Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

DOI: 10.1039/C9NA00599D
(Paper)
Nanoscale Adv., 2019, Advance Article

A. Prajapati^{a} and
G. Shalev*^{ab}
^{a}School of Electrical & Computer Engineering, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 8410501, Israel. E-mail: glshalev@bgu.ac.il
^{b}The Ilse-Katz Institute for Nanoscale Science & Technology, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 8410501, Israel

Received
21st September 2019
, Accepted 14th October 2019

First published on 15th October 2019

Texturing the front surface of thin film photovoltaic cells with ordered or disordered arrangements of subwavelength structures is beneficial in terms of efficient light harvesting as well as efficient carrier extraction. Previous studies demonstrated efficient broadband absorption of solar radiation with surface arrays of subwavelength inverted cones (light funnels – LFs). In the current work, we use three-dimensional finite-difference time-domain electromagnetic calculations as well as three-dimensional device calculations to examine carrier extraction from photovoltaic cells that are composed of LF arrays on top of underlying substrates. For the selected geometry under examination, we show a broadband absorption enhancement of 14% for the LF photovoltaic cell compared with a cell based on the respective optically optimized nanopillar arrays. However, we show that the nominal power conversion efficiency is 60% higher in the LF cell which is due to the enhancement of both open-circuit voltage and short-circuit current. The higher open-circuit voltage in the LF cell is due to the higher injection of photocarriers, and the higher short-circuit current is a result of the unique LF geometry that supports efficient carrier extraction due to the naturally occurring gradients of the quasi-Fermi levels and minority carrier conductivity that allow for enhanced contact selectivity. We believe that this work paves the way towards a new approach for carrier collection in photonic devices for energy applications.

Photovoltaic cells based on subwavelength features were experimentally realized with carrier extraction schemes such as all back-side contacts, radial configurations in which the p–n junction is aligned in a core–shell geometry, and axial configurations where the junction is positioned along the height of the subwavelength structure.^{28,29,41,42,44–46} Kayes et al. presented a model for pillar arrays with radial junctions and demonstrated carrier extraction enhancement due to the short collection lengths.^{47} Christesen et al. examined horizontally aligned silicon nanowires and showed that the open-circuit voltage (V_{oc}) of axial and radial junctions can reach values of V_{oc} > 0.7 V and that radial junctions are more immune to surface recombination.^{48} Wong et al. calculated the electrical properties of a thin-film photovoltaic cell based on silicon NP arrays and argued the superiority of radial junctions in terms of power conversion efficiency (PCE).^{20} Wang et al. numerically studied axial NP and nanohole arrays and showed that a thin and highly doped emitter provides the optimum compromise in terms of short circuit current (J_{sc}) and V_{oc}.^{49} Li et al. analytically modeled NP and nanohole arrays and showed that a high junction depth entails a decrease in efficiency.^{50} Shalev numerically explored the effect of junction depth in axial and radial NP arrays and showed that the optimal junction depth in the axial configuration is an interplay between minority carrier diffusion lengths and carrier extraction lengths.^{35}

In a recent series of publications we introduced light funnel (LF) arrays for efficient light trapping and numerically demonstrated broadband absorption of solar radiation.^{12,51–55} Fabrication of silicon LF arrays was also demonstrated.^{12} With LFs we refer to subwavelength structures that are inverted with respect to the incoming illumination, such as inverted cones, inverted hyperboloids, parabolic light concentrators (CPC), etc. We showed that LF arrays serve as anti-transmission layers in contrast to NP arrays that function as efficient anti-reflection layers.^{26,36} We suggested that the origin of the superior absorption is due to the unique inverted LF geometry that provides enhanced probability for photonic mode excitation and enables strong coupling of the incoming illumination with the individual LFs.

In the present work we numerically examine carrier collection in photovoltaic cells based on LF arrays. Specifically, we consider a photovoltaic cell based on LF arrays on top of a thin substrate with an axial configuration. We show that although the broadband absorption enhancement of the LF cell is 14% higher than that of an equivalent NP photovoltaic cell, the overall photovoltaic efficiency is 60% higher due to geometry-driven efficient carrier extraction in photovoltaic cells based on LF arrays.

