Chapter 6 Solar Cells (Supplementary)

1 Chapter 6 olar Cells (upplementary) Chapter 6... 1 olar Cells... 1 6.1.1... 6.1.... 6.1.3... 6.1.4 Effect of Minority Electron Lifetime on Efficiency... 6.1.5 Effect of Minority hole Lifetime on Efficiency... 3 6.1.6 Effect of Back urface Recombination Velocity on Efficiency... 4 6.1.7 Effect of Base Width on Efficiency... 5 6.1.8 Effect of Emitter Width W N on Efficiency... 7 6.1.9 Effect of Acceptor Concentration on Efficiency... 8 6.1.1 Effect of Donor Concentration on Efficiency... 1 6.1.11 Band Gap Energy with Teerature... 11 6.1.1 Effect of Teerature on Efficiency... 11 6.11 Additional Topics... 13 6.11.1 Parasitic Resistance Effects (Ohmic Losses)... 13 6.11. Quantum Efficiency... 15 6.11.3 Ideal olar Cell Efficiency... 17 6.1 Modeling... 4 6.1.1 Modeling for a ilicon olar Cell... 4 6.1. Coarison of the olar Cell Model with a Commercial Product... 4 Exale 6.1.1 olar Cell Design... 43 6.13 Design of a olar Cell... 45 6.13.1 olar Cell Geometry with urface Recombination Velocities... 45 6.13. Donor and Acceptor Concentrations... 46 6.13.3 Minority Carrier Diffusion Lifetimes... 46 6.13.4 Grid pacing... 47 Exale 6.13.1 Grid pacing... 51 References... 53 Problems... 54 Table 6.9 ilicon solar cell model parameters used at 3 K for AM1.5 Global spectrum.... Table 6.1 Band gap energy teerature coefficients for Ge, i, and GaAs materials (Pierret [1])... 11 Table 6.11 Coarison between the present solar model and a commercial module.... 4

Table 6.9 ilicon solar cell model parameters used at 3 K for AM1.5 Global spectrum. Parameter Value Parameter Value A(Area) 1 cm W N.35 µm N D 1 x 1 18 cm -3 W P 3 µm N A 1 x 1 15 cm -3 τ n 35 µs F,eff 3 x 1 4 cm/s τ p 1 µs BF 1 cm/s (shade factor).5 6.1.4 Effect of Minority Electron Lifetime on Efficiency The minority electron concentration was previously discussed in ection 6.9.. The minority electron concentration gradient in p-type region is of great iortance for the efficiency of a solar cell, which was visualized in Figure 6.34. The minority electron concentration gradient depends primarily on the recombination taking place in the quasineutron region or, in other words, depends on the minority electron lifetime. The effect of the minority electron lifetime τ n on the performance of a silicon solar cell is illustrated in Figure 6.38 in terms of the figures of merit such as FF, I C, and V OC from the results of the coutations using Equation (6.9). The efficiency of the solar cell is shown in Figure 6.39. These indicate that the sort-circuit current I C and the opencircuit voltage V OC increase with increasing the minority electron lifetime τ n. ince FF is a function of V OC, the FF increases as well with increasing τ n..9.8 FF 4 3.5 V oc FF ( V ).7.6 I C 3 V oc I C ( A).5 L n < W p L n > W p.5.4 1 1 1 1 1 3 1 1 4 τ n ( µs) Figure 6.38 Effect of minority electron lifetime on FF, I C, and V OC for a solar cell at W P =3 µm and BF =1 cm/s.

3 ince the minority electron diffusion length is defined as L = τ in Equation n D n (6.8), the base width W P is used to calculate a specific lifetime ( τ n = W P Dn ) that lies at 13 µs in the figure. There are two aspects. When L n < W p, the diffusion length in the base is much less than the base thickness and the electrons created deeper than the diffusion length are unlikely to be collected. This consequently reduces the short-circuit current. In this case the back surface recombination velocity BF has no effect on the dark saturation current I. When L n > W p, the minority electron diffusion length is much larger than the base thickness and the electrons created anywhere in the base can either transfer to the emitter region through the depletion region or move to the back surface and annihilate by the surface recombination (ection 6.7.1) unless a special handling of keeping away from the back surface is applied. This can be controlled by the effective back surface recombination velocity, BF (ee ection 6.9.1). The iact of the minority electron lifetime on the power conversion efficiency is depicted in Figure 6.39.The efficiency drastically increases with increasing the minority electron lifetime. The lifetime of 35 µs used in the present other figures is a large value and requires a high-quality semiconductor material. n.18.16 η.14.1.1 1 1 1 1 1 3 1 1 4 τ n ( µs) Figure 6.39 Effect of minority electron lifetime on the efficiency η for a solar cell at W P =3 µm and BF =1 cm/s. 6.1.5 Effect of Minority hole Lifetime on Efficiency The minority hole lifetime τ p is usually small due to the short emitter width W N. Figure 6.4 shows that the effect of τ p on the efficiency is minimal and the effect of τ p does not change with varying the emitter width from.1 µm to 1. µm. It is noticed that the efficiency is greatly affected by the emitter width.

4.164 W N =.1 µm.16 W N =.35 µm η.16.158 W N =1. µm.156.5 1 1.5 τ p (µm) Figure 6.4 Effect of minority hole lifetime on the efficiency. 6.1.6 Effect of Back urface Recombination Velocity on Efficiency The trend in the solar cell industry is to move to thin solar cells (below 5 µm) for the reduction of the material volume and thus the lower price. In general a thin solar cell faces three problems: the decrease of photon absorptivity, the increase of back surface recombination, and the brittleness of the material (mechanical) due to the thinness. When a solar cell is thin, some minority electrons can reach the back surface, which usually annihilate due to the surface recombination. However, the electrons that reach the back surface can be turned away from the back surface by means of the back surface field (BF), which is achieved by non-uniform heavy doping near the back surface. The effect of the back-surface recombination velocity BF on the efficiency is illustrated with two plots for base widths of W P =5 µm and W P =3 µm, as shown in Figure 6.41. We see that both the efficiencies increase with decreasing BF and the gap between the two base widths gradually becomes close together with decreasing BF. This ilies that, when BF has a low value, the thin solar cell at W P =5 µm would have the similar efficiency with the typical solar cell at W P =3 µm. This is an advantage for the thin solar cells with a low BF. The figures of merit such as FF, I C, and V OC are plotted in Figure 6.4 at W P =5 µm with respect to BF, showing that both I C and V OC increase with decreasing BF. It is good to know whether I C and V OC are in the same direction of increasing.

