Modeling of InGaN p-n junction solar cells

InGaN p-n junction solar cells with various indium composition and thickness of upper p-InGaN and lower n-InGaN junctions are investigated theoretically. The physical properties of InGaN p-n junction solar cells, such as the short circuit current density (JSC), open circuit voltage (Voc), fill factor (FF), and conversion efficiency (η), are theoretically calculated and simulated by varying the device structures, position of the depletion region, indium content, and photon penetration depth. The results indicate that an In0.6Ga0.4N solar cell, with optimal device parameters, can have a JSC ~31.8 mA/cm, Voc ~0.874 volt, FF ~0.775, and η ~21.5%. It clearly demonstrates that medium-indium-content InGaN materials have the potential to realize high efficiency solar cells. Furthermore, the simulation results, with various thicknesses of the pInGaN junction but a fixed thickness of the n-InGaN junction, shows that the performance of InGaN solar cells is determined by the upper p-InGaN junction rather than the n-InGaN substrate. This is attributed to the different amount of light absorption in the depletion region and the variation of the collection efficiency of minority carriers. © 2013 Optical Society of America OCIS codes: (040.5350) Photovoltaic; (350.6050) Solar energy. References and links 1. E. F. Schubert, Light Emitting Diodes (Cambridge University Press, 2006). 2. A. Yamamoto, M. R. Islam, T.-T. Kang, and A. Hashimoto, “Recent advances in InN-based solar cells: status and challenges in InGaN and InAlN solar cells,” Phys. Status Solidi C 7(5), 1309–1316 (2010). 3. S. W. Feng, C. M. Lai, C. H. Chen, W. C. Sun, and L. W. Tu, “Theoretical simulations of the effects of the indium content, thickness, and defect density of the i-layer on the performance of p-i-n InGaN single homojunction solar cells,” J. Appl. Phys. 108(9), 093118 (2010). 4. X. Zhang, X. L. Wang, H. L. Xiao, C. B. Yang, J. X. Ran, C. M. Wang, Q. F. Hou, and J. M. Li, “Simulation of In0.65Ga0.35N single-junction solar cell,” J. Phys. D Appl. Phys. 40(23), 7335–7338 (2007). 5. L. Hsu and W. Walukiewicz, “Modeling of InGaN/Si tandem solar cells,” J. Appl. Phys. 104(2), 024507 (2008). 6. X. Shen, S. Lin, F. Li, Y. Wei, S. Zhong, H. Wan, and J. Li, “Simulation of the InGaN-based tandem solar cells,” Proc. SPIE 7045, 70450E, 70450E-8 (2008). 7. O. Jani, I. Ferguson, C. Honsberg, and S. Kurtz, “Design and characterization of GaN/InGaN solar cells,” Appl. Phys. Lett. 91(13), 132117 (2007). 8. J. R. Lang, C. J. Neufeld, C. A. Hurni, S. C. Cruz, E. Matioli, U. K. Mishra, and J. S. 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Introduction
The bandgap of InGaN wide-bandgap semiconductors, ranging from 0.7 to 3.4 eV, can fit the full solar spectrum [1].This provides InGaN with a great potential for photovoltaic applications; especially, when they are used in multi-junction tandem solar cells in which a bandgap between 0.7 and 2.5 eV can be selected, by changing their compositions, to optimize the devices' efficiency and performance [2].Although InGaN solar cells are still not fully developed, various theoretical models and numerical simulations have been conducted to investigate the performance of single-and multiple-junction InGaN solar cells [3][4][5][6].Our previous simulation results show that the performance of InGaN p-i-n solar cells critically depends on the indium content, thickness, and defect density of the i-layer [3] and a highquality In 0.75 Ga 0.25 N solar cell with a 4 μm i-layer thickness can exhibit 23% conversion efficiency.Other works have shown that an In 0.65 Ga 0.35 N p-n junction solar cell with optimized doping concentration and thickness can have 20% conversion efficiency [4].It also has been shown that a high quality InGaN/Si tandem solar cell with optimized InGaN bandgap and Si thickness was estimated to have 30-32% conversion efficiency [5].Single-, double-, and triple-junction InGaN solar cells were calculated to exhibit 24.95, 34.44, and 41.76% conversion efficiencies, respectively [6].
