Hierarchical Ellipsoidal Patterned Solar Cell

• The paper introduces a novel HEPSC design that integrates layer-by-layer half-ellipsoidal patterning with a band-engineered double HTL to achieve a PCE increase from 24.32% to 26.38%.
• It leverages multi-scale morphological optimization and FDTD optical analysis to enhance broadband light trapping, reducing reflection losses and improving photocurrent generation.
• Electrical and thermal multiphysics simulations demonstrate a ~2.04 mA/cm² gain in JSC and stable VOC and FF despite increased absorption-induced heating.

The Hierarchical Ellipsoidal Patterned Solar Cell (HEPSC) is, in (Rahman et al., 3 Sep 2025), a specifically engineered MAPbI$_3$ perovskite solar cell in which the entire device stack is conformally sculpted into a periodic array of half-ellipsoids and combined with a band-engineered double hole transport layer (HTL). The architecture is introduced to address interfacial energy misalignment, suboptimal light absorption, and thermal instability in MAPbI$_3$-based perovskite solar cells (PSCs) by integrating morphological engineering with a CuO / I$_2$O$_5$-doped Spiro-OMeTAD double HTL, FDTD optical analysis, and FEM-based electrical and thermal modeling (Rahman et al., 3 Sep 2025).

1. Concept and distinguishing characteristics

HEPSC denotes more than a textured front surface. The defining feature is that all optically and electrically active layers are hierarchically patterned, so that the morphology becomes a coherent, vertically aligned, periodic half-ellipsoid pattern running through the stack. In this formulation, the conventional planar sequence MgF$_2$ / ITO / TiO$_2$ / MAPbI$_3$ / CuO / I$_2$O$_5$-doped Spiro-OMeTAD / Au is first optimized electronically and then reshaped geometrically (Rahman et al., 3 Sep 2025).

“Hierarchical” has two specific meanings in this design. First, the patterning is layer-by-layer and cumulative: ellipsoids are first introduced on ITO (Str. I), then TiO$_2$ (Str. II), then MAPbI$_3$0 (Str. III), then CuO (Str. IV), and finally Spiro-OMeTAD (Str. V), while preserving the morphology of previously patterned layers. Second, the patterning is multi-scale: the ellipsoids have different radii and heights in different layers, producing a gradient of feature sizes and curvatures from the MgF$_3$1 antireflection layer to the HTL. This architecture is presented as a balanced alternative to planar cells, which have limited light trapping especially in the near-UV and near-IR, and to more aggressive nanostructures, which can improve optical confinement but increase fabrication complexity (Rahman et al., 3 Sep 2025).

A common misconception is that HEPSC is simply a nanostructured ARC or front electrode. The paper explicitly distinguishes it from such approaches: the entire device stack is conformally patterned, and the purpose is to enhance broadband light trapping while preserving realistic morphology, conformal coating, and good electronic interfaces. Another important clarification is that the optimized HEPSC stops patterning at the Spiro-OMeTAD layer; the Au back contact remains planar because a trial structure with Au nanospheres (Str. VI) gave negligible or even detrimental electrical benefit (Rahman et al., 3 Sep 2025).

2. Device stack and band-engineered double-HTL configuration

The base device is built on glass and employs MgF$_3$2 as the ARC, ITO as the front electrode, compact anatase TiO$_3$3 as the ETL, MAPbI$_3$4 as the absorber, CuO plus I$_3$5O$_3$6-doped Spiro-OMeTAD as the double HTL, and Au as the back electrode. In the optimized HEPSC case, the ARC is typically $_3$7 nm, ITO is optimized at 50 nm, TiO$_3$8 is 150 nm, MAPbI$_3$9 is 200 nm, CuO is optimally about 300 nm, Spiro-OMeTAD is maintained thin, and Au is 100 nm. The material parameters reported include ITO work function $_2$0–$_2$1 eV and electron mobility $_2$2 cm$_2$3/V$_2$4s; TiO$_2$5 bandgap $_2$6 eV, electron affinity $_2$7 eV, electron mobility 20 cm$_2$8/V$_2$9s, donor density $_5$0; and MAPbI$_5$1 direct bandgap $_5$2 eV, electron affinity $_5$3 eV, mobilities $_5$4 cm$_5$5/V$_5$6s for electrons and holes, acceptor density $_5$7, SRH lifetime 8 ns, radiative coefficient $_5$8, and Auger coefficient $_5$9 (Rahman et al., 3 Sep 2025).