5.18.16 W P =3 µm η.14 W P =5 µm.1.1 1 1 1 1 3 1 1 4 1 1 5 ( cm s) BF / Figure 6.41 Effect of BF on the efficiency of a solar cell for W P =5 µm and W P =3 µm.9.8 FF 3. FF Voc[ V ].7 I C 3.8 I C [ A].6 V oc.6.5.4 1 1 1 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 ( cm s) BF / Figure 6.4 Effect of BF on the performance of a solar cell at W P =5 µm. 6.1.7 Effect of Base Width on Efficiency As said before, the back surface recombination velocity BF is an iortant parameter on a solar cell thickness that is mostly the base width since the emitter width is very thin. The effect of base width on the efficiency as a function of BF is illustrated in Figure 6.43. It is seen that the base width significantly affects the performance of a solar cell due primarily to the silicon material limits of the absorption coefficient. Notice that there is nearly no effect of BF on the efficiency at W p = 1 µm in Figure 6.43. Conclusively,

6 the effect of the back surface recombination velocity becomes significant as the base width decreases..18.16 BF =1 cm/s η.14 BF =1 cm/s.1 BF =1 cm/s.1 1 1 1 1 3 W P ( µm) Figure 6.43 Effect of base width on the efficiency as a function of BF. The effect of base width W p on the efficiency of a solar cell as a function of the minority electron diffusion length L n or lifetime τ n is illustrated in Figure 6.44. Obviously, the longer lifetimes show an iroved efficiency. It is interesting to note that the efficiency has a maximum value for a certain base width that depends on the minority electron lifetime. Figure 6.45 shows the effect of W p on FF, I C, and V OC at τ n =15 µs. As W p increases, I C increases but V OC decreases. This produces a maximum point that can be an iortant design concept of a solar cell. Conclusively, as the base width increases, the photogenerated current increases but the open-circuit voltage decreases..17.16 η.15.14.13.1 Ln = 1 µm (τn = 15 µs) Ln = µm (τn = 6 µs) Ln = 4 µm (τn = 35 µs).11 1 1 1 1 3 W P ( µm) Figure 6.44 Effect of base width on the efficiency as a function of the minority electron diffusion length or lifetime at BF = 1 cm/s.

7.9 FF 4.8 3.5 FF V oc ( V ).7 I C 3 I C ( A).6 V oc.5.5 1 1 1 1 3 W P µm Figure 6.45 Effect of base width on FF, I C, and V OC for a solar cell at τ n =15 µs. 6.1.8 Effect of Emitter Width W N on Efficiency As we discussed previously, the emitter width W N is very small coared to the base width W P. However, its effect on the performance should not be belittled. The effect of the emitter width on the short-circuit current I C and the open-circuit voltage V OC is shown in Figure 6.46. I C decreases as W N increases, while V OC remains constant as W N increases. When we look at Equation (6.94), we deduce that the dark saturation current I must decrease at the same rate of I C in order for V OC to be constant. The dark saturation current occurs due to the recombination in the quasi-neutral region. Therefore, we conclude that, with decreasing the emitter width, not only the photogenerated current increases but also the recombination in the n-type quasi-neutral region decreases. This should be an iortant feature and ilemented into the design concept of a solar cell..6 V OC 3.35 3.3 V oc [ V ] I C [ A].55 3.5 I C.5..4.6.8 3. 1 W N (µm) Figure 6.46 Effect of emitter width on I C and V OC.

8 The effect of the emitter width W N on the efficiency is a function of the front surface recombination velocity Feff, which is shown in Figure 6.47. When Feff decreases from Feff =3x1 4 cm/s to Feff =1x1 3 cm/s, its effect becomes negligible. The effect of Feff on the efficiency is shown in Figure 6.48. We find that Feff of about 3x1 3 cm/s appears to be a good number for a minimum passivation process with various emitter widths..166 Feff =1 3 cm/s.164 η.16 Feff =3x1 4 cm/s.16.158..4.6.8 1 W N (µm) Figure 6.47 Effect of Feff on the efficiency as a function of W N..165 W N =.1µm η.16 W N =.7µm.155.15 1 1 1 1 3 1 1 4 1 1 5 Feff (cm/s) Figure 6.48 Effect of emitter width on the efficiency as a function of Feff. 6.1.9 Effect of Acceptor Concentration on Efficiency The accepters are dopants producing holes in p-type region. The effect of the acceptor concentration N A on FF, I C, and V OC is presented in Figure 6.49. The figure of merit of Equation (6.36) tells us that the power conversion efficiency η is strongly related to only both I C and V OC. We see in Figure 6.49 that, as the acceptor concentration N A increases, the open-circuit voltage V OC increases while the short-circuit current I C

9 decreases. The net effect forms a maximum point that is shown in Figure 6.5 for the efficiency. For the given input data in Table 6.9 used for the present coutations, a maximum efficiency is found at about N A =1 x 1 17 cm -3 in the figure. The decrease in the short-circuit current in Figure 6.49 must be mainly associated with the minority electron diffusion coefficient D n in the p-type region, which is shown in Equation (6.87). From Equation (6.16), D n is proportional to the electron mobility µ n. From Figure 6.3, the mobility µ n decreases with increasing N A. Therefore, D n must decrease with increasing N A. The decrease of D n results in lowering the short-circuit current. Consequently, the short-circuit current I C decreases with increasing the acceptors concentration N A, which is illustrated in Figure 6.49. The open-circuit voltage interestingly increases with increasing N A as shown in Figure 6.49. The open-circuit voltage was expressed in Equation (6.94). V OC increases logarithmically with increasing the ratio I C /I. ince we know that I C decreases with increasing N A in the foregoing paragraph, I must decrease at a rate faster than I C in order to have the increase of the ratio. I is the dark-saturation current due to the recombination in the quasi-neutral region, which was presented in Equation (6.89), where I decreases directly with increasing N A. Conclusively, as acceptor concentration increases, the photo generated current decreases while the open-circuit voltage increases..9 3.3 FF 3.3 FF and V oc (V).8.7 I C 3.8 3.6 3.4 I C (A) V oc 3..6 3. 1 1 15 1 1 16 1 1 17 1 1 18 1 1 19 1 1 N A (cm -3 ) Figure 6.49 FF, I C, and V OC versus N A for a solar cell.

