The Shockley-Queisser limit for nanostructured solar cells
TThe Shockley-Queisser limit for nanostructured solar cells
Yunlu Xu,
1, 2
Tao Gong,
1, 2 and Jeremy N. Munday
1, 2, a)1)
Department of Electrical and Computer Engineering, University of Maryland,College Park, MD 20740, USA Institute for Research in Electronics and Applied Physics, University of Maryland,College Park, MD 20740, USA (Dated: 4 December 2014)
The Shockley-Queisser limit describes the maximum solar energy conver-sion efficiency achievable for a particular material and is the standard bywhich new photovoltaic technologies are compared. This limit is basedon the principle of detailed balance, which equates the photon flux intoa device to the particle flux (photons or electrons) out of that device.Nanostructured solar cells represent a new class of photovoltaic devices,and questions have been raised about whether or not they can exceedthe Shockley-Queisser limit. Here we show that single-junction nanos-tructured solar cells have a theoretical maximum efficiency of 42% underAM 1.5 solar illumination. While this exceeds the efficiency of a non-concentrating planar device, it does not exceed the Shockley-Queisserlimit for a planar device with optical concentration. We conclude thatnanostructured solar cells offer an important route towards higher effi-ciency photovoltaic devices through a built-in optical concentration. a) Electronic mail: [email protected] a r X i v : . [ phy s i c s . op ti c s ] D ec n 1961, Shockley and Queisser developed a theoretical framework for determining thelimiting efficiency of a single junction solar cell based on the principle of detailed balanceequating the incoming and outgoing fluxes of photons for a device at open circuit conditions. This model incorporates various light management and trapping techniques including pho-ton recycling, optical concentration, and emission angle restriction.
It was recently sug-gested that a nanowire solar cell could exceed the Shockley-Queisser (SQ) limit based onits geometry; however, without exploiting 3rd generation PV concepts which break the as-sumptions of Shockley and Queisser (e.g. multi-exciton generation, hot carrier collection,etc), even nanowire solar cells should be bounded by the SQ limit. Here we show thatfor nanostructured solar cells, the limiting efficiency is identical to that of a planar solarcell with concentrating optics and that the improvement results strictly from an increasein the open circuit voltage. This formalism leads to a maximum efficiency of 42% for ananostructured semiconductor with a bandgap energy of 1.43 eV (e.g. GaAs) under AM1.5G illumination. The SQ limit is reached by applying the principle of detailed balance to the particle fluxinto and out of the semiconductor. For every above bandgap photon that is absorbed bythe semiconductor, one electron-hole pair is generated. The maximum possible efficiency isachieved when non-radiative recombination is absent, and all generated carriers are eithercollected as current in the leads or recombine, emitting a single photon per electron-holepair. The total generated current is: I total = q [ N abs − N emit ( V )] (1)where q is the charge of an electron, and N abs and N emit are the numbers of photons perunit time that are absorbed or emitted by the photovoltaic device, respectively. These ratescan be calculated as: N ( θ max , V, T ) = (cid:90) ∞ (cid:90) πφ =0 (cid:90) θ max θ =0 σ abs ( θ, φ, E ) × F ( E, T, V ) cos ( θ ) sin ( θ ) dφdθdE (2)where σ abs ( θ, φ, E ) is the absorption cross-section, F ( E, T, V ) is the spectral photon flux,and θ max is the maximum angle for absorption (for N abs ) or emission (for N emit ). For a bulkplanar cell, the absorption cross-section is given by σ abs ( θ, φ, E ) = A cell × a ( θ, φ, E ), where2 cell is the top illuminated surface area of the cell and a ( θ, φ, E ) is the angle dependentprobability of photon absorption for incident photons of energy E . In the simplest case, a ( θ, φ, E ) is a step-function going from 0 for E < E g to 1 for E ≥ E g . The spectral photonflux can be obtained from the generalized Planck blackbody law: F ( E, T, V ) = 2 n h c E e E − qVkbT − h is Planck’s constant, k b is Boltzmann’s constant, c is the speed of light, n is therefractive index of the surroundings, which is usually taken to be vacuum ( n = 1), and qV characterizes the quasi-Fermi level splitting when describing emission from the cell.The incoming flux from the sun can be obtained from experimental data (e.g. AM 1.5solar spectrum ) or from the blackbody expression above with V = 0 and where θ max = θ s = 0 . ◦ is the acceptance half-angle for incident light from the sun at temperature T = T s = 5760 K . The outgoing flux from the cell is given by Eq. [2] for a cell temperature T c = 300 K , operating voltage V , and emission half-angle θ max = θ c = 90 ◦ . At open circuitconditions, there is no current extracted, and the current balance equation becomes0 = qN ( θ s , T s , V = 0) + qN ( θ c , T c , V = 0) − qN ( θ c , T c , V = V oc ) (4)where