The double HTL is central to the HEPSC concept. CuO, adjacent to MAPbI$_2$0, is described as a wide-gap p-type inorganic HTL with some additional absorption in the 550–830 nm range and higher thermal and environmental stability than purely organic HTLs. Pristine Spiro-OMeTAD has $_2$1 eV and $_2$2 eV, corresponding to HOMO $_2$3 eV and LUMO $_2$4 eV. After I$_2$5O$_2$6 doping, the reported values become $_2$7 eV, HOMO $_2$8 eV, LUMO $_2$9 eV, and $_2$0. The reported mobility is $_2$1 cm$_2$2/V$_2$3s for holes and electrons (Rahman et al., 3 Sep 2025).

The interfacial energetics are quantified through conduction- and valence-band offsets. At the front interface,

$_2$4

which gives $_2$5, described as favorable for electron extraction. At the CuO/Spiro interface, the valence-band offset is reported to be about $_2$6 eV for pristine Spiro and about $_2$7 eV after I$_2$8O$_2$9 doping, reducing the hole extraction barrier and aligning better with Au. Thickness optimization over 20–300 nm for CuO and Spiro-OMeTAD gives an optimal planar double-HTL thickness of CuO 300 nm and Spiro 60 nm; doping sweeps over $_3$0–$_3$1 cm$_3$2 identify a best trade-off at CuO $_3$3 and Spiro $_3$4. At those values, planar PCE reaches 22.78% before MgF$_3$5 ARC and 24.32% after ARC (Rahman et al., 3 Sep 2025).

3. Ellipsoidal geometry and staged morphological optimization

The ellipsoidal geometry is defined in three dimensions by

$_3$6

with major radius $_3$7 in $_3$8 and minor radius $_3$9 in $_2$0. HEPSC uses the upper half of this ellipsoid, truncated at depth $_2$1 below the top. The effective major radius and the height above the planar reference are

$_2$2

and

$_2$3

Periodic boundary conditions in $_2$4 and $_2$5 make the dome array periodic laterally, while vertical alignment yields the stacked ellipsoid morphology (Rahman et al., 3 Sep 2025).

The optimized radii differ by layer. For the MgF$_2$6 ARC top ellipsoid, the reported values are $_2$7 nm, $_2$8 nm, $_2$9 nm, giving $_5$0 nm and $_5$1 nm. The ITO ellipsoid is optimized at $_5$2 nm, $_5$3 nm, $_5$4 nm; TiO$_5$5 at $_5$6 nm, $_5$7 nm; MAPbI$_5$8 at $_5$9 nm, $_2$0 nm; CuO at $_2$1 nm, $_2$2 nm, $_2$3 nm; and Spiro-OMeTAD, in the final Str. V HEPSC, at major 115 nm, minor 110 nm, and $_2$4 nm (Rahman et al., 3 Sep 2025).

The optimization is explicitly staged. The paper reports that efficiency is fairly flat around the optimum radii, and symmetric ellipsoids with major $_2$5 minor often give the best FF and $_2$6. The cumulative performance progression across Str. I–V is as follows:

Stage	Patterned through	Max efficiency
Str. I	ITO	24.15 %
Str. II	ITO + TiO$_2$7	24.25 %
Str. III	ITO + TiO$_2$8 + MAPbI$_2$9	24.79 %
Str. IV	ITO + TiO$_3$00 + MAPbI$_3$01 + CuO	25.26 %
Str. V	ITO + TiO$_3$02 + MAPbI$_3$03 + CuO + Spiro-OMeTAD	26.38 %

This sequence shows that the performance increase is cumulative rather than dominated by a single patterned layer. It also clarifies the role of Str. VI: adding Au nanospheres with radius 10–110 nm at the back contact did not define the optimized architecture, because the final HEPSC is Str. V with planar Au (Rahman et al., 3 Sep 2025).