On the other hand, device fabrications of various InGaN solar cells have been conducted with some interesting and promising results.For example, p-GaN/i-InGaN/n-GaN heterojunction [7][8][9], p-InGaN/i-InGaN/n-InGaN homojunction [10], p-InGaN/n-InGaN homojunction [11], and InGaN/GaN multiple quantum well [12,13] solar cells have been demonstrated to show good photovoltaic effects.However, due to high densities of threading dislocations, stacking faults, and V-shaped defects, the conversion efficiencies of those solar cells are lower than 2% [7][8][9][10][11][12][13]. Piezoelectric polarization effects can also reduce the efficiencies of InGaN/GaN solar cells [14,15].Possible solutions to the challenges in InGaN solar cells have been proposed: conductive and transparent substrates, high quality film growth, p-type doping, and cell design [2].Although the experimental efforts on InGaN solar cells are still in the initial stages, theoretical studies could provide useful insight and possible guidelines to optimize their performance.For example, the influences of various device structures and the indium composition on the performance of InGaN p-n junction solar cells InGaN solar cells deserve careful investigations.Although, previous theoretical works have been conducted and some interesting results have been obtained, several important factors are still not fully investigated, such as the position of the junction.Therefore, in this study, the physical properties of InGaN p-n junction solar cells, such as the short circuit current density, open circuit voltage, fill factor, and conversion efficiency, are theoretically calculated and simulated by varying the device structures, position of the depletion region, indium content, and photon penetration depth.
This paper is organized as follows: In section 2, theoretical modelling is described.In section 3, simulation results of the performance of InGaN p-n junction solar cells are discussed.Finally, conclusions are drawn in section 4.  In the numerical simulations, the theoretical model is used to design the structures of InGaN p-n junction solar cells with various widths and indium compositions.The firstprinciples-continuity and Poisson's equations are combined to analyze the transport behavior of the solar cells [16].The photovoltaic function of an InGaN p-n junction can be analyzed by solving a set of coupled differential equations for the electron density, hole density, and electrostatic potential [16].Carrier and current densities can be analytically obtained to determine the current-voltage (J-V) curve of the InGaN p-n junction solar cells.The total current density, J, in the InGaN p-n junction solar cells can be expressed as [16]: where J SCP is the hole diffusion current density in the p-InGaN junction, J SCN is the electron diffusion current density in the n-InGaN junction, and J G,D is the drift current density in the depletion region.J DP , J DN , and J DD are the dark current densities in the p-InGaN junction, n-InGaN junction, and depletion region, respectively.V a is the built-in potential.Each term of J SCP , J SCN , J G,D , J DP , J DN , and J DD in Eq. ( 1) can be obtained in Reference [16].From Eq.

J e
− is the recombination current in the depletion region.Details of the calculations of total current density, J, are described in Reference [16].
Assuming that the recombination current in the depletion region The fill factor, FF, is defined as: The power conversion efficiency of a solar cell, η, is defined as: The intrinsic carrier concentration, n i , can be described by [3]: The band-gap energy, E g (x), for In x Ga 1-x N is expressed as [1]: Donor and acceptor concentrations for InN and GaN are both set at 5 × 10 17 cm −3 [4].Hole and electron surface recombination velocities for InN and GaN are both set at 10 3 (cm‧s −1 ).Except for the band gap energy, the physical parameters of In x Ga 1-x N are expressed as the linear interpolation formula of wurtzite InN and GaN and are listed in Table 1 [1,4,[17][18][19][20][21].
80 [21] 39 [4] Operation mechanisms of InGaN p-n junction solar cells are explored through the calculation of characteristic parameters such as the short circuit current density (J sc ), open circuit voltage (V oc ), fill factor (FF), and conversion efficiency (η).Two situations are considered for theoretical simulation: (I) Situation I: the dependence on the width (w p = 50-4,000nm) and the indium composition (x p = 0-1) of the upper p-InGaN junction.The width of the lower n-InGaN substrate is set at 1,000 nm.