the middle term corresponds to absorption due to emission from the ambient sur-roundings, also at T = 300 K ; however, this term is much smaller than the flux from thesun. Thus, the light generated current is given by I L = qN ( θ s , T s , V = 0) and the darkcurrent, in the radiative limit, is given by I = I R exp( qVk B T c ) = qN ( θ c , T c , V ), where I R is thereverse saturation current. Solving the above expression for the voltage yields the commonexpression for the open circuit voltage: V oc = k B T c q ln (cid:18) I L I R + 1 (cid:19) ≈ k B T c q ln (cid:18) I L I R (cid:19) (5)which is valid for both bulk planar solar cells and nanostructured solar cells with the appro-priate absorption cross-sections as described in the next section.3 esultsNanostructured solar cells with built-in optical concentration. To achieve themaximum efficiency, we need to increase the light generated current compared to its bulkform or reduce the reverse saturation current to increase V oc . For any absorbing structure,Eqs.[2,3,4,5] can be used to determine the resulting V oc numerically; however, for the lim-iting case, we will consider a simple analytical expression. For maximum V oc , we want theabsorption cross-section to be maximized for angles near normal incidence 0 ≤ θ ≤ θ m andminimized for all other angles θ m ≤ θ ≤ θ c , where θ s ≤ θ ≤ θ c and θ m is some angle definedby the structure. We can define this piece-wise function for the absorption cross-section as: σ abs ( θ : 0 → θ m ) = σ max and σ abs ( θ : θ m → θ c ) = σ min , which allows us to perform the solidangle integration to determine the light and dark currents: I L = qN ( θ s , T s , V = 0)= σ max (cid:90) ∞ E g (cid:90) πφ =0 (cid:90) θ s θ =0 F ( E, T s , V = 0) × cos ( θ ) sin ( θ ) dφdθdE = σ max A cell I L, (6)where σ abs = 0 for E < E g , I L, is the light generated current for a bulk cell of area A cell ,and I R = qN ( θ c , T c , V = 0)= πqσ min θ m ) − cos (2 θ c )] (cid:90) ∞ E g F ( E, T c , V = 0) dE + πqσ max − cos (2 θ m )] (cid:90) ∞ E g F ( E, T c , V = 0) dE = σ max + σ min + ( σ min − σ max ) × cos (2 θ m )2 A cell I R, (7)where I R, is the reverse saturation current for a bulk cell. Substituting these expressionsinto Eq. [5], we have V oc ≈ k B T c q ln (cid:20) σ max σ max + σ min + ( σ min − σ max ) cos(2 θ m ) (cid:21) + k B T c q ln (cid:20) I L, I R, (cid:21) = k B T c q (cid:20) ln (cid:18) I L, I R, (cid:19) + ln ( X ) (cid:21) (8)4here X = 2 σ max σ max + σ min + ( σ min − σ max ) cos(2 θ m ) (9)Thus, the open circuit voltage for a nanostructured device takes on the same form as theopen circuit voltage for a macroscopic concentrating system, where X is the concentrationfactor. For maximum concentration, we consider the limit as θ m → θ s and σ min → X = 21 − cos(2 θ s ) ≈ ,
050 (10)which is the same as the maximum concentration factor that is obtained for a macroscaleconcentrator and results in a maximum solar energy conversion efficiency of ∼ σ min corresponding to thegeometric cross-section of the device, σ min → σ geo . For this case, and with cos(2 θ m ) =cos(2 θ s ) ≈
1, we get X = σ max /σ geo , and the open circuit voltage reduces to: V oc = k B T c q ln (cid:20) σ max σ geo (cid:18) I L, I R, (cid:19)(cid:21) (11)Finally, the power conversion efficiency is given by η = I L V oc F F/P in , where F F is thefill-factor, which can be obtained from the I − V characteristic defined by Eq. [1], and P in , isthe incident power from the sun. We note that the area used to calculate P in is determinedby the illumination area and not the geometric cross-section, which would lead to undercounting the number of incident photons. In general, optical concentration can be achievedusing lenses, mirrors, or unique optical nanostructures (see Fig. 1(a)). A nanostructuredsolar cell can result in optical concentration that is similar to the concentration obtainedusing lens or parabolic mirrors but relies on the wave nature of light. Fig. 1 (b) showsthe power conversion efficiency of recently reported vertically aligned nanowire-based PVcells. The optical and geometrical cross-sections are extracted from the current densitydata and from the geometrical information provided within the references. The vast majorityof the experiments are focused on Si, GaAs and InP radial or axial junction nanowirearrays fabricated with various techniques, such as MBE, MOVCD, reactive-ion etching,5tc. Generally, X = σ max σ geo is found to fall in the range of 1-10 for these structures; however,the actual concentration factor is likely significantly smaller if σ min > σ geo . Additionally, thereduced efficiency in these nanowire structures compared to the theoretical limit is due tosignificant surface recombination and device and material constraints that could be improvedwith further experimental development. The effect of entropic losses on V oc . Next we consider an alternative, but equivalent,approach to understanding the maximum efficiency of a nanostructured PV device by con-sidering the energetic and entropic loss mechanisms.