4. Optical modeling and broadband light-trapping mechanisms

The optical analysis uses finite-difference time-domain (FDTD) simulations in Ansys Lumerical with complex refractive indices $_3$04 from literature for ITO, TiO$_3$05, MAPbI$_3$06, CuO, Spiro-OMeTAD, Au, and MgF$_3$07. In 2D, the reported boundary conditions are periodic boundary conditions in $_3$08 and perfectly matched layer in $_3$09; in 3D, periodic boundary conditions are used in $_3$10 and $_3$11, and perfectly matched layer in $_3$12. The source is the AM1.5G spectrum from 300–1000 nm, incident from the top, while photo-generation is integrated over 300–830 nm to exclude parasitic intraband absorption. The outputs include layer-resolved power absorption, reflectance, transmission, and volumetric generation rate, with absorptance given by

$_3$13

(Rahman et al., 3 Sep 2025)

Four optical mechanisms are emphasized. First, the MgF$_3$14/ITO/TiO$_3$15 ellipsoids create a graded-index antireflection profile from air to the perovskite. In the planar case, adding a 100 nm MgF$_3$16 ARC reduces average reflection from $_3$17% to 9.4% over 300–830 nm. Second, the ellipsoids act as mini-lenses, reshaping the incident wavefront and focusing light into MAPbI$_3$18 and CuO; electric-field intensity maps of $_3$19 show focusing at the ellipsoid tips. Third, the periodic pattern supports guided-mode and quasi-resonant coupling. At $_3$20 nm, the paper reports distributed confinement across several ellipsoidal layers; at $_3$21 nm, a resonant mode with field enhancement $_3$22 the incident field, primarily near the MAPbI$_3$23/CuO interface; and at $_3$24 nm, hotspots $_3$25 in the lower ellipsoids spanning MAPbI$_3$26, CuO, and Spiro. Fourth, the architecture increases broadband path length by scattering photons into oblique directions in the thin 200 nm MAPbI$_3$27 and 300 nm CuO layers (Rahman et al., 3 Sep 2025).

These mechanisms are reflected in the spectral response. In the planar DHLSC with ARC, MAPbI$_3$28 absorbs most photons from $_3$29–750 nm and CuO absorbs significantly from 550–830 nm, but absorption beyond $_3$30 nm is weak. In the HEPSC, normalized power absorption is higher across the full 300–1000 nm range, with especially marked gains from 600–830 nm and into the 880–990 nm range. The cumulative spectral current increases from 27.25 mA/cm$_3$31 for the optimized planar DHLSC with ARC to 29.29 mA/cm$_3$32 for the HEPSC, corresponding to roughly a 7.5% gain in $_3$33 mainly from visible-to-near-IR enhancement (Rahman et al., 3 Sep 2025).

5. Electrical and thermal multiphysics behavior

The electrical model solves Poisson, drift-diffusion, and continuity equations with SRH, radiative, and Auger recombination:

$_3$34

$_3$35

$_3$36

The Einstein relation $_3$37 is used, periodic boundaries are treated with Neumann conditions for current, and metal-semiconductor interfaces use Dirichlet conditions for electrostatic potential. A crucial modeling step is that the spatially resolved generation rate $_3$38 from FDTD is fed directly into the continuity equations. In the HEPSC, $_3$39 is highly non-uniform, with peaks near ellipsoid apexes and interfaces; the center of the periodic cell shows substantially higher generation at the top of the MAPbI$_3$40 ellipsoid than the edge, which more closely resembles the planar cell (Rahman et al., 3 Sep 2025).

The electrical consequence is that optical gains translate almost directly into device-level gains. Although the hierarchical geometry increases interfacial area and curvature, the simulated $_3$41 remains high and FF remains essentially unchanged relative to the planar DHLSC, which the paper attributes to good band alignment and relatively low defect densities. The final optimized device metrics are:

Metric	Planar DHLSC	HEPSC
$_3$42	27.25 mA/cm$_3$43	29.29 mA/cm$_3$44
$_3$45	1.066 V	1.074 V
FF	83.74 %	83.87 %
PCE	24.32 %	26.38 %

Relative to the best planar design with the same material stack, the reported gains are $_3$46 mA/cm$_3$47 in $_3$48, $_3$49 mV in $_3$50, a slight $_3$51% in FF, and $_3$52 absolute percentage points in PCE, or $_3$53% relative (Rahman et al., 3 Sep 2025).