In the following photovoltaic analysis, we construct LF and NP photovoltaic cells from single unit cells in the respective complexes, as presented in the inset in Fig. 1a for the LF cell. Fig. 2a shows the optical generation profiles along the vertical axes of the NP and LF complexes under both J_{sc} and V_{oc} conditions, where the higher optical generation in the LF complex is evident. Fig. 2b presents current–voltage (I–V) curves for an absorber acceptor doping level (N_{A}) of 10^{18} cm^{−3} for both NP and LF photovoltaic cells, where the photocurrent enhancement of the LF cell is a direct consequence of the enhanced optical generation. We next follow the transition from a NP cell into a LF cell by gradually decreasing the NP bottom diameter from D_{b} = 400 nm to D_{b} = 350 nm, 300 nm, 200 nm and 100 nm. The absorption and the optical generation for each geometry were calculated (not shown). Fig. 2c shows the dependency of J_{sc} and V_{oc} on N_{A} for different D_{b} values. A gradual increase in both V_{oc} and J_{sc} is noticeable with the decrease in D_{b} for the full range of N_{A} values. Note that V_{oc} reaches a maximum for N_{A} = 10^{18} cm^{−3} which is a direct consequence of the saturation current dependency on N_{A} and minority carrier lifetimes which also depend on N_{A}. The respective electron diffusion lengths (L_{n}) are also presented in Fig. 2c. Fig. 2d presents the dependency of the nominal power conversion efficiency (nPCE), defined as J_{sc} × V_{oc}/P_{in} where P_{in} is the power of the solar spectrum at AM 1.5 G, on N_{A} for all the considered D_{b} values. The nPCE values of all cells peak at N_{A} = 10^{17} cm^{−3}. The highest nPCE = 8% is obtained for the smallest D_{b} = 100 nm as compared with the NP cell with an nPCE of 5% which reflects an nPCE enhancement of 60%. This is surprising as the η_{BB} enhancement is only 14%. In order to understand the origin of the 60% nPCE enhancement we next carefully examine the behavior of J_{sc} and V_{oc} of both LF and NP cells.

The V_{oc} enhancement of the LF cell over the NP cell decreases with an increase in N_{A} with a highest value of 15% for N_{A} = 10^{15} cm^{−3}, and with a corresponding J_{sc} enhancement of 23% (see Fig. 2c). V_{oc} is equal to the splitting of the quasi-Fermi levels (ε_{fc} − ε_{fv}, where ε_{fc} is the electron quasi-Fermi level and ε_{fv} is the hole quasi-Fermi level), between the top emitter electron contact and the bottom absorber hole contact, and it is equal to the free energy delivered to the load. Fig. 3a and b present the energy band diagrams along the vertical axes of the LF cell and the NP cell for N_{A} = 10^{15} cm^{−3} and 10^{19} cm^{−3}, respectively. The cross-sections of LF and NP cells above the band diagrams visualize the considered orientation. The V_{oc} values extracted from the energy band diagrams for N_{A} = 10^{15} cm^{−3} are 419 mV and 364 mV for the LF and the NP cells, respectively, and similarly for N_{A} = 10^{19} cm^{−3} the V_{oc} values are 537 mV and 527 mV. The V_{oc} values extracted from the energy band diagrams are in agreement with the V_{oc} values extracted from the I–V curves (Fig. 2c). Next, we examine the origin of the quasi-fermi level splitting and its dependency on N_{A}. For a p-type absorber (1) V_{oc} = kTln(1 + Δn_{e}/n_{e0}), where k is the Boltzmann constant, T is the cell temperature (300 K), Δn_{e} is the injected photoelectron density and n_{e0} is the equilibrium electron density.^{56} Fig. 3c and d show the Δn_{e} profiles where for N_{A} = 10^{15} cm^{−3} the densities are higher than for N_{A} = 10^{19} cm^{−3} due to the doping-dependent lifetimes (see methodology). Also, while for N_{A} = 10^{15} cm^{−3} Δn_{e} of the LF is significantly higher, for N_{A} = 10^{19} cm^{−3} Δn_{e} values of the LF and NP are similar. Fig. 3e and f present the respective calculated spatially resolved V_{oc} profile for a p-type absorber according to (1). The origin of the higher V_{oc} of the LF cell for N_{A} = 10^{15} cm^{−3} is now apparent; the higher broadband absorption and the optical generation of the LF cell entail a higher population of electron minority carriers in the absorber which, for most of the 1D profile, is about one order of magnitude higher than that in the NP cell. This considerably higher excitation of the absorber directly translates into a higher V_{oc}. This difference in absorber excited minority carriers disappears for N_{A} = 10^{19} cm^{−3} due to the higher Shockley–Read–Hall (SRH) recombination. However, note that a higher N_{A} presents a higher V_{oc} due to the absorber n_{e0} which is three orders of magnitude smaller for N_{A} = 10^{19} cm^{−3}. The enhanced absorber photoelectron density for N_{A} = 10^{15} cm^{−3} is also apparent in the 2D cross-section presented in Fig. 3a. Therefore, the higher V_{oc} at N_{A} = 10^{15} cm^{−3} in the LF cell is a combination of a higher carrier injection and a lower level of recombination.