(II) Situation II: the dependence on the width (w n = 50-4,000nm) and the indium composition (x n = 0-1) of the lower n-InGaN substrate.The p-InGaN width is set at 300 nm.

The effects of the width and the indium composition of the upper p-InGaN on the performance of InGaN p-n junction solar cells
In simulation I, the effects of the width (w p = 50-4,000nm) and indium composition (x p = 0-1) of the upper p-InGaN junction on the performance of InGaN p-n junction solar cells are investigated.Figure 2 shows the J-V characteristic of InGaN p-n junction solar cells with 300 nm p-InGaN and 1,000 nm n-InGaN for various indium compositions.Dramatic variations of the J-V characteristic, depending on indium content, are obtained.As the indium content decreases, V oc increases, while J sc and FF decrease.The operation mechanism of InGaN solar cells will be discussed later.Figures 3(a) and 3(b) show the short circuit current densities, J sc (w p , x p ), of InGaN p-n junction solar cells as a function of p-InGaN width (w p ) and indium composition (x p ), respectively.In Fig. 3(a), simulation results show that as the w p increases, J sc (w p , x p ) very slightly decreases and then droops at around 1,000 nm.The varied width of the p-InGaN layer changes not only the amount of light absorbed in the depletion region, but also the drift length to the p-contact.Although a thicker p-InGaN layer enhances the absorption and generates more carriers in the p-InGaN, a longer drift length makes it harder for the photogenerated carriers to drift to the p-contact.J sc (w p , x p ) slightly decreases.As the w p is above 1,000 nm, fewer carriers are collected in the p-contact due to both less absorption in the depletion region and longer drift length.The smaller absorption in the n-InGaN layer also makes photongenerated carriers contribute less to photocurrent.Those factors lead to J sc (w p , x p ) drooping above 1,000 nm.Also, J sc (w p , x p ) shows a strong dependence on w p and x p of the solar cells.A larger J sc (w p , x p ) in the high-indium-content InGaN solar cells strongly depends on w p , while smaller J sc (w p , x p ) in the low-indium-content ones weakly depends on it.Due to the photons being incident from the p-InGaN side of the solar cell, photon-generated carriers in the p-InGaN layer of the solar cell contribute more current than those in the n-InGaN.Also, because the penetration depth of longer-wavelength photons in the high-indium-content InGaN solar cells is larger than that of shorter-wavelength photons in low-indium-content ones, photo-generated carriers in the n-InGaN of the high-indium-content solar cells would contribute more current than those of the low-indium-content ones.Those factors lead to the fact that the wider the p-InGaN in the high-indium-content InGaN solar cells, the larger the variation of J sc (w p , x p ).In addition, in Fig. 3(b), J sc (w p , x p ) increases with increasing indium content.J sc (w p , x p ) of the InN solar cell corresponding to the same width of the p-InGaN is about one order of magnitude larger than that of the GaN solar cell.With a small bandgap in the high-indium solar cell, more photons with energy larger than the bandgap can be absorbed.The larger penetration depth of infrared photons also enhances the photon absorption.Smaller bandgap and larger penetration depth of long-wavelength photons enhance absorption such that more photocurrent is generated in the high-indium-content InGaN solar cells.A lower J sc (w p , x p ) of the GaN and low-indium-content InGaN solar cells will lead to a lower conversion efficiency.Furthermore, the reported J sc (0.26 mA/cm 2 ) of p-In 0.168 Ga 0.832 N(35nm)/n-In 0.148 Ga 0.852 N(45nm) junction solar cell under AM 1.5 illumination are plotted with (*) in Figs.3(a) and 3(b) for comparison [11].The reported J sc is much lower than that of the simulation result (in the range of 1.62-4.37mA/cm 2 for In content 10-20%).Because the InGaN p-n junction structure is sandwiched between p-and n-GaN layers, some photons are absorbed by the p-contact and p-GaN layer.Due to high densities of threading dislocations, stacking faults, strain-induced piezoelectric polarization effects, and V-shaped defects, the collection efficiency of the InGaN p-n junction solar cell becomes lower [7][8][9][10][11][12][13][14][15].These factors lead to the reported J sc showing a lower value in the actual fabricated solar cells.
