Perovskite LED (PeLED)

1. Theory

Definition: A perovskite LED (PeLED) is a thin-film diode whose emitter is a lead halide perovskite ( $AB X_{3}$ ) film or nanocrystal layer between charge-transport layers.
Crystal stability: The Goldschmidt tolerance factor $t = (r_{A} + r_{X}) / 2 (r_{B} + r_{X})$ predicts whether the $AB X_{3}$ structure forms.
Halide-tuned bandgap: Mixing iodide, bromide and chloride tunes emission across the visible, but reported bowing parameters vary widely, so composition-to-wavelength is uncertain.
Exciton binding: 3D perovskites have binding energies near $k_{B} T$ (about 20 to 50 meV), so emission mixes free-carrier and excitonic recombination.
Layered (quasi-2D) perovskites: Ruddlesden-Popper phases confine excitons in thin layers, raising binding energy and luminescence yield.
Defect tolerance: Sharp absorption edges and high photoluminescence yields are achieved even from low-temperature solution processing.

2. Working Principle

Injection and recombination: Carriers injected from transport layers recombine via trap-assisted ( $k_{1} n$ ), radiative ( $k_{2} n^{2}$ ) and Auger ( $k_{3} n^{3}$ ) channels.
Photon recycling: Trapped photons are re-absorbed and re-emitted, boosting external efficiency strongly when internal yield is high.
Energy funnelling: In quasi-2D films, excitons flow from wide-gap thin layers to narrow-gap thick layers, concentrating emission.
Efficiency: External quantum efficiencies above 20% have been reported, matching good OLEDs.
Stability limits: Ion migration, halide segregation and degradation limit lifetime (covered on the reliability pages).

3. Applications

Current status: Research devices; operating lifetime and lead toxicity block commercialization.
Displays and lighting (target): Low-cost, solution-processed, narrow-band emitters for wide-colour-gamut displays.