The J_{sc} enhancement of the LF cell compared with the NP cell increases with N_{A} from 24% to 61% for N_{A} = 10^{15} cm^{−3} to 10^{19} cm^{−3}, respectively (see Fig. 2c). The generation of photocurrent reflects the conversion efficiency of chemical energy into electrical energy. The electron current is described by J_{e} = σ_{e}/q grad(ε_{fn}), where grad(ε_{fn}) is the gradient of the electron quasi-fermi level, q is the elementary charge and σ_{e} is the electron conductivity equal to n_{e}qμ_{e} where n_{e} is the electron density and μ_{e} is the electron mobility. Hence a higher σ_{e} and grad(ε_{fn}) along the electron path towards the emitter contact will induce greater emitter selectivity towards electrons and therefore a higher J_{e}. Fig. 4a and b show the energy band diagrams of the NP and LF cells under J_{sc} conditions for N_{A} = 10^{15} cm^{−3} and 10^{19} cm^{−3}, respectively, Fig. 4c and d present the respective n_{e} which is linearly proportional to σ_{e}, and Fig. 4e and f present the respective grad(ε_{fn}) for which it is positive. For N_{A} = 10^{15} cm^{−3} n_{e}, and hence σ_{e}, in the LF cell increases towards the emitter contact more appreciably which provides an enhanced emitter selectivity towards electrons in the LF cell compared with that of the NP cell. grad(ε_{fn}) for N_{A} = 10^{15} cm^{−3} is positive for both cells, which indicates a decrease in ε_{fn} towards the emitter, but it is not necessarily higher in the LF cell as shown in Fig. 4e. Overall for N_{A} = 10^{15} cm^{−3} the combined effect of n_{e} and grad(ε_{fn}) results in 24% J_{sc} enhancement. For N_{A} = 10^{19} cm^{−3} (Fig. 4f) both σ_{e} and grad(ε_{fn}) (i.e. where grad(ε_{fn}) > 0) are higher in the LF cell. Therefore, the combined enhancement of σ_{e} and grad(ε_{fn}) results in an overall J_{sc} enhancement of 61% in the LF cell compared with the NP cell. Importantly, the origin of the higher grad(ε_{fn}) in the LF cell is due to the gradient in the electron chemical potential which is due to the higher n_{e} at the LF bottom part which is a direct consequence of the unique LF inverted geometry.

R^{A}_{net} = (C_{n}n + C_{p}p) (np − n_{i,eff}^{2}),
| (1) |

(2) |

(3) |

(4) |

(5) |

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