The thermal model is solved self-consistently with the electrical equations through

$_3$54

where $_3$55 includes Joule heating, non-radiative recombination heat, and carrier thermalization. Under 1 sun illumination at ambient 300 K, the planar DHLSC with I$_3$56O$_3$57-doped Spiro and MgF$_3$58 ARC rises to about 50 $_3$59C (323 K), while the HEPSC reaches about 52 $_3$60C (325 K), slightly higher because more light is absorbed. For the planar DHLSC, PCE drops from 24.32% to 23.02%, corresponding to about 94.7% retention. For the HEPSC, $_3$61 changes from 29.29 to 29.39 mA/cm$_3$62, $_3$63 falls from 1.074 to 1.026 V, FF decreases from 83.87% to 82.69%, and PCE decreases from 26.38% to 24.93%, corresponding to about 94.5% retention. The paper therefore characterizes the HEPSC as thermally robust in the sense that enhanced absorption does not produce a disproportionate thermal penalty (Rahman et al., 3 Sep 2025).

6. Fabrication routes, limitations, and interpretive boundaries

The proposed fabrication sequence is designed around conformal processing of the hierarchical morphology. For the glass substrate and MgF$_3$64 ARC, the suggested routes are nanoimprint lithography or laser interference lithography, followed by reactive ion etching or HF-based wet etch, and then MgF$_3$65 deposition by thermal evaporation or ALD. Patterned ITO is deposited by DC magnetron sputtering and nanostructured through RIE with Ar/H$_3$66 plasma. Compact TiO$_3$67 is either etched directly or conformally deposited onto structured ITO so that the ellipsoid contours are preserved. For MAPbI$_3$68, the recommended approaches are hybrid vapor-solution processing or co-evaporation, specifically because they are expected to provide better uniformity on non-planar surfaces than simple spin coating; fine patterning, if required, may use FIB lithography with XeF$_3$69/I$_3$70 gas assist for research-scale prototypes. CuO$_3$71 is grown by low-temperature pulsed CVD, refined by mild aqueous acetic acid etching at about 35 $_3$72C, and Spiro-OMeTAD is applied by ultrasonic spray coating with I$_3$73O$_3$74 doping in solution. The Au contact is then formed by thermal or e-beam evaporation as a planar back electrode (Rahman et al., 3 Sep 2025).

Several limitations are explicit. The HEPSC study is theoretical; long-term degradation pathways such as ion migration, phase segregation, moisture exposure, and long-term thermal cycling are not simulated. Surface recombination and defect densities on curved interfaces are treated as comparable to the planar case, so process-induced damage in an experimental realization would require control. A further practical boundary is that not all morphological complexity is beneficial: the paper explicitly rejects aggressive patterning of Au because it tends to reduce PCE, indicating that electrode continuity remains a constraint. At the same time, the optimization maps show broad maxima in PCE as functions of $_3$75 and $_3$76, with no sharp efficiency collapse for $_3$77–15 nm deviations in radii; this suggests reasonable tolerance to lithographic and etch variation (Rahman et al., 3 Sep 2025).

Within those boundaries, HEPSC is presented as an integrated architecture in which a band-engineered CuO / I$_3$78O$_3$79-doped Spiro-OMeTAD double HTL and a vertically aligned half-ellipsoidal nano-texture act jointly. The first component improves hole extraction, carrier selectivity, and interfacial energy alignment; the second provides broadband antireflection, forward scattering, resonant confinement, and enhanced optical path length in an ultrathin 200 nm MAPbI$_3$80 absorber. A plausible implication is that the HEPSC concept is best understood not as a purely photonic modification or a purely interfacial one, but as a coupled optoelectronic design strategy whose reported advantages depend on both elements remaining simultaneously optimized (Rahman et al., 3 Sep 2025).