Design for strong absorption in a nanowire array tandem solar cell

Semiconductor nanowires are a promising candidate for next-generation solar cells. However, the optical response of nanowires is, due to diffraction effects, complicated to optimize. Here, we optimize through optical modeling the absorption in a dual-junction nanowire-array solar cell in terms of the Shockley-Quessier detailed balance efficiency limit. We identify efficiency maxima that originate from resonant absorption of photons through the HE11 and the HE12 waveguide modes in the top cell. An efficiency limit above 40% is reached in the band gap optimized Al0.10Ga0.90As/In0.34Ga0.66As system when we allow for different diameter for the top and the bottom nanowire subcell. However, for experiments, equal diameter for the top and the bottom cell might be easier to realize. In this case, we find in our modeling a modest 1–2% drop in the efficiency limit. In the Ga0.51In0.49P/InP system, an efficiency limit of η = 37.3% could be reached. These efficiencies, which include reflection losses and sub-optimal absorption, are well above the 31.0% limit of a perfectly-absorbing, idealized single-junction bulk cell, and close to the 42.0% limit of the idealized dual-junction bulk cell. Our results offer guidance in the choice of materials and dimensions for nanowires with potential for high efficiency tandem solar cells.


Direct iteration method to optimize geometry
In general, for an optimization problem of a function η with variables x 1 , x 2 …x n , (such as the L bot , D top , D bot , P, and L top in the case of our nanowire array), the necessary condition for a maximum point is ∀ , ƞ = 0, unless the maximum resides at the boundary of the independent variable domain.
Many of the conventional optimization methods, such as the Newton's method, are based on first and/or second order derivatives. However, the efficiency η of the dual-junction nanowire solar cell shows a dependence on L top , L bot , D top , D bot , and P that makes the optimization to appear unsuitable for such derivative based methods. First, the efficiency tends to increase monotonously with both L top and L bot (see Figures S3-S14). Second, for a fixed L top and L bot , the efficiency can show a rugged landscape with many local maxima as a function of D top , D bot , and P (see Figures S3-S14).
Here, we use a direct iteration method where we vary one geometrical parameter at a time. We choose to perform the iteration for a fixed L top . Regarding the L bot , for which η tends to increase monotonously, we allow the iteration to stop if dη/dL bot < 0.001 μm -1 in the iteration. See Algorithm 1 below for a brief pseudocode description of the iteration method.

Algorithm 1
Choose

Efficiency analysis
The most important property of a solar cell is η, the conversion efficiency of sun light into electrical energy. To theoretically study η, the Shockley-Queisser detailed balance analysis gives a general framework. There, a balance between photo generation of electron-hole pairs, extracted current, and recombination of electron-hole pairs is used [1][2][3] . This analysis is of general character and is applicable also for p-i-n junction tandem solar cells [3][4][5] .
By considering optical absorption and radiative recombination, we find an upper limit for the efficiency of a solar cell design 2,5-9 . Non-radiative recombination and ohmic losses reduce the efficiency from this radiatively limited efficiency 3,6,10 . In this way, we study the prospect of the nanowire tandem solar cell as a platform for next-generation photovoltaics, in the case when material properties, such as non-radiative recombination and ohmic losses, are minimized.
Let us consider a dual-junction tandem solar cell where we by cell 1 denote the top cell and by cell 2 the bottom cell ( Figure 1). For such a solar cell under AM1.5D solar illumination, the current density as a function of voltage in each subcell is given by 1-3 (S1) Here, j 1(2) is the total current in the top (bottom) cell, j ph1(ph2) is the photogenerated current density under the AM1.5D solar spectrum, and j rec1(rec2) , which depends on the voltage 1(2) over subcell 1 (2), is the decrease of current due to the varying types of recombination losses. We consider here a series connected tandem cell where the same current j = j 1 = j 2 flows through both cells. Note that in this case of a series-connected subcells, the voltages V 1 and V 2 must be coupled in such a way that the condition j 1 = j 2 is fulfilled. The output power of the tandem cell is given by P = (V 1 +V 2 )j. Note that this power is maximized at V 1 = V 1max and V 2 = V 2max , where the current is j = j max and the output power P = P max = (V 1max + V 2max ) j max . Note that in this analysis, we vary both V 1 and V 2 in order to find V 1max and V 2max , under the constraint that j 1 = j 2 .
The Shockley-Queisser detailed balance efficiency, η, is in turn given by the ratio between this P max and the total incident solar power 1 : Here, I AM1.5 (λ) is the 1 sun direct and circumsolar AM1.5D solar spectrum 5 .
We start by considering the photogenerated current j ph1(ph2) in each cell. Here, we assume that each absorbed photon with energy above the band gap energy generates one electron-hole pair in the solar cell. Notice that each photogenerated electron-hole pair contributes one charge carrier to this current since the electron and hole are split by the p-i-n junction in different directions 3,9 . The incident intensity multiplied by the absorption probability A(λ) gives the amount of absorbed energy at wavelength λ. By dividing this absorbed energy with 2 ћ / , we obtain the rate of photons absorbed at energy λ. By integrating this rate over wavelength, we find the generation rate of electron-hole pairs, and by multiplying with the elementary charge e, we find the photogenerated current (density): Here, λ 1(2) =hc/E 1 (2) corresponds to the wavelength of photons at the band gap energy E 1 (2)  Now that we know how to analyze j ph1(ph2) , we turn to consider the radiative recombination. Here, since we consider the case of a solar cell limited by the radiative recombination, j rec1(rec2) = j rad1(rad2) in Eq. (S1). In a semiconductor solar cell, the radiative recombination is typically given by 2,3,6 1( 2) ( 1(2) ) = ( 0, ,1(2) + 0, ,1(2) )[exp ( 1(2) ) − 1] where the radiative emission rate at thermal equilibrium is given by: properties of the solar cell, and therefore also the efficiency limit. For example, for a single-junction solar cell, an absolute increase in the efficiency limit by 1-2% has been predicted compared with planar cells 2 . However, an analysis of the emission properties of nanowire tandem solar cells is beyond the scope and computational resources of this work. Instead, we believe that by using the simplifying approximation e TE/TM,up(down),1(2) =1, we obtain a reasonable comparison between solar cells constructed from different nanowire geometries and materials. In this way, we assume maximum possible emission from the solar cell. Notice that this assumption gives a lower bound for the efficiency of the solar cell in the radiative limit. With this assumption of e =1, the radiative recombination rate at thermal equilibrium simplifies to: Here, ñ up(down),1(2) is the effective refractive index for emission and depends on how light can couple out from the top and the bottom cell to the top air side and into the substrate 11 . We assume that the emission into the substrate can occur from the top and bottom cell at all angles (see Figure S1(a)). This gives ñ down,1(2) = n substrate . For the emission into the top air side, we are limited to propagation angles θ < 90⁰ in the air. This limitation on emission angles translates into ñ up,1(2) =n air =1 in Eq. (S6) 11 .     Table S1).  Table S1).  Table S1).  Table S1).  Table S1).  Table S1).  Table S1).  Table S1).  Table S1).  Table S1).  Table S1). Figure S14. Dependence of the efficiency on D bot = D top , P, and L bot when L top = 8000 nm for the GaInP/InP dual junction nanowire solar cell. For the separate variation of each parameter shown here, the other three parameters are kept at their optimized value (D bot = D top = 160 nm, P = 420 nm, and L bot = 8800 nm give the optimized η = 37.5 %; see Table S1).