Recovered Dipole Orientation of a Perovskite Nanocrystal Superlattice Film

Kumar et al. (2022): p-polarized angle-resolved photoluminescence of a weak-cavity film and the horizontal dipole ratio recovered from it

AuthorLuke Cole

Anisotropic nanocrystal superlattices overcoming intrinsic light outcoupling efficiency limit in perovskite quantum dot light-emitting diodes

Authors: Sudhir Kumar, Tommaso Marcato, Frank Krumeich, Yen-Ting Li, Yu-Cheng Chiu, et al. (corresponding author Chih-Jen Shih)

Journal: Nature Communications 13, 2106 (2022) · Comparison target: Figure 3a

Whether the dipoles in an emissive layer lie flat or stand up decides how much light the same device stack can extract. The standard way to measure it is to index-match the film to a hemispherical glass lens, record p-polarized photoluminescence angle by angle, and invert that curve for the orientation ratio with a thin-film optical model.

This case builds a two-layer weak-cavity model — 18 nm of hole-transport layer plus a 30 nm emissive layer between glass and air — runs one angle sweep to get the curve of paper Figure 3a, and then recovers the horizontal dipole ratio from the 18 measured points in that figure.

Kumar Figure 3a — p-polarized photoluminescence intensity of the film versus viewing angle; circles are measurements and the four calculated curves correspond to horizontal dipole ratios of 0.67, 0.72, 0.91, and 1.00.Kumar et al., Nature Communications 13, 2106 (2022), Figure 3a CC BY

The measurements are normalized to 1 at 0°, fall monotonically to a minimum of 0.498 at 40°, jump past the glass–air critical angle to 1.034 at 50°, and then decay to 0.042 at 85°. The curve for a horizontal dipole ratio of 1 goes exactly to zero at 41.9°; the lower the ratio, the shallower the minimum near 41° and the larger the share above 41° in substrate modes.

Background

The hemispherical lens and the index-matching fluid remove the back-surface interface, so the stage reading equals the emission angle inside the glass and the substrate can be treated optically as a semi-infinite medium. That makes the substrate-mode region above 41° measurable, where a conventional measurement without a hemispherical lens traps that light by total internal reflection.

The shape of the p-polarized curve is sensitive to the spatial orientation of the transition dipole moment: dipoles parallel to the film concentrate energy at small angles, dipoles perpendicular to it send energy to large angles. The sharp feature near 41° is total internal reflection at the glass–air interface, with a critical angle of 41.14° for glass at $n=1.52$ . Beyond it the air side becomes evanescent, p-polarized emission from horizontal dipoles nearly vanishes, and vertical dipoles couple strongly into the substrate at the same angle. That is the mechanism the paper uses to resolve orientation.

The published

\Theta_{\mathrm H}

is the horizontal fraction, while the Vertical Fraction field is the vertical fraction; the two sum to 1, so the published 0.72 is entered as 0.28. Before using any orientation value from the literature, check whether it reports the horizontal fraction, the vertical fraction, or the order parameter.

Structure

The stack is just air, the emissive layer, and the hole-transport layer, plus glass treated as a semi-infinite medium. There is no metal mirror, so this is a weak-cavity system: the result is insensitive to absolute efficiency and emission depth and depends only on the two indices, the two thicknesses, and the orientation ratio.

Position	Material	Thickness	Optical input
`Incidence Medium` (detection side)	Glass	Semi-infinite	$n=1.52$ , $k=0$
Layer 1	X-F6-TAPC hole-transport layer	`18 nm`	Wavelength-resolved $n(\lambda)$ , 1.5648 at `528 nm`; $k=0$
Layer 2	Nanocrystal superlattice emissive layer	`30 nm`	$n$ below; $k=0.03548$
`Transmission Medium`	Air	Semi-infinite	$n=1$ , $k=0$

Putting glass in the Incidence Medium is the key step of this case: angular distributions are defined inside the incidence medium, so with glass there the viewing angle reaches 90° and covers the whole substrate-mode region the paper measures with its hemispherical lens. Moving glass to the Transmission Medium gives the air-side distribution instead, with no data above 41° and no critical-angle feature.