Figures 3(c) and 3(d) show the open circuit voltage, V oc (w p , x p ), of InGaN p-n junction solar cells as a function of p-InGaN width (w p ) and indium composition (x p ), respectively.Simulation results show that the V oc (w p , x p ) depends on x p , but only slightly on w p .The V oc (w p , x p ) is determined by the difference of the Fermi energies (ΔE F ) of the electron and the hole in the depletion region, which in turn is affected by the bandgap energy [16].In Fig. 3(c), except for the wider cells, the V oc is nearly independent of w p .Note that V oc (w p , x p ) starts to decrease slightly in the wider cells, due to the larger saturation current, J 0 , in the wider cells.In Fig. 3(d), because V oc is determined by the bandgap energy of the subcell [17], smaller V oc in the high-In-content InGaN solar cell is expected.Furthermore, the reported V oc (1.47 volt) of p-In 0.168 Ga 0.832 N(35nm)/n-In 0.148 Ga 0.852 N(45nm) junction solar cell under AM 1.5 illumination is plotted with () in Figs.3(c) and 3(d) for comparison [11].The reported V oc is lower than that of the simulation result (in the range of 1.97-2.29 V for In content 10-20%).According to Eq. ( 8), V oc is proportional to J sc .Due to a smaller J sc , a smaller reported V oc is expected.
Figures 4(a) and 4(b) show the fill factor, FF(w p , x p ), of InGaN p-n junction solar cells as a function of p-InGaN width (w p ) and indium composition (x p ), respectively.For GaN solar cells in Fig. 4(a), it is found that the FF(w p , x p ) can be as high as 0.90 and decreases slightly in the wider cells.This is due to the fact that the V oc (w p , x p ) of GaN solar cells shows larger variations with varying p-InGaN width, as shown in Fig. 3(c).In addition, in Fig. 4(b), as the indium content increases, the FF(w p , x p ) decreases and has slightly larger variation with varying p-InGaN width, in particular for InN solar cells.This is due to a larger variation of J sc (w p , x p ) with varying p-InGaN width in the high-indium-content solar cells, as shown in Fig. 3(a).As the indium content increases, the steeply increasing J sc (w p , x p ) 3(b)) and smoothly decreasing V oc (w p , x p ) (Fig. 3(d)) lead to a smaller FF(w p , x p ) in the high-indiumcontent InGaN solar cells, which basically verifies the fact that V oc (w p , x p ) is roughly proportional to FF(w p , x p ) [16].Currently, the reported FF (0.57) of p-In 0.168 Ga 0.832 N(35nm)/n-In 0.148 Ga 0.852 N(45nm) junction solar cell under AM 1.5 illumination is plotted with (*) in Figs.4(a) and 4(b) for comparison [11].Without consideration of the effects of the current leakage and shunt resistance, the simulated FF can be higher than those of the actual fabricated solar cells.