The generalized Planck equationcan be used to determine the open circuit voltage of a solar cell operating at the maximumefficiency limit: V oc = E g q (cid:18) − T c T s (cid:19) + k B T c q ln (cid:18) γ s γ c (cid:19) − k B T c q ln (cid:18) Ω emit Ω abs (cid:19) (12)where γ s and γ c are blackbody radiation flux terms that depend on E g , T s , and T c . Thefirst term represents a voltage drop related to the conversion of thermal energy into work(sometimes called the Carnot factor). The second term occurs from the mismatch betweenBoltzmann distributions at T c and T s . The third term is the voltage loss due to entropygeneration as a result of a mismatch between the absorption solid angle and the emissionsolid angle of the cell. This third term represents a voltage drop of ∼ .
28 V, which can berecovered if Ω emit = Ω abs . The most common way to recover the entropy loss due to the mismatch between theabsorption and emission solid angles is through optical concentration (Fig. 2(a)). For aplanar solar cell without optical concentration, the absorption solid angle corresponds tothe sun’s angular extent, i.e. Ω abs = 2 π (1 − cos ( θ s )) = 6 . × − sr. However, emissionfrom the cell occurs over Ω emit = 4 π . The addition of a back reflector reduces the emissionsolid angle to Ω emit = 2 π , resulting in a slight voltage improvement. For more substantialvoltage improvements, optical concentration is necessary. Optical concentration enables theabsorption solid angle to exceed the sun’s solid angle and approach the cell’s emission solidangle (Fig. 2(a)), which could largely increase the V oc .Properly designed photovoltaic nanostructures can have the same effect, reducing theentropy generation by either increasing Ω abs or by reducing Ω emit in an attempt to achieve6 emit = Ω abs (Fig. 2(b)). From a device point-of-view, Ω abs is related to the light generatedcurrent density, J L = I L /A , and Ω emit is related to the reverse saturation current density, J R = I R /A . Because the V oc depends on their ratio (see Eq. [5]), increasing Ω abs will havethe same affect as decreasing Ω emit . Thus, the voltage improvement can equivalently be seenfrom the thermodynamics of reduced entropy generation or from the device aspects of thepn-junction.According to Kirchhoff’s law, the emissivity and absorptivity of a solar cell are equal inthermal equilibrium. For a standard cell without back reflector, the device can absorbthe incident power from all directions and hence will emit in all directions (Fig. 3(a)). Theaddition of a back reflector reduces both absorption and emission from the back surface (Fig.3(b)); however, this has no effect on the absorption of the incident solar power because noillumination is coming from the back. Thus, I L is unaffected by the addition of the backreflector but I R is reduced. An ideal nanostructure would allow for absorption only overthe range of angles corresponding to the incident illumination of the source, i.e. the sun (Fig.3(c)). The current-voltage characteristics for these devices show that a back reflector yieldsa ∼
2% increase in efficiency over the traditional planar device, and an ideal nanostructureyields a ∼
11% improvement, resulting in a ∼
42% efficient device.
Numerical simulation of nanowire PV.