1.18.175 η.17.165.16 1 1 15 1 1 16 1 1 17 1 1 18 1 1 19 1 1 N A (cm -3 ) Figure 6.5 Efficiency versus acceptor concentration for a solar cell. 6.1.1 Effect of Donor Concentration on Efficiency It is interesting how the coutational results show the effect of donor concentration on the short-circuit current and the open-circuit voltage. As for the acceptor concentration in the p-type region, the donor concentration in the n-type region behaves in the similar way. As N D increases, V OC increases but I C decreases. This produces a maximum point in the efficiency. This is shown in Figures 6.51 and 6.5. We find an optimum donor concentration N D at about 1x1 18 cm -3 toward a little lower part of the maximum from the figure because the high donor concentration causes other problems such as the increase of the lateral sheet resistance in the emitter (see ection 6.13.4) and the increase of Auger recombination (see ection 6.7.1)..6 3.35 V oc.61 V oc (V) 3.3 I C (A).6 I C 3.5.59 3. 1 1 17 1 1 18 1 1 19 1 1 1 1 1 N D (cm -3 ) Figure 6.51 the effect of N D on I C and V OC.

11.164.163 η.16.161.16 1 1 17 1 1 18 1 1 19 1 1 1 1 1 N D (cm -3 ) Figure 6.5 The effect of N D on the efficiency. 6.1.11 Band Gap Energy with Teerature The band gap energy E g is perhaps the most iortant parameter in semiconductor physics. With increasing teerature a contraction of the crystal lattice usually leads to a strengthening of the inter-atomic bonds and an associated increase in the bans gap energy. To a very good approximation, the cited variation of band gap energy with teerature can be modeled by the universal eirical relationship E g ( T ) = E g αt (6.38) T + β where α and β are constants chosen to obtain the best fit to experimental data and E g ( K) is the limiting value of the band gap at zero Kelvin. The constants and the limiting values are found in Table 6.1. Table 6.1 Band gap energy teerature coefficients for Ge, i, and GaAs materials (Pierret [1]). Material E g (3 K) E g ( K) α β Ge.663.7437 4.774 x 1-4 35 i 1.15 1.17 4.73 x 1-4 636 GaAs 1.4 1.519 5.45 x 1-4 4 6.1.1 Effect of Teerature on Efficiency olar cells deliver only a small part of the absorbed energy as electrical energy to a load. The remainder is dissipated as heat and the solar cell must therefore have a higher teerature than the environment. Heating reduces the size of the band gap energy. The absorbed photocurrent increases, leading to a slight increase in the short-circuit current. The heating has a detrimental effect on the open-circuit voltage, which is shown in Figure 6.37.

1 Many parameters in Equation (6.9) actually depend on teerature such as band gap energy, diffusion coefficients, diffusion lengths, built-in voltage, depletion region width, and intrinsic carrier concentration. It is difficult to explicitly elucidate the effect of teerature term by term. However, the coutational results give us a solid solution for the effect of teerature on FF, I C, and V OC as shown in Figure 6.5. With increasing teerature, I C increases almost linearly while V OC decreases as expected. The net result is shown in Figure 6.53 for the effect of teerature on the efficiency. Cell teerature is of great iortance for the energy conversion efficiency..9 FF 3.4.8 3.35 FF V oc [ V ].7.6 I C V OC 3.3 I C [ A].5 3.5.4 3. 4 6 8 3 3 34 36 Cell Teerature ( C) Figure 6.53 FF, I C, and V OC versus teerature for a solar cell...18 η.16.14.1 4 6 8 3 3 34 36 Cell Teerature ( C) Figure 6.54 Efficiency versus teerature for a solar cell.

13 6.11 Additional Topics 6.11.1 Parasitic Resistance Effects (Ohmic Losses) In real solar cells, power is dissipated through the resistance of the contacts and through leakage currents around the sides of the device. These effects are equivalent electrically to two parasitic resistances in series (R s ) and in parallel (R sh ) with the solar cell. The solar cell two-diode circuit model now including the two parasitic resistances is presented in Figure 6.55. Figure 6.55 olar cell two-diode circuit model including the parasitic series and shunt resistances. The series resistance arises from the bulk resistance of the semiconductor and the resistance of contacts and interconnection. eries resistance is a particular problem at high current, for instance, under concentrated light. The parallel or shunt resistance arises from leakage current around the edges of the solar cell and extended lattice defects in the depletion region. eries and parallel resistances reduce the fill factor. For an efficient solar cell we want R s to be as small and R sh to be as large as possible. When parasitic resistances are included, the total current is modified from Equation (6.9) as I ( V IR ) q( V IR ) q + + V + IR = I C I I 1 s s s 1 exp 1 exp (6.39) kt kt Rsh The effect of these parasitic resistances on the I-V characteristic is shown in Figures 6.56 and 6.57. As can also be seen in Equation (6.39), the series resistance R s has no effect on the open-circuit voltage, but reduces the short-circuit current, which is seen in Figure 6.56. Conversely, the shunt resistance R sh has no effect on the short-circuit current, but reduces the open-circuit voltage, which is seen in Figure 6.48. ources of series resistance include the metal contacts, particularly the front grid, and the transverse flow of current in solar cell emitter to the front grid. In particular, as shown in Figure 6.56, the series resistance influences the gradient of the I-V characteristic near the open-circuit voltage and can be graphically determined in a silified model by

14 R s dv (6.31) di V = V OC As shown in Figure 6.57, the shunt resistance influences the gradient of the I-V characteristic within the area of the short-circuit current and can be determined graphically approximately by dv R sh (6.311) di V = Figure 6.56 Effect of series resistances on the I-V characteristic of a solar cell ( R ) sh Figure 6.57 Effect of parallel resistances on the I-V characteristic of a solar cell ( R ) s

15 The short circuit current will be degraded by series resistance. Any series resistance will bias the dark diode in Figure 6.55, and cause the current through the load to be less than the short circuit current (photocurrent) in Equation (6.39). The open-circuit voltage will be degraded by shunt resistance. When the circuit is opened at the load, the current path through the shunt resistance lowers the bias across the dark diode in Figure 6.55 thereby degrading the open-circuit voltage. 6.11. Quantum Efficiency Quantum efficiency (QE), often called spectral response (R), is the probability that an incident photon of energy E ph will deliver one electron to the external circuit. The external quantum efficiency (EQE) of a solar cell includes the effect of optical losses such as grid shading and reflection. However, it is often useful to look at the quantum efficiency of the light left after the reflected and shaded light has been lost. Internal quantum efficiency (IQE) refers to the efficiency with which photons that are not reflected or shaded out of the cell can generate collectable carriers. By measuring the reflection and grid-shading factor of a device, the external quantum efficiency curve can be corrected to obtain the internal quantum efficiency. Hence, we can write the shortcircuit current I C in terms of EQE defined above as ince I I C = qa f ( ) EQE( ) d (6.31) = I C C ) ( d, (6.313) The external quantum efficiency is expressed by I C ( ) EQE( ) = (6.314) qaf where f () is the incident spectral photon flux. And the internal quantum efficiency (IQE) can be expressed by EQE( ) IQE( ) = (6.315) ( 1 s)( 1 r( ) ) where s is the grid-shading factor and r() is the spectral reflection. The quantum efficiency (QE) depends upon the absorption coefficient of the solar cell material, the efficiency of charge separation and the efficiency of charge collection in the device but does not depend on the incident spectrum. It is therefore a key quantity in describing solar cell performance in coarison with the spectrum of solar photons. The external quantum efficiency is plotted along with the incident spectral power using