Figures 4(c) and 4(d) show the conversion efficiency, η(w p , x p ), of InGaN p-n junction solar cells as a function of p-InGaN width (w p ) and indium composition (x p ), respectively.Simulation results help us to better understand the operation mechanisms of and provide important information for the structure design of InGaN p-n junction solar cells.In Fig. 4(c), high η(w p , x p ) solar cells show strong dependence on the p-layer width for longer p-layer solar cells, while low η(w p , x p ) ones show slight dependence on that.This trend is similar to the fact that the wider the p-layer width, the larger the variation of the J sc (w p , x p ). Device structures and photon penetration depth proposed to explain the trend of J sc (w p , x p ) can explain that of η(w p , x p ) as well.In addition, in Fig. 4(d), as the indium content in the cell increases, absorption and J sc (w p , x p ) increase, while V oc (w p , x p ) and FF(w p , x p ) decrease.A high η(w p , x p ) in the medium-indium-content InGaN solar cell is expected.The combined effects of the absorption, J sc (w p , x p ), V oc (w p , x p ), and FF(w p , x p ) lead to η(w p , x p ) showing an increasing and then decreasing trend with indium content.Furthermore, η(w p , x p ) can be classified into three regions: (I) low-indium-content (x = 0~0.3),(II) medium-indium-content (x = 0.3~0.7),and (III) high-indium-content (x = 0.7~1).(I) Due to very lower η(w p , x p ), GaN and low-indium-content InGaN solar cells are not suitable for application in solar cells.This explains that the reported low-indium-content InGaN solar cells show low conversion efficiencies [7][8][9][10][11][12][13]. (II) The combined effects of the J sc , V oc , and FF lead to an optimized η in the medium-indium-content InGaN solar cell.Medium-indium-content InGaN solar cells with high η(w p , x p ) are useful for photovoltaic applications.With p-layer width less than 500 nm and 1,000 nm n-InGaN, a high-quality In 0.6 Ga 0.4 N solar cell can exhibit as high a η(w p , x p ) as ~22%.This demonstrates that medium-indium-content InGaN is an appealing candidate to realize a high efficiency solar cell.However, the difficulty of high quality devices would be a potential obstacle to fabricate such solar cells.Growth of In-rich InGaN can be obtained by using high-pressure chemical vapor deposition [22,23].(III) InN and high-indium-content InGaN solar cells could be useful for making tandem solar cells with other material such as Si, provided that it can be grown on that other material.Also, nonpolar and semipolar GaN substrates were studied for the growth of high-indium-content InGaN/GaN multiple quantum wells (MQWs).It has been shown that growth of high-indium InGaN/GaN MQWs on the (1122) and (2021) semipolar planes exhibits a higher indium incorporation [24].Highindium-content InGaN solar cells can be fabricated on the (1122) and (2021) semipolar GaN substrates.Furthermore, the reported η (0.05%) of p-In 0.168 Ga 0.832 N(35nm)/n-In 0.148 Ga 0.852 N(45nm) junction solar cells under AM 1.5 illumination is plotted using () in Figs.4(c) and 4(d) for comparison [11].Also, the conversion efficiencies of p-GaN/i-InGaN/n-GaN heterojunction [7][8][9], p-InGaN/i-InGaN/n-InGaN homojunction [10], and InGaN/GaN MQW [12,13] solar cells with a low-indium-content are lower than 2%.The reported η is much lower than that of the simulation result (in the range of 3.26-7.49%for In content 10-20%).Without consideration of the effects of the current leakage and shunt resistance, the simulated η are higher than those of the actual fabricated solar cells.High densities of threading dislocations, stacking faults, and strain-induced piezoelectric polarization effect can reduce the efficiencies of InGaN/GaN solar cells [7][8][9][10][11][12][13][14][15].

The effects of the width and the indium composition of the n-InGaN substrate on the performance of InGaN p-n junction solar cells
In simulation II, the effects of the width (w n = 50-4,000nm) and the indium composition (x n = 0-1) of the lower n-InGaN substrate on the performance of InGaN p-n junction solar cells are investigated.The p-InGaN width is set at 300 nm.Figures 5(a) and 5(b) show the short circuit current densities, J sc (w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN width (w n ) and indium composition (x n ), respectively.In Fig. 5(a), simulation results show that the J sc very slightly increases with increasing w n .Although a thicker n-InGaN enhances the absorption and generates more carriers in the n-InGaN, a longer drift length makes it harder for the photogenerated carriers to drift to the n-contact.The nearly constant J sc (w n , x n ) as a function of w n can be the balance of a larger absorption and a longer drift length.Also, regardless of the n-InGaN width, J sc (w n , x n ) is equal to J sc (w p , x p ) corresponding to the same indium composition in Fig. 3(a) when the p-InGaN width is smaller than a few hundred nm.Due to larger absorption by InGaN, most light is absorbed by the p-InGaN junction, and less light is absorbed by the n-InGaN substrate.This shows that the J sc of InGaN solar cells is determined by the J sc (w p , x p ) of the upper p-InGaN junction.In addition, in Fig. 5(b), J sc (w n , x n ) increases with increasing indium content.J sc (w n , x n ) of the InN solar cell corresponding to the same width of the n-InGaN is about one order of magnitude larger than that of the GaN solar cell.Both smaller bandgap and larger penetration depth of long-wavelength photons enhance absorption such that more photocurrent is generated in the high-indium-content InGaN solar cells.[16], the V oc (w n , x n ) is nearly independent of w n .Similar to the J sc of InGaN solar cells, the V oc (w n , x n ) is equal to the V oc (w p , x p ) corresponding to the same indium composition in Fig. 3(c) when the p-InGaN width is smaller than a few hundred nm.It is reasonable that the V oc of InGaN solar cells is also determined by the V oc (w p , x p ) of the upper p-InGaN junction.In addition, in Fig. 5(d), V oc (w n , x n ) becomes smaller in the high-indium-content InGaN solar cells.This is due to the difference of the Fermi energies (ΔE F ) of the electron and the hole in the depletion region, which in turn is affected by the bandgap energy [16].
