the incoming light is the higher the possible concentration is, the maximum concentration can be reached if θ out is 90, the maximum concentration for a two-dimensional ideal concentrator is 212x and for a three-dimensional ideal concentrator around 45000x, where n = 1, θ out = 90, θ in = 0.27 (aperture angle of the sun). θin pdirection θout nin 2ain x nout 2aout loss-free concentrator Figure 2.6: Scheme of a loss-free two-dimensional concentrator. An often used parameter to characterise a real concentrator system is the acceptance angle θ acc. θ acc is defined as the angle of incident light at which 90% of the maximum signal at the exit aperture is detected by the receiver. Signal(θ ) 0. 9 Equation 2.65 acc Signal max Tracking modes Concentrator systems can only use the direct light of the sun. Thus, concentrator systems have to be tracked with the sun. The two favourite tracking concepts are: Two-axes tracked concentrator systems which have a vertical and horizontal tracking axis. Because the aperture angle of the sun is θ s = ± 0.27, all the rays of the aperture angle have to reach the solar cell and so, these systems need a minimal vertical and horizontal acceptance angle θ acc,v,h of ± Standard two-axes tracking systems using e. g. lenses or dishes reach a high geometrical concentration of 250x up to 500x. One-axis tracked concentrator systems. These systems are tracked in most of the cases around the polar axis which is tilted by the degree of latitude. Due to the aperture angle of the sun, these systems have also a horizontal acceptance angle of θ acc,h = ± 0.27, while the vertical acceptance angle θ acc,v is ± 23.5 because the angle of incident sun irradiation onto a polar tracked system changes between ± 23.5 from the summer to the winter solstice as illustrated in Figure 2.7. Standard 24

35 2 Basics of solar cells and concentrator systems one-axis tracking systems using parabolic trough mirror or linear Fresnel lenses and reach a geometrical concentration from 2x up to 50x. Figure 2.7: All-season ecliptic of the sun at Fraunhofer ISE, Germany. The system is tilted by the angle of latitude, which is 48 in Freiburg. The angle of incidence onto a polar tracked concentrator system is 23.5 at the summer solstice and 23.5 at the winter solstice. A theoretical comparison between both systems shows that the annual irradiation density is 980 kwh/m 2 a for the two-axes tracked system and is 930 kwh/m 2 a for a polar tracked system at Freiburg (48 degree of latitude) [19]. 25

36

37 3 Cells at high concentration levels At high incident power densities the concentration of the generated carriers in solar cells exceed the base doping concentration ( n or p >> N D or N A ) and the concentration of free electrons equals the concentration of free holes ( n = p). In this case the cells are in high-level injection. In this chapter the recombination losses, the conductivity and the band gap narrowing in the highly injected case of the cell are presented, and the effects of high current densities, edge losses and temperature losses on the cell performance are discussed for high incident power densities. 3.1 Recombination Taking the equation for the output current density J out (Equation 2.23) and the output voltage V out (Equation 2.27) into account, the current voltage characteristics can be derived under high-injection conditions. If the base is in high-level injection, the number of free electrons equals the number of the holes. This implies that 2 2 qv pn = n = ni exp, Equation 3.1 kt 1 qv n = ni exp. Equation kt Under high-level injection conditions the number of light-generated free carriers n is much larger than the number of thermal generated carriers n 0, so that n n 0 n. Equation 3.3 SRH recombination in the base In the base the SRH recombination rate is under high-level injection R SRH n =. Equation 3.4 τ SRH The recombination current due to the SRH recombination is then by using Equation 2.41 and Equation 3.2 J rec, SRH q t = τ c SRH n i 1 exp 2 qv kt. Equation

38 3 Cells at high concentration levels Thus, the ideality factor for the SRH recombination is 2 under high-level injection conditions. Auger recombination in the base The Auger recombination rate is in the base under high-level injection R Auger = Cnn + C pn = C An, Equation 3.6 where C A = C n + C p (ambipolar Auger coefficient). Using Equation 3.2 and Equation 3.4, the Auger recombination current is J rec 3 3 qv, Auger = q tc C Ani e. Equation kt Thus, the ideality factor for the Auger recombination is 2/3 under high-level injection. Recombination in the highly doped regions The recombination in the highly doped regions can be treated as the SRHrecombination in the base under low-level injection. This is due to the fact that the highly doped regions are under low-level injection even for very high irradiation. So, the ideality factor of the recombination in the highly doped regions is always 1 (see Chapter 2.4). Recombination in the depletion region In the depletion region the number of electrons equals nearly the number of holes. Thus, the recombination in the depletion region can be seen (if the recombination centre is in the middle of the band gap) as the SRH recombination in the base under high-level injection. So, the ideality factor is 2. Surface recombination The surface recombination mechanisms vary along the rear side of the cell cause of the different local carrier concentrations underneath the rear side. In the red and yellow marked regions in Figure 3.1 the dopant concentration is larger than the concentration of the generated carriers, if the cell is in low-level injection. This leads to an ideality factor of 1 (see Chapter 2.3.5). In the yellow marked region, the surface is under lowlevel injection even for high concentration levels. This leads to an ideality factor of 1 independent of the concentration. Under high-level conditions in the base the generated carrier density in the red marked area exceeds the dopant concentration leading to high level injection effects in this region underneath the surface. 28