We simulated a bulk (80 µm thick) GaAs solarcell and a nanowire solar cell with the same thickness using S4 to solve the detailed balanceexpression numerically. For simplicity, we used the blackbody spectrum in the followingcalculations. The nanowires are embedded within an index of 2.66 and both the nanowireand planar structures are coated with a double-layer antireflection coating (52 nm of n=2.66and 98 nm of 1.46). The antireflection coating is designed to maximize the efficiency of thebulk GaAs cell. The integrated short circuit current density is almost identical for both cases( <
1% difference); however, the emitted power density is significantly different. Because alarge amount of the radiated power is near the bandgap, the lower absorption rate near thebandgap that occurs with the nanowire structure leads to a decrease in emission. This effectis demonstrated in Fig. 4(d), where the bulk cell has a higher reverse saturation currentdensity compared to the nanowire cell with same thickness. The reverse saturation currentof the nanowire cell decreases by 3.46%, and the absorption increases by 0.38%. As a result,the V oc increases by 10 mV due to these combined effects in the nanowire device, and thus,7he nanowire solar cell has a slightly higher efficiency than the bulk device (28.22% vs.28.09%).Ideally, an optical structure should be designed to minimize absorption for angles greaterthan θ s , particularly near the semiconductor bandgap, which is where the emission is peaked.To emphasize this effect, we consider a smaller radius nanowire (40 nm), which will haveincreased optical concentration. In order to minimize the loss in photogenerated current,the periodicity is decreased to 200 nm, and the nanowire length is set to 2 µ m, which is areasonable thickness for a GaAs cell. Fig. 4(c) shows this device whose absorption near thebandgap is limited so that the reverse saturation current density is one order of magnitudesmaller than that of the bulk cell (Fig. 4(d)). This nanostructuring leads to the reversesaturation current decreasing from 8 . × − to 9 . × − A/m . Although theabsorption is also decreased ( J L decreased from 362 .
68 to 237 .
55 A/m ), the V oc is increasedfrom 1 .
169 V to 1 .
214 V, showing an improvement of 45 mV in V oc . This result suggeststhat nanostructures that incorporate more complexity may be needed to yield higher V oc ’swithout loss in I L . Discussion
While the overall performance of nanostructed solar cells is still bounded by the SQ limit,one must consider the built-in optical concentration when applying this theory. Recently anInP nanowire solar cell was found to have a V oc in excess of the record InP planar device. This improvement is likely the result of the built-in optical concentration, which leads tohigher carrier densities and hence a higher V oc . Although the best devices to date are < there is great potential for improvement, which could allow nanowire solarcells to exceed 40% solar power efficiency. Here we have shown that besides the possibilityof improved carrier collection that has been previously reported, another key advantageof nanostructured solar cells over planar ones is that the optical concentration is alreadybuilt-in, yielding the possibility of higher efficiencies than planar devices.In conclusion, we have used the principle of detailed balance to determine the maximumefficiency for nanostructured photovoltaic devices. The ideal nanostructured devices resultin an efficiency of 42%, which is equivalent to the result of Shockley and Queisser whenconsidering full optical concentration. This improvement comes strictly from an improve-8ent of the open circuit voltage, and not from an improvement in the current. For futurenanostructured devices to take advantage of these benefits, high quality surface passivationand