16 Equation (6.314) with the aid of Equation (6.85) through Equation (6.88), as shown in Figure 6.58. Notice that the quantum efficiency shows zero after the wavelength of 1.1 µs because no light is absorbed below the band gap (1.1 ev ~ 1.1 µs by E ph =hc/). External Quantum Efficiency (EQE) 1.5 EQE Incident Power 1.678 1 9 8.39 1 8 Incident Power (W/m^micron) 1 3 Wavelength (µm) Figure 6.58 External quantum efficiency and incident spectral power. 1 External Quantum Efficiency (EQE).8.6.4...4.6.8 1 1. Wavelength (µm) Feff =3 x 1 5 cm/s, BF =1 cm/s Feff =1 cm/s, BF =1 cm/s Feff =3 x 1 5 cm/s, BF =1 7 cm/s Figure 6.59 External quantum efficiency of a silicon solar cell.

17 The effects of both effective front surface recombination velocity Feff and back surface recombination velocity BF on external quantum efficiency are illustrated in Figure 6.59. The short wavelength (violet spectrum) response iroves dramatically by decreasing Feff from 3 x 1 15 cm/s to 1 cm/s since the high energy (short wavelength) photon are absorbed at the vicinity of the front surface in the emitter region. This decrease can be practically achieved by passivation (coating) at the front surface. The long wavelength (red spectrum) response iroves by decreasing BF from 1 7 cm/s to 1 cm/s since the low energy (long wavelength) photon are absorbed deep in the base region. These are shown in Figure 6.59. This also can be achieved by passivation at the back surface. The differential space between the unity and the external quantum efficiency indicates the quantities of the losses such as the reflection/transmission and the shading factor at the front surface and the recombination at the vicinities of the front and back surfaces. 6.11.3 Ideal olar Cell Efficiency The ideal solar cell efficiency or the limit of solar cell efficiency is a fundamental question that has been driven since the invention of solar cells. There are many approaches reported which includes thermodynamic limits with a blackbody source of sun. However, we rather herein try to obtain more practical limits of a solar cell for the AM1.5G spectral irradiance (Figure 6.6) for 1 sun and 1 sun (concentrated solar cell). We have seen in the previous sections that the photocurrent is greatly limited by the band gap energy E g. As a result of it, the photocurrent reduces with increasing the band gap energy (see Figure 6.5). The photocurrent is equal to the short-circuit current I C with assuming small ohmic losses. Conversely, the high band gap energy causes the high output voltage which results in the high open-circuit voltage. As a result, the open-circuit voltage V OC increases with increasing the band gap energy. When we considering the definition of the energy conversion efficiency in Equation (6.36), the efficiency is dependent only of both the short-circuit current and the open-circuit voltage since the fill factor FF is a function of the open-circuit voltage as shown in Equation (6.35). Therefore, the two counteracting features allow the maximum efficiency with respect to the band gap energy. We are herein going to develop these two parameters in terms of the band gap energy for an ideal solar cell using the already developed equations. The ideal solar cell would be achieved by primarily assuming that the external quantum efficiency is unity. This means that every incident photon that has greater energy than the band gap energy creates an electron-hole pair. Therefore, from Equation (6.314) we have the spectral short-circuit current as I C ( ) = qaf ( ) (6.316) Then, the ideal short-circuit current becomes I C = qaf ( ) d (6.317) By converting the band gap energy into the wavelength using Equations (6.1) and (6.), the band gap wavelength is expressed by

18 hc BG ( E g ) = (6.318) E g We rewrite Equation (6.137) in terms of this as I C ( E ) g = qaf BG E g ( d (6.319) ) This indicates that I C is a function of E g. Now we take a look at the open-circuit voltage in Equation (6.94a) kt I C V = ln OC (6.94a) q I And from Equation (6.9a), qv I = I C I exp 1 (6.9a) kt J is found in Equation (6.89). Multiplying J by the area gives the dark saturation current I as qd p YP ni qdn YN ni I = A + (6.3) Lp X P N D Ln X N N A Using Equation (6.66) for the intrinsic carrier concentration n i, we rewrite Equation (6.3). D p Y E P 1 Dn YN 1 g I = qan cn v + exp (6.31) Lp X P N D Ln X N N A kt For the ideal solar cell, some parasitic assutions deduced from the primary assution of the unity of the EQE come to follow. First we assume that the base width W P is infinite and the depletion width W D is zero so that the absorption is perfect and no recombination exists in the depletion region. W = (6.3a) P W =, x = and x = (6.3b) D N P

19 The ideal solar cell has no front and back surface recombination, so we have F, eff = BF = (6.3c) It is assumed for the ideal solar cell that the diffusion length is sufficiently large and uniform in both the emitter and base regions. L = L L (6.3d) n p = Likewise, the carrier lifetimes are long and the diffusion lengths are large, so that we have τ n = τ p = τ and Dn = Dp = D (6.3e) We can reasonably assume that the emitter width W N is much smaller than the diffusion length L. W N << L (6.3f) Also the product of the high absorption coefficient and the large diffusion length lead to α L >> 1 (6.3g) where α is the absorption coefficient. Actually, if we conduct the derivation for the shortcircuit current I C using Equations (6.86) to (6.88) with the above assutions, we can come up with the unity of the external quantum efficiency as we already assumed in the earlier part of this section. Equations (6.74) to (6.77), after some algebra, lead to YN WP = tanh 1 (6.33) X L N YP WN = tanh (6.34) X L P Equation (6.31) becomes D Eg Eg I = = qan cn v exp C exp LN (6.35) A kt kt where C is the coefficient as