Figures 6 (a) and 6(b) show the fill factor, FF(w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN width (w n ) and indium composition (x n ), respectively.In Fig. 6(a), because the J sc (w n , x n ) (Fig. 5(a)) and V oc (w n , x n ) (Fig. 5(c)) are nearly independent of w n , the FF(w n , x n ) is also nearly independent of w n .Similar to the J sc of InGaN solar cells, the FF(w n , x n ) is nearly equal to the FF(w p , x p ) corresponding to the same indium composition in Fig. 4(a) when the p-InGaN width is smaller than a few hundred nm.The FF of InGaN solar cells is also determined by the FF(w p , x p ) of the upper p-InGaN junction.Also, in Fig. 6(b), the FF(w n , x n ) decreases as the In composition increases.The steeply increasing J sc (w n , x n ) (Fig. 5(b)) and the smoothly decreasing V oc (w n , x n ) (Fig. 5(d)) lead to a smaller FF(w n , x n ), in the high-indium-content InGaN solar cells.Figures 6(c) and 6(d) show the conversion efficiency, η(w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN width (w n ) and indium composition (x n ), respectively.The conversion efficiency represents the combined effects of the J sc , V oc , and FF.In Fig. 6(c), because J sc (w n , x n ), V oc (w n , x n ), and FF(w n , x n ) are nearly independent of the n-InGaN width, it is reasonable that the η(w n , x n ) is also nearly independent of w n .In addition, in Fig. 6(d), the combined effects of the absorption, J sc (w n , x n ), V oc (w n , x n ), and FF(w n , x n ) lead to η(w n , x n ) showing an increasing and then decreasing trend with the indium content.With 300 nm player width and a few hundred nm n-InGaN, a high-quality In 0.6 Ga 0.4 N solar cell can exhibit as high a η(w n , x n ) as ~22%.Also, the relationship between η(w n , x n ) and x n in Fig. 6(d) is similar to that between η(w p , x p ) and x p in Fig. 4(d), but the relationship between η(w n , x n ) and x n is almost independent of w n .Similar arguments can be used to explain this relationship.Because most light is absorbed by the p-InGaN junction, η of InGaN p-n junction solar cells is determined by the η(w p , x p ) in the upper p-InGaN layer, not the η(w n , x n ) in the n-InGaN substrate.

Conclusions
In summary, the J sc , V oc , FF, and η are shown to strongly depend on the indium content and width of the upper p-InGaN junction in the InGaN solar cells.With proper thicknesses of pand n-InGaN, an In 0.6 Ga 0.4 N solar cell can exhibit a J sc ~31.8 mA/cm 2 , V oc ~0.874 volt, FF ~0.775, and η ~21.5%, demonstrating that medium-indium-content InGaN is an appealing candidate to realize a high efficiency solar cell.Also, InN and high-indium-content InGaN solar cells could be useful for making tandem cells with other materials.In addition, due to larger absorption of p-InGaN junction, the performance of InGaN solar cells is determined by the upper p-InGaN junction, not the n-InGaN substrate.By appropriate design of the devices, the performance of InGaN p-n junction solar cells can be optimized.