reduced non-radiative recombination are needed. From an optical design point-of-view,nanostructures should be created that have limited absorption for angles and wavelengthsthat do not match the incident illumination. When this condition is achieved, new highefficiency nanostructured PV devices will be possible.9 EFERENCES W. Shockley, and H. J. Queisser, J. Appl. Phys. , 510 (1961) A. Marti, J. L. Balenzategui, and R. F. Reyna, J. Appl. Phys. , 4067 (1997) J. N. Munday, J. Appl. Phys. , 064501 (2012) P. Krogstrup et al. , Nat. Photon. , 306 (2013) M. A. Green, Progress in Photovoltaics: Research and Applications, , 123 (2001) A. Luque S. Hegedus, Handbook of Photovoltaic Science and Engineering, John Wiley andSons, United Kingdom (2011) P. Wurfel, S. Finkbeiner and E. Daub, A. Phys. A , 67(1995) M. C. Putnam et al. , Energ. Environ. Sci. , 1037 (2010) C. Yang et al. , Chin. Phys. Lett. , 035202 (2011) J. Wang, Z. Li, N. Singh, S. Lee, Opt. Express , 23078 (2011) J. Y. Jung, K. Zhou, J. H. Bang, J. H. Lee, J. Phys. Chem. C , 12409 (2012) B. R. Huang, Y. K. Yang, T. C. Lin, W. L. Yang, Solar Energ. Mat. Solar Cells , 357(2012) C. E. Kendrick et al. , Appl. Phys. Lett. , 143108 (2010) H. P. T. Nguyen, Y. L. Chang, I. Shih, Z. Mi, IEEE. J. Sel. Top. Quantum Electron. ,1062 (2011) G. Mariani, A. C. Scofield, C. H. Hung, D. L. Huffaker, Nat. Commun. , 1497 (2013) G. E. Cirlin et al. , Nanoscale Res. Lett. , 360 (2010) E. Nakai, M. Yoshimura, K. Tomioka, T. Fukui, Jap. J. Appl. Phys. , 055002 (2013) G. Mariani et al. , Nano Lett. , 2490 (2011) J. Wallentin et al. , Science , 1057 (2013) Y. Cui et al. , Nano Lett. , 4113 (2013) M. Yoshimura, E. Nakai, K. Tomioka, T. Fukui, Appl. Phys. Express , 052301 (2013) H. Goto et al. , Appl. Phys. Express , 035004 (2009) L. C. Hirst and N. J. Ekins-Daukes, Progress in Photovoltaics: Research and Applications, , 286 (2011) A. Polman, H. A. Atwater, Nat. Mater. , 174 (2012) U. Rau, U. W. Paetzold, and T. Kirchartz, Phys. Rev. B. , 035211 (2014) C. Henry, J. Appl. Phys. , 4494 (1980) 10 W. Ruppel and P. Wurfel, Electron Devices, IEEE Transactions , 877 (1980) T. Markvart, Appl. Phys. Lett. , 064102 (2007) A. Braun, E. A. Katz, D. Feuermann, B. M. Kayes, J. M. Gordon, Energy & EnvironmentalScience , 1499-1503 (2013) E. D. Kosten, B. M. Kayes, H. A. Atwater, Energy & Environmental Science , 1907-1912(2014) G. L. Arajo1 and A. Mart1 Appl. Phys. Lett. , 894 (1995) Technically I L could be increased due to an increased path length in thin or low absorptionmaterials, resulting in a further increase in V oc . V. Liu and S. Fan, Comp. Phys. Commun , 2233-2244 (2012) S. Sandhu, Z. Yu, S. Fan, Opt. Exp. , 1209-1217 (2013) S. Sandhu, Z. Yu, S. Fan, Nano Lett. , 1011-1015 (2014) M. A. Green, K. Emery, Y. Hishikawa, W. Warta, E. D. Dunlop, Progress in Photovoltaics:Research and Applications , 1 (2014) B. M. Kayes, H. A. Atwater and N. S. Lewis J. Appl. Phys. , 114302 (2005) C. Colombo et al.
Appl. Phys. Lett. , 173108 (2009) M. D. Kelzenberg et al. , Nat. Mater. , 239 (2010)11 E f fic i en cy ( % ) Optical concentration factor, X
Si GaAs InP InN a ! Geometric ! cross-section ! Optical ! cross-section ! Geometric ! cross-section ! Concentrating Lens ! Parabolic Mirror ! Optical ! cross-section ! Geometric ! cross-section ! Nanostructure ! Optical ! cross-section ! b ! [16] (cid:1) [4,8(15,*17(22] (cid:1) Op/cal*concentra/on*factor,*X (cid:1) E ffi c i e n c y * ( % ) (cid:1) * * * * *