C = qan D cn v (6.35a) LN A The fraction term mostly affect the coefficient C O (the smaller the C O, the larger the V OC, see Equation (6.37)) and can be expressed using Equations (6.7) and (6.159) by D LN A = N A D = Dτ 1 N A D = τ 1 N A kt µ q τ = kt q µ τ N A (6.35b) The lifetimes of electrons and holes are intrinsically limited by the radiative combination in the semiconductor. One can assign favorable values to the mobility and lifetime and calculate the efficiency as a function of the band gap energy. Combining this with Equation (6.94a), the open-circuit voltage is derived by V = OC = kt I ln q I kt I ln q C C C = kt q Eg exp = kt ln C kt q I C E exp kt I ln C C g E g + ln exp kt (6.36) Finally, we have an expression for the open-circuit voltage as a function of E g. ( E ) 1 I C g V = + ln OC Eg Eg kt (6.37) q C V OC is usually less than E g so that I C should be less than C O (it is true). Hence, the logarithm has the negative values. Now we recall the definition of the fill factor FF in Equation (6.35) and rewrite as FF ( V ) ( E ) qv ( E ) qvoc g OC g ln + 1 kt kt OC (6.38) qvoc ( Eg ) + 1 kt The power conversion efficiency defined in Equation (6.36) is expressed as

in ( V ) I ( E ) V ( E ) I V FF OC C g OC g η ( Eg ) = = (6.39) P P in where P in is the incident power that is obtained by Equation (6.37) as 1 P in = A hν ( ) f ( d (6.37) ) The summary for the power conversion efficiency for the AM1.5G spectral irradiance with 1 sun at 3 K is presented as follows in ( V ) I ( E ) V ( E ) I V FF OC C g OC g η ( Eg ) = = (6.33a) P P I C ( E ) g = qa in f ( d (6.33b) BG E g ) P in = A hν ( ) f ( d (6.33c) ) ( E ) 1 I C g V = + ln OC Eg Eg kt (6.33d) q C D C = qancn v (6.33e) LN FF ( V ) A ( E ) qv ( E ) qvoc g OC g ln + 1 kt kt OC (6.33f) qvoc ( Eg ) + 1 kt We now consider the concentration of 1 sun on the ideal solar cell. The power conversion efficiency can be easily obtained by applying that the number of incident photons f () is sily multiplied by 1 due to the concentration of the solar irradiation, which is shown in Equations (6.331b) and (6.331c), respectively. The summary for the power conversion efficiency for the AM1.5G spectral irradiance with 1 sun at 3 K is presented here. in ( V ) I ( E ) V ( E ) I V FF OC C g OC g η ( Eg ) = = (6.331a) P P in

I C 1 ) d (6.331b) BG E g E = qa [ f ( ] g [ f ( ] P in = A hν ( ) 1 d (6.331c) ) ( E ) 1 I C g V = + ln OC Eg Eg kt (6.331d) q C C = qan N D c v (6.331e) LN A FF ( V ) ( E ) qv ( E ) qvoc g OC g ln + 1 kt kt OC (6.331f) qvoc ( Eg ) + 1 kt The ideal solar cell efficiencies for the AM1.5G spectral irradiance with 1 sun and 1 sun at 3 K are presented in Figure 6.6. In considering the AM1.5G 1 sun, the maximum possible efficiency of about 31% occurs between 1. ev and 1.5 ev in band gap energy for a non-concentrated ideal solar cell. More interestingly, the GaAs band gap of 1.4 ev closely matches the band gap of the ideal maximum possible efficiency. The i band gap of 1.1 ev shows about 3% efficiency that is slightly less than the maximum. However, these two materials show a good selection as solar cell. Actually the most efficient single-junction solar cell, made of GaAs, has achieved an efficiency of 5.1% of AM1.5G spectrum by Green [19] in 1. The world record of a singlejunction crystalline i solar cell was reported to be 4.7% by UNW (ydney, Australia) in 1999.

3 4 35 AM1.5G 1 sun 3 GaAs i 5 η (%) 15 Ge AM1.5G 1 sun 1 5.5 1 1.5.5 3 3.5 4 Band gap energy E g (ev) Figure 6.6 Ideal solar cell efficiency as a function of band gap energy for spectral distribution AM1.5 with 1 sun and 1 sun at 3 K. ince the ratio I C /P in in Equation (6.331a) is independent of the concentration of the incident power (1 is cancelled in the fraction), only V OC in Equation (6.331a) contributes to the increase of η. Equation (6.331) suggests that the power conversion efficiency can be increased by enhancing the intensity of the incident sunlight. The increase is most effective near the maximum. Figure 6.6 shows that the maximum increases to about 37%. In 1961, hockley and Queisser [] studied the efficiency upper limit of an ideal solar cell. Their results for both AM1.5G and AM1.5D spectrum with 1 sun are in good agreement with the present work although both used different methods in analysis.

4 6.1 Modeling 6.1.1 Modeling for a ilicon olar Cell We want to develop a MathCAD model to solve the solar cell equations derived in the preceding sections, so that we can study the current-voltage characteristic of a solar cell and the figures of merit, and develop the design concept of a solar cell. Typically, nine parameters are linked each other and govern the performance of a solar cell, which are cell teerature, minority carrier lifetimes, emitter and base widths, front and back surface recombination velocities, and donor and acceptor concentrations. In this model, we want to express the cell current as a function of only cell teerature among the nine parameters. We consider a silicon solar cell with typical operating conditions. MathCAD format solution: We first define the physical constants for a silicon solar cell. The equation number begins newly with model number as M.1. e c := 1.6177331 19 C Electron charge k B 8.6173861 5 e c V := Boltzmann constant K q := e Electron charge c m e := 9.19391 31 kg Electron mass h P := 6.6681 34 J s Planck's constant m n m p := 1.18 m Effective mass of electron e :=.81 m Effective mass of hole e ε = 8.854 1 14 C V 1 cm 1 Permittivity of vacuum (default) K s := 11.8 emiconductor dielectric constant of i ε p := K s ε Electric permittivity for i (definition) (M.1) The following silicon solar cell model parameters are defined as A cell 1cm := olar cell illumination area N D := 1 1 18 cm 3 Number of donors per unit volume (M.) N A := 1 1 15 cm 3 Number of acceptors per unit volume

5 Feff 3 1 4 cm := Effective front surface recombination velocity s BF := 1 cm Effective back surface recombination velocity s W N :=.35µm Emitter width of a solar cell W P := 3µm Base width of a solar cell τ n := 35µs Lifetimes of electrons τ p := 1µs Lifetime of holes s f :=.5 Grid shading factor Blackbody olar Irradiance The un has an effective blackbody surface teerature of 5777K. T sol := 5777K The un's blackbody emittance (W/m µ m) is obtained by the Planck's law. (M.3) f b ( ) := 5 exp π h P c h P c k B T sol 1 (M.4) The un's radius and the Earth's orbit radius are known as r s r o := 6.95981 8 m un's radius := 1.49597891 11 m Earth's orbit radius The Earth receives a small fraction of the un's energy. The fraction is (M.5) (M.6) r s F E := F r E =.164 1 5 o (M.7) Extraterrestrial blackbody solar irradiation between the un and the Earth is called solar constant G sc. We calculated the solar constant to be 1367 W/m that is an irradiation at space outside the Earth s atmosphere.