2 .
Figure 1 shows the structure of InGaN p-n junction solar cells used for the theoretical simulation.w p and w n are the widths of the p-and n-InGaN junctions, respectively.d p and d n are the widths of the depletion region in the p-and n-InGaN junctions, respectively.The solar cells are under solar radiation AM 1.5G illumination (100 mW/cm 2 ).Photons are assumed to be incident from the p-InGaN side of the InGaN solar cells.

Fig. 1 .
Fig. 1.The structure of InGaN p-n junction solar cells used for theoretical simulation.Light is incident from the p-InGaN side.w p and w n are the widths of the p-and n-InGaN junctions, respectively.d p and d n are the widths of the depletion region in the p-and n-InGaN junctions, respectively.

Fig. 2 .
Fig. 2. The J-V characteristic of InGaN p-n junction solar cells with 300 nm p-InGaN and 1,000 nm n-InGaN for various indium compositions.

ΔFig. 3 .
Fig. 3. Short circuit current density, J sc (w p , x p ), of InGaN p-n junction solar cells as a function of p-InGaN (a) width (w p ) and (b) indium composition (x p ). Open circuit voltage, V oc (w p , x p ), of InGaN p-n junction solar cells as a function of p-InGaN (c) width (w p ) and (d) indium composition (x p ).The reported J sc (* in Figs.3(a) and 3(b)) and V oc ( in Figs.3(c) and 3(d)) of p-In 0.168 Ga 0.832 N(35nm)/n-In 0.148 Ga 0.852 N(45nm) junction solar cell under AM 1.5 illumination are plotted for comparison [11].

Fig. 5 .
Fig. 5. Short circuit current density, J sc (w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN (a) width (w n ) and (b) indium composition (x n ).Open circuit voltage, V oc (w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN (c) width (w n ) and (d) indium composition (x n ).

Figures 5 (
Figures5(c) and5(d)  show the open circuit voltage, V oc (w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN width (w n ) and indium composition (x n ), respectively.In Fig.5(c), because V oc (w n , x n ) is determined by the material bandgap[16], the V oc (w n , x n ) is nearly independent of w n .Similar to the J sc of InGaN solar cells, the V oc (w n , x n ) is equal to the V oc (w p , x p ) corresponding to the same indium composition in Fig.3(c) when the p-InGaN width is smaller than a few hundred nm.It is reasonable that the V oc of InGaN solar cells is also determined by the V oc (w p , x p ) of the upper p-InGaN junction.In addition, in Fig.5(d),V oc (w n , x n ) becomes smaller in the high-indium-content InGaN solar cells.This is due to the difference of the Fermi energies (ΔE F ) of the electron and the hole in the depletion region, which in turn is affected by the bandgap energy[16].Figures6 (a) and 6(b) show the fill factor, FF(w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN width (w n ) and indium composition (x n ), respectively.In Fig.6(a), because the J sc (w n , x n ) (Fig.5(a)) and V oc (w n , x n ) (Fig.5(c)) are nearly independent of w n , the FF(w n , x n ) is also nearly independent of w n .Similar to the J sc of InGaN solar cells, the FF(w n , x n ) is nearly equal to the FF(w p , x p ) corresponding to the same indium composition in Fig.4(a) when the p-InGaN width is smaller than a few hundred nm.The FF of InGaN solar cells is also determined by the FF(w p , x p ) of the upper p-InGaN junction.Also, in Fig.6(b), the FF(w n , x n ) decreases as the In composition increases.The steeply increasing J sc (w n , x n ) (Fig.5(b)) and the smoothly decreasing V oc (w n , x n ) (Fig.5(d)) lead to a smaller FF(w n , x n ), in the high-indium-content InGaN solar cells.
n-InGaN w n (nm) Thickness of n-InGaN w n (nm)

Fig. 6 .
Fig. 6.Fill factor, FF(w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN (a) width (w n ) and (b) indium composition (x n ).Conversion efficiency, η(w n , x n ), of InGaN p-n junction solar cells as a function of n-InGaN (c) width (w n ) and (d) indium composition (x n ).