The emissive-layer extinction coefficient comes from the published 528 nm optical density of 0.011 over 30 nm, giving 0.0355. The refractive index has two published values, given in two different places:

Value at `528 nm`	Where the paper gives it	Validity
`1.760`	$n=1.76\pm0.05$ at `530 nm` in the main text and supplementary information, from ellipsometry	Stated directly, at a wavelength adjacent to the `528 nm` used here
`1.680`	Supplementary dispersion parameters evaluated at `528 nm`	That fit used only non-absorbing data over `550–850 nm`, so `528 nm` is an extrapolation

This case takes 1.760 for its main result — the paper's own supplementary efficiency calculation uses the same value. Both numbers are printed in the paper, so choosing one introduces nothing from outside it, and results for both are reported below.

The structure table holds two finite-thickness films between glass and air, with the emissive layer at a constant 1.76:

Structure page of the model using the index from the text, with a glass incidence medium, 18 nm X-F6-TAPC, a 30 nm emissive layer, and an air transmission medium

The two models using the supplementary index differ only in the emissive-layer row, switched to wavelength-resolved data from the dispersion parameters:

$Structure page of the model using the supplementary index, with the emissive layer switched to a wavelength-resolved refractive index$

Optical and Emitter Settings

The five models differ in only two places: the emissive-layer index and the Vertical Fraction.

Model	Emissive-layer $n$ at `528 nm`	`Vertical Fraction`	$\Theta_{\mathrm H}$
Published orientation, index from the text	1.760	`0.28`	0.72
Recovered orientation, index from the text	1.760	`0.24655`	0.753
Pure horizontal dipole, index from the text	1.760	`0`	1.00
Published orientation, supplementary index	1.68005	`0.28`	0.72
Recovered orientation, supplementary index	1.68005	`0.2192`	0.781

Everything else is shared:

Setting	Value
Emission spectrum	Gaussian, peak `527.4 nm`, FWHM `23.2 nm`, over `490–570 nm`
Dipole position and distribution	Relative position `0.5`, `Delta`
Detector	Emission `Intensity` (one Propagation detector is also required; this case uses `R`)
Wavelength mode	`Single`, `528 nm`
Angle mode	`Sweep`, `0°–89.5°`, step `0.5°`
Comparison quantity	The `TM` curve of `Normalized Angular Distribution`

The emitter panel should show Custom orientation, the matching Vertical Fraction, relative position 0.5, and a Delta distribution:

$Emitter settings of the model at the published orientation, showing a Custom orientation with a vertical fraction of 0.28$

The angle range must reach 89.5° to cover the whole substrate-mode region:

Intensity detector settings at the single wavelength 528 nm with an angle sweep from 0 to 89.5 degrees

Before normalization the angular distribution is strictly linear in the orientation. To compare a batch of values, run Vertical Fraction at 0 and 1 once each and combine them linearly by fraction, instead of rerunning every value.

Simulation Results and Comparison with Figure 3a

The published figure:

The four published model curves correspond to horizontal-dipole ratios of 0.67, 0.72, 0.91, and 1.00. Entering each of them as a Vertical Fraction and running gives four curves that can be drawn on the same axis as the published ones:

The four model curves and experimental points of paper Figure 3a against the Dreapex TMM angular runs on one axis — Dashed lines with open squares are independent readings of paper Figure 3a; solid lines are the real Dreapex TMM runs; green open circles are the published experimental points.Paper data from Kumar et al. Figure 3a; Dreapex TMM data from the real runs of this case; the graphic is drawn independently CC BY 4.0 (independently drawn)

All four curves reproduce the published trend: the lower the orientation ratio, the shallower the minimum near 41° and the higher the substrate-mode peak near 52°.

Horizontal-dipole ratio $\Theta_{\mathrm H}$	RMS deviation from the same published curve
`0.67`	`0.038`
`0.72`	`0.087`
`0.91`	`0.080`
`1.00`	`0.072`

The raw software results follow.

