Graphene OLED Mode Partition

Adding 55 nm TiO₂ beneath the graphene OLED anode raises the paper's air-outcoupling fraction from 31.5% to 43.4%; replacing GraHIL, the paper's low-index hole-injection layer (HIL), with PEDOT:PSS raises evanescent loss from 22.4% to 28.6%. The three-anode mode budgets in Supplementary Table 2 are the primary comparison target. The GraHIL/PEDOT:PSS control in Supplementary Table 5 extends the comparison to HIL-driven mode transfer. Dreapex TMM compares the definitions and values of the paper's six channels with its seven channels at 550 nm.

Every row closes to 100%. TiO₂/graphene/GraHIL has the highest air fraction, graphene/GraHIL has the largest waveguide fraction, and ITO/GraHIL has the largest absorption fraction.

Background

In a conventional bottom-emitting OLED, generated light enters air, the thick glass substrate, organic waveguide modes, and evanescent modes near the metal cathode. Lee et al. place high-index TiO₂ beneath graphene to strengthen cavity-assisted outcoupling. Low-index GraHIL sits between graphene and the organic stack to reduce evanescent loss near the metal cathode. Supplementary Table 2 expresses the anode effect as a directly comparable monochromatic power budget; Supplementary Table 5 adds the effect of the HIL material on evanescent loss.

The paper's optical analysis includes the Purcell effect, dipole orientation, waveguide modes, and surface-plasmon loss. This case uses the Mode result to perform the same class of complete mode accounting; an externally incident propagation calculation is not treated as an emission result.

Mapping the Paper to the TMM Model

The Methods state that all organic layers outside the HIL were merged into a homogeneous optical layer while preserving total thickness. Figure 2c uses an optimized organic thickness for each TiO₂ device, and Supplementary Table 3 explicitly gives a 90 nm organic layer for the corresponding TiO₂/graphene/GraHIL model. This case uses that 90 nm equivalent organic layer as the primary comparison input. The fabricated organic stack 1 (OS1) totals 80 nm and is reserved for a later sensitivity check.

Primary model (output side → Al)

The three anode and HIL controls change only the entries below; all other settings remain fixed.

The paper does not publish the exact complex ITO index at 550 nm. This case fixes a transparent-conducting-oxide proxy of 1.90 + 0.02i before execution, so the ITO row tests the trend and error envelope rather than a point-fitted material.

Mode-channel mapping

The paper uses an intrinsic radiative quantum efficiency of $q_0=0.97$; $q_0$ is the emitter's radiative quantum efficiency before cavity modification. Its approximately 2% Other losses are consistent in scale with a Purcell-corrected non-radiative fraction. The energy closure below supports only this order-of-magnitude correspondence. If Bottom Outcoupling is nonzero or Other cannot be mapped uniquely, the terms remain separate.

Reproduction Target and Acceptance Criteria

1. All Dreapex TMM channels must close to 100% within the displayed precision.
2. The directly matched air, substrate, waveguide, evanescent, and absorption channels must be within 5 percentage points of the paper.
3. TiO₂/graphene/GraHIL must have higher Top Outcoupling than the graphene/GraHIL and ITO/GraHIL controls.
4. PEDOT:PSS must increase the Evanescent fraction relative to GraHIL, reproducing the mode-transfer direction in Supplementary Table 5.
5. Other losses enter quantitative acceptance only if the channel definitions match one-to-one; otherwise they are reported separately.

The same criteria apply to all four models. The 90 nm organic layer, emitter position, orientation, and optical constants remain fixed across the variants.

Modeling Path in Dreapex TMM

Structure

Import the corresponding model to rebuild the fixed inputs above:

• TiO₂ / graphene / GraHIL model
• Graphene / GraHIL model
• ITO / GraHIL model
• TiO₂ / graphene / PEDOT:PSS model

Set Glass as incoherent and enable Emis. on the equivalent organic layer. The stack runs from the Glass output side to the Al cathode.

Optical Settings

Enable only Mode in the Emission detector lane, set Wavelength Mode to Single and the wavelength to 550 nm, then run the calculation. All four models use the same emitter and detector settings.