Name	Formula	Related Formulae	Variables/Constants
Structural stability of the ABX3 halide perovskite	$t = \frac{r _{A} + r _{X}}{2 ( r _{B} + r _{X} )}$	$μ_{oc t} = \frac{r _{B}}{r _{X}}$ $0.8 \leq t \leq 1.0, 0.44 \leq μ_{oc t} \leq 0.9$	$t : G o l d sc hmi d t t o l er an ce f a c t or o f t h e p er o v s k i t e A B X_{3}$ $r_{A}, r_{B}, r_{X} : i o ni c r a d ii o f t h e A c a t i o n (ces i u m, m e t h y l amm o ni u m or f or mami d ini u m), t h e B c a t i o n (l e a d or t in) an d t h e ha l i d e X (i o d i d e, b r o mi d e or c h l or i d e), t ak e n f or t h e coor d ina t i o n n u mb er s 12, 6 an d 6$ $μ_{oc t} : oc t ah e d r a l f a c t or, w hi c h m e a s u r es w h e t h er t h e B c a t i o n f i t s in t h e oc t ah e d r o n o f t h e ha l i d es$ $(a p er o v s k i t e w i t h a t o l er an ce f a c t or n e a r o n e i s c u bi c, a v a l u e b e tw ee n 0.8 an d ab o u t 0.9 g i v es t h e t i l t e d t e t r a g o na l, or t h or h o mbi c an d r h o mb o h e d r a l p ha ses, an d t h e r an g es a r e e m p i r i c a l r u l es f or t h e f or ma t i o n o f t h e s t r u c t u r e)$
Bandgap tuning by mixing the halides	$E_{g} (x) = (1 - x) E_{g}^{I} + x E_{g}^{B r} - b x (1 - x)$	$λ = \frac{1239.8}{E _{g} ( e V )} nm$	$E_{g} (x) : ban d g a p o f t h e mi x e d - ha l i d e p er o v s k i t e s u c h a s M A P b (I_{1 - x} B r_{x})_{3}$ $x : b r o mi d e f r a c t i o n o f t h e ha l i d e s i t e$ $E_{g}^{I}, E_{g}^{B r} : ban d g a p s o f t h e p u r e i o d i d e an d b r o mi d e co m p o u n d s, r e p or t e d n e a r 1.55 t o 1.65 e V an d n e a r 2.2 t o 2.4 e V f or t h e m e t h y l amm o ni u m l e a d co m p o u n d s, d e p e n d in g o n t h e r e p or t$ $b : b o w in g p a r am e t er, f or w hi c h t h e r e p or t s d i f f er f r o m a l m os t z er o t o a s i g ni f i c an t v a l u e (a c a l c u l a t i o n g a v e 0.49 e V, f r o m a f i t o v er a l imi t e d r an g e o f c a l c u l a t e d g a p s, n o t f r o m e x p er im e n t)$ $λ : e mi ss i o n w a v e l e n g t h n e a r t h e ban d g a p, in nm f or t h e e n er g y in e V$ $(t h e e mi ss i o n p e ak o f t h e f i l m l i es s l i g h tl y b e l o w t h e ab sor pt i o n e d g e, an d t h e mi x e d - ha l i d e co m p os i t i o n s c an se p a r a t e in t o i o d i d e - r i c h an d b r o mi d e - r i c h r e g i o n s u n d er l i g h t an d f i e l d, w hi c h i s n o t co v er e d h er e)$
Exciton binding energy and the free carrier fraction	$E_{b} = \frac{μ}{m _{0}} \frac{R _{y}}{ε _{e f f}^{2}}$	$\frac{x ^{2}}{1 - x} = \frac{1}{n _{t o t}} (\frac{2 π μ k _{B} T}{h ^{2}})^{3/2} exp (- \frac{E _{b}}{k _{B} T})$ $E_{b}^{2 D} \approx 4 E_{b}^{3 D}$	$E_{b} : bin d in g e n er g y o f t h e W anni er e x c i t o n in t h e h y d r o g e ni c m o d e l$ $μ : r e d u ce d ma ss o f t h e e l ec t r o n - h o l e p ai r (ab o u t 0.1 m_{0} in t h e l e a d i o d i d e an d b r o mi d e p er o v s k i t es)$ $m_{0} : f r ee e l ec t r o n ma ss$ $R_{y} : R y d b er g e n er g y, 13.6 e V$ $ε_{e f f} : e f f ec t i v e r e l a t i v e p er mi tt i v i t y, b e tw ee n t h e hi g h - f r e q u e n cy an d t h e s t a t i c v a l u e b ec a u se t h e l a tt i ce scr ee n s t h e C o u l o mb in t er a c t i o n o n l y p a r tl y, so t h e s t a t i c v a l u e u n d er es t ima t es t h e bin d in g$ $x : f r a c t i o n o f t h e p h o t oe x c i t e d e l ec t r o n - h o l e p ai r s t ha t a r e f r ee c a r r i er s a t t h er ma l e q u i l ib r i u m (t h e S aha e q u a t i o n f or e x c i t o n s)$ $n_{t o t} : t o t a l d e n s i t y o f e l ec t r o n - h o l e p ai r s (b o u n d an d f r ee)$ $k_{B}, T, h : B o l t z mann co n s t an t, t e m p er a t u r e an d P l an c k co n s t an t$ $E_{b}^{2 D}, E_{b}^{3 D} : bin d in g e n er g i es o f an i d e a l tw o - d im e n s i o na l e x c i t o n an d o f a t h r ee - d im e n s i o na l o n e (t h e f a c t or o f f o u r i s f or t h e h y d r o g e ni c a t o m in tw o d im e n s i o n s, an d t h e d i e l ec t r i c co n f in e m e n t o f t h e l a y er e d p er o v s k i t es r ai ses i t f u r t h er)$