Entering the published orientation directly

Converting 0.72 into a Vertical Fraction of 0.28 and running once gives a minimum of 0.601 at 39.5° and a substrate-mode peak of 1.118 at 52.5°:

Normalized angular distribution at the published orientation with the index from the text

Releasing the orientation and fitting the 18 measured points by least squares gives 0.753 (1σ interval 0.746–0.761), with the minimum moving to 0.547 at 41.5° and the peak to 1.047 at 53.0°:

Normalized angular distribution at the recovered orientation with the index from the text

The recovered value is +0.033 from the published 0.72, and the all-angle RMS difference falls from 5.25 % to 3.57 %.

Pure horizontal dipole, with zero free parameters

Setting Vertical Fraction to 0 gives the pure horizontal-dipole curve, with no free parameters, corresponding to the $\Theta_{\mathrm H}=1$ curve of paper Figure 3a:

Normalized angular distribution for a pure horizontal dipole with the index from the text

The model zero lands at 41.5° against the paper's 41.9°, both above the theoretical critical angle of 41.14°; the offset comes from the propagation phase over the 15 nm from the dipole to the top film surface. The peak of 0.691 is at 55.5°, and the RMS difference from the paper's own curve over 28°–84° is 4.68 %.

Switching to the supplementary index

Changing the emissive-layer index to the extrapolated 1.68005 with everything else fixed raises the minimum to 0.636 at 39.0° and the peak to 1.201 at 51.5°:

Normalized angular distribution at the published orientation with the supplementary index

Releasing the orientation recovers 0.781, at which the curve itself matches the measured points well:

Normalized angular distribution at the recovered orientation with the supplementary index

Characteristic values for all five models. The reference values are: measured minimum 0.498 at 40° and substrate-mode peak 1.034 at 50°; the paper's own $\Theta_{\mathrm H}=0.72$ curve has a minimum of 0.523 at 38.7° and a peak of 1.061 at 51.9°.

Model	Minimum near `41°`	Substrate-mode peak	RMS vs. the 18 measured points
Published orientation, index from the text	`0.601` @ `39.5°`	`1.118` @ `52.5°`	`5.25 %`
Recovered orientation, index from the text	`0.547` @ `41.5°`	`1.047` @ `53.0°`	`3.57 %`
Pure horizontal dipole, index from the text	`0.000` @ `41.5°`	`0.691` @ `55.5°`	Not applicable
Published orientation, supplementary index	`0.636` @ `39.0°`	`1.201` @ `51.5°`	`8.32 %`
Recovered orientation, supplementary index	`0.555` @ `41.5°`	`1.055` @ `52.5°`	`3.10 %`

All five models move in the same direction as the paper: the lower the orientation ratio, the shallower the minimum near 41° and the higher the substrate-mode peak.

Emissive-layer $n$ at `528 nm`	Recovered $\Theta_{\mathrm H}$	1σ interval	Difference from `0.72`
1.760 (ellipsometry value in the main text)	0.753	0.746–0.761	`+0.033`
1.680 (supplementary extrapolation)	0.781	0.774–0.787	`+0.061`

The recovered value is almost entirely set by the emissive-layer index: every 0.02 of index lowers the recovered orientation ratio by about 0.007. Every other input is a weak determinant — hole-transport thickness from 5 to 35 nm, emissive-layer thickness from 24 to 40 nm, in-layer dipole position from 0.3 to 0.7, and spectral integration instead of a single wavelength together move the recovered value by only about 0.02. Digitization is not the bottleneck either: a ±0.0036 uncertainty on a normalized point propagates to a 3σ bound of only ±0.002.

Deviation Notes

The paper gives two different values for the emissive-layer index, and that is the main source of residual difference. Pixel readings of the published dispersion curve give about 1.69–1.72 at 528 nm, between the two, but the paper provides no fit parameters for that curve. The measured band the paper itself quotes is $1.76\pm0.05$ .

Over the 52°–68° substrate-mode region, the paper-to-model intensity ratio is 0.918 for the horizontal channel and 0.887 for the vertical, so the vertical channel is 2.5 percentage points weaker. That asymmetry lets this model reproduce the same substrate-mode peak height with fewer vertical dipoles, which biases the recovered orientation ratio upward.

The hole-transport thickness of the photoluminescence sample is not stated separately, so the device value of 18 nm is used. The emissive-layer extinction coefficient is a constant recovered from the optical density with no wavelength dispersion, while 528 nm sits right against that material's absorption edge.

Back to Case Studies