6 G sc ( ) := F E f b ( ) 1µm G sc := G sc ( ) d G sc = 1367 W.1µm m (M.8) (M.9) Incident Photon Flux f ( ) Measured incident photon fluxes are iorted from a structured ACII data file, where f 1 and f are AM1.5 Global and AM1.5 Direct, respectively. READPRN (MathCAD commend) reads the file and returns a matrix M. By default, ORIGIN is, but we want to change its value to 1. The green underline under the ORIGIN indicates when we redefine an already defined quantity in MathCAD. ORIGIN:= 1 M := READPRN ("solar radiation AM1-5.txt" ) (M.1) (M.11) A portion of the data matrix is displayed by default and the full data can be seen by clicking and scrolling the matrix in MathCAD. M = 1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 AM1.5G AM1.5D 1 3 35 9.5 3.4 31 4.3 15.6 315 17.8 41.1 3 181 71. 35 46.8 1. 33 395.3 15.4 335 39.1 155.6 34 435.3 179.4 345 438.9 186.7 35 483.7 1 36 5.3 4.5 37 666. 34 38 71.5 36.4 39 7.7 381.7 4 1.1313 556 41 1.15813... Figure M.1 Iorted data for AM1.5G spectrum. tx reads column 1 and adds the unit of nanometer (nm). The superscript symbol can be obtained in the Programming Toolbar icon and the number in the symbol indicates the column number in the matrix M.

7 tx M 1 := nm (M.1) The cubic spline interpolation is applied for a smooth interpolation between two data points. And the unit of the spectral incident power is added. hy M := W m µm 1 hs := lspline( tx, hy) (M.13) (M.14) f 1 ( ) interpolates the AM1.5G spectrum data for a specific wavelength. f 1 ( ) := interp( hs, tx, hy, ) (M.15) The ranges of the wavelengths for the AM1.5G spectrum and the un's blackbody irradiances are defined for plotting 1 M 1 := nm.1µm,.1µm.. 3µm := (M.16) Incident Power (W/m^micron) G sc ( ) W m µm 1 f 1 ( 1 ) W m µm 1 1 3 1 1 3 Blackbody at 5777 K AM1.5 Global.5 1 1.5.5 3 1, µm µm Wavelength (µm) Figure M. olar spectra, blackbody irradiance at 5777 K and AM1.5G spectrum. The AM1.5G spectrum measured on Earth shows good agreement with the theoretical blackbody irradiation. The more detailed description for the figure may be found in ection 6.1.4.

8 The accidental incident photon flux less than zero from either the Matrix data or the cubic spline interpolation has no physical meaning (actually deteriorates the results) and forced to be zero for correct numerical coutations. f 1 ( ) := f 1 ( ) if f 1 ( ) > W m µm 1 W m µm 1 otherwise (M.17) We can obtain the incident photon flux (number of photon per unit area per second per unit wavelength) by dividing the AM1.5G data f 1 () by the photon energy shown in Equation (6.1). The incident photon flux is written by f ( ) := f 1 ( ) c h P (M.18) Assume that the front surface of the solar cell is passivated with anti-reflective material, which gives approximately the low reflectance. The reflectance r() is given by r( ) :=.1 (M.19) Band Gap Energy Dependence on Teerature The band gap energy is a function of teerature as shown in Equation (6.38). The band gap coefficients for silicon semiconductor are found in Table 6.1. Eg := 1.17 e c V α 1 4.73 1 4 e c V := β K 1 := 636K (M.) From Equation (6.38), α 1 T o Eg E g T o := E T o + β g ( 3K) = 1.15 e c V 1 (M.1) Absorption Coefficient α (cm -1 ) We herein develop a semi-eirical correlation based on the theory for modeling and coaring it with the measured absorption coefficients. The following semi-eirical correlation is obtained by curve fitting.

9 1 A d.3 1 6 cm 1 := e c V E gd :=.9 e c V A i 1391cm 1 := e c V hν 1 := 1.87 1 e c V c h P E gd α d ( ) := A d c h P α in (, T o ) := c h P exp 3 E g ( T o ) + hν 1 hν 1 k B T o 1 + c h P E g ( T o ) hν 1 1 exp hν 1 k B T o (M.) (M.3) (M.4) (M.5) The spectral absorption coefficient as a function of both wavelength and teerature is expressed by := Re( α d ( ) ) + A i α in (, T o ) α, T o (M.6) The first term in Equation (M.6) takes only the real part of the term for the curve fitting. The absorption coefficient that its photon energy is less than the band gap energy is vanished. c := α(, T o ) if h P α, T o cm 1 otherwise > E g ( T o ) (M.7) Experimental data of absorption coefficients are iorted by reading an external ACII file for ilicon and coared with the semi-eirical correlation. Z := READPRN ("absorption_coeff_c_i.txt" ) (M.8) data Z 1 := µm αdata Z 1 := cm (M.9) (M.3) 1 :=.1µm,.1µm.. 1.3µm (M.31)

3 The semi-eirical correlation developed is coared with the measurement with good agreement as shown in Figure M.3 below, so we have a realistic absorption coefficient for our model. There was actually an available correlation in the literature [13] but somehow no agreement between their proposed correlation and the measurements was found in the present work. Therefore, a new correlation had to be developed for this model. Absorption Coefficient [1/cm] 1 1 7 1 1 6 1 1 5 1 1 4 1 1 3 1 1 Correlation Experiment 1..4.6.8 1 1. Wavelength [ µm] Figure M.3 Absorption coefficient: semi-eirical correlation and measurements. Intrinsic Carrier Concentration The intrinsic carrier concentration was derived and presented in Equation (6.66). Also the effective density of state of the conduction band is found in Equation (6.6). N c T o π m n k B T o := N c ( 3K) = 3.17 1 19 cm 3 h P 3 (M.3) The effective density of state of the valence band is found in Equation (6.63) N v T o π m p k B T o := N v ( 3K) = 1.89 1 19 cm 3 h P 3 (M.33) The intrinsic carrier concentration from Equation (6.66) is written here.