Example Setup

All four models share the emitter, equivalent organic layer, LiF, Al, and optical settings. Their complete layer sequences are listed below; optical constants not repeated in the final column follow the primary-model and control tables above.

The following UI state belongs to the TiO₂/graphene/GraHIL primary model. The other three models replace the anode or HIL according to the table.

The structure screenshot verifies the 55/1.36/70/90/1/150 nm sequence and the incoherent Glass setting.

These three control screenshots verify the actual anode and HIL substitutions; the equivalent organic layer, LiF, and Al remain unchanged.

Relative position 0.2222 places the dipole plane 70 nm from Al; 0.23 is the vertical-dipole fraction.

Single-wavelength Mode does not read a broadband emitter spectrum, so the Ir(ppy)₂acac EL curve in Supplementary Figure 3 does not need to be digitized for this target.

Simulation Results and Comparison with Supplementary Tables 2 and 5

Supplementary Table 2 tests mode redistribution between anodes. Supplementary Table 5 separately tests how the low-index HIL changes evanescent loss.

Supplementary Table 2: anode structures

Both failures come from the split between Waveguide and Evanescent. Combining those adjacent high-in-plane-wave-vector channels gives simulated/paper totals of 26.74%/29.1% and 31.74%/32.4%, reducing the differences to 2.36 and 0.66 percentage points. This aggregate locates the discrepancy; it does not replace the channel-by-channel criterion. All five direct channels pass the ±5 pp threshold for graphene / GraHIL.

Supplementary Table 5: GraHIL versus PEDOT:PSS

All five directly mapped PEDOT:PSS channels pass, with a largest difference of 1.77 pp. Replacing the HIL increases Evanescent and lowers Top Outcoupling in both the paper and the simulation.

All four seven-channel budgets close to 100.0000%, and Bottom Outcoupling is below 0.000001%. Non-radiative spans 1.93–2.24%, within 0.17 pp of the paper's 2.1–2.3% Other losses. The magnitudes support the mapping, while the distinct physical labels are retained. The final verdict is therefore: energy closure, both structure-ordering checks, and the HIL-transfer trend pass; two of four variants pass every channel-level numerical threshold, while the Waveguide/Evanescent split fails for TiO₂/GraHIL and ITO/GraHIL.

Deviation Analysis

1. The paper publishes the optical constants of TiO₂, GraHIL, PEDOT:PSS, and Al only as plotted curves. Single-wavelength reading error changes both cavity phase and absorption partition.
2. Four-layer graphene uses the paper-cited effective optical model. Interlayer contamination, doping, and transfer residue in a real electrode are not represented by the homogeneous 1.36 nm film.
3. The ITO row uses a disclosed point proxy and therefore has the largest uncertainty in absorption and cavity phase.
4. The primary model follows the paper's optimized 90 nm equivalent layer, whereas fabricated OS1 totals 80 nm. The latter answers a thickness-sensitivity question rather than replacing the primary Supplementary Table 2 input.
5. The emitter region is one delta plane. The paper reports that spatial-distribution effects were minor, but a finite recombination-zone width would still smooth mode resonances slightly.
6. The largest discrepancies are confined to the Waveguide/Evanescent boundary, while their combined difference is only 0.66–2.36 pp. Total high-in-plane-wave-vector power is therefore stable, whereas its exact split is more sensitive to the paper's plotted material curves, effective-layer model, and channel boundaries.
7. Dreapex TMM lists Bottom Outcoupling and Non-radiative separately; the paper compresses the budget into six terms. Fractions without a demonstrated definition match are excluded from the pass decision.

Further Extensions

1. Keep all other inputs fixed, change the equivalent organic layer to the fabricated OS1 total of 80 nm, and quantify the structural-convention error.
2. Sweep the vertical-dipole fraction from 0.21 to 0.25 to cover the paper's 0.23 ± 0.02 range.
3. Digitize the complete Ir(ppy)₂acac EL and material dispersion in Supplementary Figure 3, then extend the comparison to spectrum-weighted Mode.
4. Enable Power Dissipation and map the integrated channels to the in-plane-wave-vector peaks in Supplementary Figures 5 and 8.

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