0* 10 ************10 ************10 ***********10 (cid:1) Experimental data: ! wire geometry ! S h o c k l e y - Q u e i s s e r l i m i t f o r n a n o s t r u c t u r e s ! Figure 1 | The Shockley-Queisser limit for nanostructures. ( a ) Schematic of theoptical concentration implemented by a concentrating lens, parabolic mirror, and using ananostructure itself (self concentration). ( b ) The efficiencies of cells with optical concentra-tion. The solid line is the theoretical limit of nanostructured PV devices based on detailedbalance, whereas individual dots represents experimental data reported in the literature. raditional cell with concentrator ! Ω abs% Ω emit% Acceptance%angle%(related%to% J L )% Emission%angle%(related%to% J R )% Nanostructures ! Acceptance%angle%(related%to% J L )% Emission%angle%(related%to% J R )% Ω S =Ω abs% Ω emit% a" b" Ω S% Figure 2 | Nanostructures can reduce the mismatch between absorption andemission angles. ( a ) A traditional planar solar cell with concentrator increases Ω abs to ap-proach Ω emit , thus reducing the entropy generation caused by their mismatch. ( b ) Similarly,a nanostructured solar cell can reduce the difference between Ω abs and Ω emit .13 x - C u rr e n t ' ( m A / c m ) ' !a! !b! !c! Standard'cell' Mirrored'back' Ideal'nanostructure' !d! W a v e l e n g t h ' Angle'(°)' θ S' ' ' W a v e l e n g t h ' Angle'(°)' θ S' ' ' W a v e l e n g t h ' Angle'(°)' θ S' ' ' AbsorpEon/'Emission' θ emit =180° ' θ emit =90° ' AbsorpEon/'Emission' AbsorpEon/'Emission' θ emit =θ S' c!!!!! b!!!! a!!!! W a v e l e n g t h * Angle*θ S* * * W a v e l e n g t h * Angle*θ S* * * W a v e l e n g t h * Angle*θ S* * * Absorp8on* θ emit =180° * θ emit =90° *Absorp8on* Absorp8on* θ emit =θ S* W a v e l e n g t h * Angle*θ S* * * W a v e l e n g t h * Angle*θ S* * * W a v e l e n g t h * Angle*θ S* * * Absorp8on* θ emit =180° * θ emit =90° *Absorp8on* Absorp8on* θ emit =θ S* W a v e l e n g t h * Angle*θ S* * * W a v e l e n g t h * Angle*θ S* * * W a v e l e n g t h * Angle*θ S* * * Absorp8on* θ emit =180° * θ emit =90° *Absorp8on* Absorp8on* θ emit =θ S* Figure 3 | Modification of absorption and emission results in an ideal PV nanos-tructure achieving >
40% power conversion efficiency.
Emission and absorption for( a ) slab without back reflector, ( b ) slab with back reflector and ( c ) ideal nanostructuredcell. The emission and absorption are represented in terms of their half-angle, θ . Ab-sorption/emission over all angles (standard cell) corresponds to θ = 180 ◦ ; however, theillumination from the sun is only over a subset of half-angles from 0 to θ s . Thus, the mis-match between θ s and θ emit results in a decreased voltage. ( d ) I-V curves corresponding tothe three structures (a-c). All structures are illuminated with the AM1.5G spectrum andshow increased V oc as θ emit → θ s . 14 """"""""20""""""""40""""""""60""""""""80 (cid:1) " " " " " (cid:1) W a v e l e n g t h " ( n m ) (cid:1) a (cid:1) Angle"(°) (cid:1) " " " " " (cid:1) (cid:1) W a v e l e n g t h " ( n m ) (cid:1) Angle"(°) (cid:1) c (cid:1) (cid:1) " " " " " (cid:1) W a v e l e n g t h " ( n m ) (cid:1) Angle"(°) (cid:1) " " " " " (cid:1) b (cid:1) Absorp?vity (cid:1) x - (cid:1) " " " " " " (cid:1) J R / Ω " ( A F m G s r G ) " x " G " " (cid:1) Angle"(°) (cid:1) c$$$$
NWs"w/o"ARC (cid:1) b """"NWs"with"ARC" a """"Bulk"with"ARC (cid:1) d (cid:1) Bulk"Cell"(2Glayer"ARC)"NW"Cell" Embedded"NW"Cell"(2Glayer"ARC)" (V oc =1.214"V) (cid:1) (V oc =1.169"V) (cid:1) (V oc =1.179"V)" Figure 4 | Reduced dark current in nanowire structures.
Angular dependence ofthe absorption spectrum for ( a ) a bulk (80 µm thick) GaAs solar cell, ( b ) a GaAs nanowiresolar cell (embedded in a dielectric) with a period of 300 nm, a radius of 75 nm, and lengthof 80 µ m, and ( c ) a GaAs nanowire solar cell with a period of 200 nm, a radius of 40 nm,and a length of 2 µ m. The devices in (a) and (b) have a double-layer ARC on top, and allcells have a perfect back reflector. The nanowire solar cells have decreased absorption (andhence emission) near the bandedge for angles > θ s . ( d ) The current density correspondingto the three structures (a-c) decreases, showing an improved V ococ