31 N c ( T o ) N v ( T o ) n i T o E g T o := exp n k B T i ( 3K) = 8.697 1 9 cm 3 o (M.34) The numerical value of the intrinsic carrier concentration for silicon semiconductor is in agreement with the measurements. Built-in Voltage V bi The built-in voltage from Equation (6.97) is written here and calculated but cannot be measured. V bi ( T o ) k B T o N D N A := ln V q bi ( 3K) =.781V n i ( T o ) (M.35) Dimensions of the Depletion Region or the pace Charge Region The schematic of the depletion region can be found in Figure 6.31. x N, x P, and W D in Equations (6.187a), (6.188a), and (6.189a) are rewritten here x N ( T o, V L ) x P T o, V L := ε p q N A N D N A + N D x N ( T o, V L ) N D := W D T o, V L ( V bi ( T o ) V L ) N A := x N ( T o, V L ) + x P ( T o, V L ) 1 (M.36) (M.37) (M.38) Assuming that V=.57 V, the numerical values are x N ( 3K,.57V ) = 5.754 1 4 µm x P ( 3K,.57V ) =.575 µm W D ( 3K,.57V ) =.576 µm (M.39) (M.4) (M.41) Carrier Mobility, µ Carrier mobility µ depends on both doping and teerature. The carrier mobility in silicon is well approximated with measurements. The eirical correlation was developed as follows. You may find more information in ection 6.8..

3 9 µ n1 T o N D1 T o µ p1 T o T o cm := µ 3K V s n T o 1.3 1 17.57 T o := cm 3 α 3K n T o 54.3 N A1 T o.4 T o cm := µ 3K V s p T o.35 1 17.57 T o := cm 3 α 3K p T o.4 := 168 :=.91 := 46.9 :=.88 T o 3K T o 3K T o 3K T o 3K.33.146.3.146 cm V s cm V s (M.4) (M.43) (M.44) (M.45) For electron mobility, := µ n1 ( T o ) µ n N D, T o For hole mobility, := µ p1 ( T o ) µ p N A, T o + + 1 + 1 + µ n ( T o ) N D N D1 T o µ p ( T o ) N A N A1 T o α n T o α p T o N1 D := 1 14 cm 3, 1 15 cm 3.. 1 cm 3 N1 A := 1 14 cm 3, 1 15 cm 3.. 1 cm 3 (M.46) (M.47) (M.48)

33 1 1 4 µ n ( N1 D, 3K) cm V s µ p N1 A, 3K cm V s 1 1 3 1 Hole Electron 1 1 1 14 1 1 15 1 1 16 1 1 17 1 1 18 1 1 19 1 1 N1 D N1 A, cm 3 cm 3 Figure M.4 Electron and hole mobility for silicon at 3 K (see Figure 6.3) We want that the carrier mobilities only as a function of teerature for convenience. := µ n ( N D, T o ) µ n T o := µ p ( N A, T o ) µ p T o (M.49) (M.5) Minority Carrier Diffusion Coefficients D n and D p The diffusion coefficients for electrons and holes are defined by the Einstein relationship in Equations (6.16) and (6.159). µ n ( T o ) k B T o D n T o D p T o := D q n ( 3K) = 6.87 cm s := D q p ( 3K) = 11.838 cm s µ p ( T o ) k B T o (M.51) (M.5) Minority Carrier Diffusion Lengths L n and L p The minority carrier diffusion lengths for electrons and holes are defined as in Equations (6.8) and (6.7). D n ( T o ) τ n L n T o := L n ( 3K) = 488.15µm (M.53)

34 D p ( T o ) τ p L p T o := L p ( 3K) = 34.46µm (M.54) ome groups of quantities related to the recombination as shown in Equations (6.74) through (6.77) are defined herein. X P ( T o, V L ) Y P ( T o, V L ) X N ( T o, V L ) Y N ( T o, V L ) Feff := D p T o Feff := D p T o sinh cosh BF := D n T o BF := D n T o sinh cosh W N W N W P W P x N T o, V L L p T o x N T o, V L L p T o x P T o, V L L n T o x P T o, V L L n T o + + + + 1 cosh L p T o 1 sinh L p T o 1 cosh L n T o 1 sinh L n T o W N W N W P W P x N T o, V L L p T o x N T o, V L L p T o x P T o, V L L n T o x P T o, V L L n T o (M.55) (M.56) (M.57) (M.58) By considering photon energy in Equation (6.1), determine the band gap wavelength that is an absorbable limit and contributes to the photocurrent. BG ( T o ) h P c := E g T o (M.59) Define the ranges of wavelength for definite integrals, treating the singularity in the denominator by dividing it into two integrals to avoid the singularity. i :=.1µm BG ( 3K) = 1.13 µm (M.6) The groups of quantities related to photogeneration were presented in Equations (6.78) through (6.81), where a singularity appears in each denominator of the equations. This causes a problem in smooth numerical coutations. Therefore, the singularity point must be found by using a MathCAD built-in function called 'root function'. And the point is eliminated by separating the integrals at the point. A guess value (any reasonable number) is needed for the MathCAD root function. You may always try a most reasonable guess value for the best result. :=.6µm (M.61)

35 The root functions and the singularity points P (T o ) and n (T o ) for holes and electrons are written by root α(, T o ) L p ( T o ) p T o n T o := 1, p ( 3K) =.888 µm root α(, T o ) L n ( T o ) := 1, n ( 3K) = 1.34 µm (M.6) (M.63) With the given parameters in the beginning of this section, the couted singularity points appear less than the band gap wavelength of 1.13 µm calculated in Equation (M.6). Therefore, the singularity points are in effective during the coutations. However, it is noted that the singularity points will be changing with changing teerature, minority carrier lifetimes or dopants concentrations. This must be ilemented into the coutations. In the present section of problem solution, since we have the fixed dopants concentrations (N A and N D ) and the fixed lifetimes (τ n and τ p ), the singularity points are dependent only on teerature. The groups of quantities related to photogeneration shown in Equations (6.78) through (6.81) are defined here as. G p1 T o, x, G p1 T o, x := := ( 1 s f ) 1 s f α, T o τ p L p ( T o ) i 1 BG ( T o ) i p ( T o ).999 G p1 T o, x, ( 1 r( ) ) f ( ) α, T o exp α, T o x + W N d G p1 T o, x, if < p ( T o ) BG T o BG ( T o ) d + p T o 1.1 G p1 T o, x, d (M.64) otherwise (M.65) The vertical line (Add Line) in Equation (M.65) allows the MathCAD built-in program where we can execute if-do-otherwise-do task. G p T o, x, := α, T o τ p L p ( T o ) 1 ( 1 r( ) ) f ( ) α, T o exp α, T o x + W N (M.66)

36 := ( 1 s f ) G p T o, x G n1 T o, x, G n1 T o, x 1 s f := := ( 1 s f ) G n T o, x, 1 s f := := ( 1 s f ) G n T o, x 1 s f i BG ( T o ) i α, T o p ( T o ).999 τ n L n ( T o ) i G p T o, x, 1 BG ( T o ) i α, T o n ( T o ).999 τ n L n ( T o ) i d G p T o, x, G n1 T o, x, 1 BG ( T o ) i n ( T o ).999 Current-Voltage Characteristic From Equation (6.86), we define if < p ( T o ) BG T o BG ( T o ) d + p T o 1.1 G p T o, x, d ( 1 r( ) ) f ( ) α, T o exp α, T o x + W N d G n1 T o, x, G n T o, x, if < n ( T o ) BG T o BG ( T o ) d + n T o 1.1 G n1 T o, x, d ( 1 r( ) ) f ( ) α, T o exp α, T o x + W N d G n T o, x, if < n ( T o ) BG T o BG ( T o ) d + n T o 1.1 otherwise (M.67) G n T o, x, d (M.68) otherwise (M.69) (M.7) otherwise (M.71) J CN1 ( T o, V L ) := L p T o q Feff X P ( T o, V L ) G p1 T o, W N + q D p ( T o ) X P ( T o, V L ) L p T o G p T o, W N (M.7)

37 J CN ( T o, V L ) J CN T o, V L X P T o V L q D p T o Y P T o, V L := G L p T o (, ) p1 T o, x N T o, V L := J CN1 ( T o, V L ) + J CN ( T o, V L ) From Equation (6.87), J CP1 ( T o, V L ) J CP ( T o, V L ) J CP T o, V L := L n T o q BF X N ( T o, V L ) q D n T o := L n T o Y N ( T o, V L ) X N ( T o, V L ) ( ) G n1 ( T o, W P ) := J CP1 ( T o, V L ) + J CP ( T o, V L ) From Equation (6.88), DEXP T o, V L, L n T o ( ) G n1 T o, x P T o, V L := exp α(, T o ) ( x N ( T o, V L ) + W N ) := q ( 1 s f ) J CD T o, V L ( ) q D p T o G p T o, x N T o, V L q D n ( T o ) X N ( T o, V L ) exp α, T o i BG ( T o ) ( 1 r( ) ) f ( ) DEXP T o, V L, G n ( T o, W P ) ( ) + q D n T o G n T o, x P T o, V L ( x P ( T o, V L ) + W N ) d (M.73) (M.74) (M.75) (M.76) (M.77) (M.78) (M.79) The short circuit current density which equals the photocurrent is the sum of the above three equations as shown in Equation (6.85). := J CN ( T o, V L ) J CP ( T o, V L ) J C T o, V L + + J CD T o, V L (M.8) We want to curiously take a look at the numerical values for the photocurrents in the n- type and p-type regions. It is interesting to note from the following values in Equations (M.8a) and (M.8b) that the photocurrent of the p-type region produces about 6 times more than that of the n-type region, while the width of the p-type region is about 1 times greater than that of the n-type region. This is associated with the solar cell parameters such as the surface recombination velocities, minority carrier lifetimes or diffusion lengths. J C ( 3K,.57V ) =.37A cm A cell J CN ( 3K,.57V ) =.437A (M.8a) (M.8b)

38 A cell J CP ( 3K,.57V ) =.448A (M.8c) The dark saturation current density due to the recombination in the quasi-neutral region shown in Equation (6.89) is defined. J 1 ( T o, V L ) := q D p T o L p T o Y P T o, V L X P T o, V L n i ( T o ) N D + q D n T o L n T o Y N T o, V L X N T o, V L n i T o N A (M.81) J 1 ( 3K,.57V ) = 1.869 1 1 A cm (M.81a) The dark saturation current density due to the recombination in the depletion region shown in Equation (6.9) is defined. J ( T o, V L ) q W D T o, V L := τ p + τ n n i ( T o ) (M.8) The cell current density which is the current produced by solar cell shown in Equation (6.84) is defined here. := J C ( T o, V L ) J 1 ( T o, V L ) exp J cell T o, V L q V L k B T o 1 exp J T o, V L q V L k B T o 1 (M.83) Currents are now obtained by multiplying the solar cell area and the numerical values for coarison. := A cell J C ( T o, V L ) I C T o, V L := A cell J 1 ( T o, V L ) I 1 T o, V L := A cell J ( T o, V L ) I T o, V L := A cell J cell ( T o, V L ) I cell T o, V L := I cell ( T o, V L ) V L Power T o, V L (M.84) (M.85) (M.86) (M.87) (M.88) The numerical values for the physical interpretations are shown here I C ( 3K,.57V ) = 3.66A (M.89)

39 I 1 ( 3K,.57V ) exp q.53v k B 3K 1 =.16A I ( 3K,.57V ) exp I cell ( 3K,.57V ) = 3.13A Power( 3K,.57V ) = 1.651W q.53v = 6.75 1 4 A k B 3K 1 (M.9) (M.91) (M.9) (M.93) Notice that the dark current density due to the depletion region I is very small and usually neglected. The range of voltages for the I-V curve is defined. We better estimate the number of iteration as dividing the full scale of 1 V by the interval of. V which gives 5 iterations. Assuming that each iteration takes 5 seconds, we have total 5 seconds (4 minutes 1 seconds) in coutations. V1 L := V,.V.. 1V (M.94) Figure M.5 Current-voltage curve for a silicon solar cell Note that the I-V characteristic varies with teerature and the maximum power point on the power curve. ee Figure 6.37 and the more discussion in ection 6.1.. Figures of Merit

4 We found numerically that the open-circuit voltage is independent of the cell voltage. o zero voltage is used to coute the open-circuit voltage. From Equation (6.94), we have V oc ( T o ) := k B T o q ln I C T o, V I 1 T o, V + 1 (M.95) The short-circuit current in Equation (6.9) is equal to the cell current when the voltage is set to zero, so that the short-circuit current is no longer a function of voltage. Therefore, we can write I cell ( T o, V) I C T o := I C ( 3K) = 3.68A (M.96) The maximum power voltage and current in Equations (6.99) and (6.31) are rewritten here := V oc ( T o ) V T o I T o k B T o q ln q V oc ( T o ) k B T o := I C ( T o ) I 1 ( T o, V ( T o )) exp The numerical values are + 1 q V oc ( T o ) k B T o ln q V oc ( T o ) k B T o + 1 (M.97) (M.98) V oc ( 3K) =.61V V ( 3K) =.57V I ( 3K) = 3.135A (M.99) (M.1) (M.11) The explicit approximate expressions such as V OC, I C, V MP, and I MP significantly reduce the coutational time with tolerable uncertainties (within about 1% error). There is usually a good agreement between the exact numerical calculations and the approximate calculations. The maximum power will be I ( T o ) V ( T o ) P T o := P ( 3K) = 1.65W (M.1) The total incident solar power is obtained using Equation (6.37) with Equation (6.).