Recombination kinetics and photoluminescence yield	$- \frac{d n}{d t} = k_{1} n + k_{2} n^{2} + k_{3} n^{3}$	$Φ_{P L} (n) = \frac{k _{2} n}{k _{1} + k _{2} n + k _{3} n ^{2}}$ $n_{p e ak} = \frac{k _{1}}{k _{3}}$	$n : d e n s i t y o f t h e p h o t oe x c i t e d or inj ec t e d c a r r i er s$ $k_{1} : m o n o m o l ec u l a r r a t e co n s t an t, t h e n o n - r a d ia t i v e r eco mbina t i o n a t t h e t r a p s (t h e S h oc k l ey - R e a d - H a l l p r ocess)$ $k_{2} : bim o l ec u l a r r a t e co n s t an t o f t h e r a d ia t i v e r eco mbina t i o n o f t h e e l ec t r o n s an d h o l es$ $k_{3} : A ug er r a t e co n s t an t$ $Φ_{P L} (n) : p h o t o l u min esce n ce y i e l d a t t h e d e n s i t y n (i t r i ses w i t h n w hi l e t h e t r a p s d o mina t e, b ec a u se t h e bim o l ec u l a r t er m g r o w s f a s t er, an d i t f a l l s a t a hi g h d e n s i t y b ec a u se o f t h e A ug er t er m)$ $n_{p e ak} : d e n s i t y o f t h e ma x im u m y i e l d (a s f or t h e A, B, C coe f f i c i e n t s o f t h e L i g h t E mi tt in g D i o d e p a g e)$ $(t h e t r a p t er m k_{1} d o mina t es b e l o w ab o u t 1 0^{16} c m^{- 3}, t h e bim o l ec u l a r t er m b e tw ee n ab o u t 1 0^{16} an d 1 0^{18} c m^{- 3} an d t h e A ug er t er m ab o v e, an d t h e d e n s i t y o f a w or k in g L E D i s in t h e r an g e o f t h e bim o l ec u l a r t er m)$
Photon recycling and the external quantum efficiency	$η_{e x t} = \frac{η _{in t} η _{esc}}{1 - η _{in t} ( 1 - η _{esc} )}$	$η_{in t} = [\frac{η _{esc}}{η _{e x t}} + (1 - η_{esc})]^{- 1}$ $η_{e x t} = \frac{η _{in t} /2 n ^{2}}{η _{in t} /2 n ^{2} + ( 1 - η _{in t} ) + L / ( α d )}$ $η_{E QE} = γ η_{e x t}$	$η_{e x t} : e x t er na l p h o t o l u min esce n ce e f f i c i e n cy o f t h e f i l m, t h e f r a c t i o n o f t h e g e n er a t e d p h o t o n s t ha t l e a v e t h e f i l m a f t er an y n u mb er o f ab sor pt i o n an d r e - e mi ss i o n e v e n t s$ $η_{in t} : in t er na l r a d ia t i v e e f f i c i e n cy, t h e p h o t o l u min esce n ce y i e l d o f t h e e mi ss i o n e v e n t$ $η_{esc} : p r o babi l i t y t ha t a p h o t o n esc a p es a t a s in g l e p a ss (r e p or t e d t o b e ab o u t 12.7% f or a g l a ss / p er o v s k i t e / ai r s am pl e; t h e pl ana r es t ima t e i s 1/2 n^{2})$ $n : r e f r a c t i v e in d e x o f t h e p er o v s k i t e f i l m$ $L : o n e min u s t h e r e f l ec t i v i t y o f t h e ba c k s u r f a ce (a p a r am e t er o f t h e p a r a s i t i c ab sor pt i o n in t h e t hi r d e x p r ess i o n)$ $α, d : ab sor pt i o n coe f f i c i e n t a t t h e e mi ss i o n w a v e l e n g t h an d t hi c k n ess o f t h e f i l m$ $γ : c ha r g e ba l an ce f a c t or o f t h e L E D (t h e e x p r ess i o n f or η_{E QE} a ss u m es t ha t t h e e l ec t r i c a l e x c i t a t i o n g i v es t h e s am e c a r r i er d i s t r ib u t i o n a s t h e o pt i c a l o n e)$ $(t h e f i r s t e x p r ess i o n i s t h e s u m o f a g eo m e t r i c ser i es o f t h e r e - e mi ss i o n e v e n t s in w hi c h e v er y p h o t o n t ha t d oes n o t esc a p e i s r e ab sor b e d b y t h e p er o v s k i t e, so i t i s t h e u pp er l imi t o f t h e e f f ec t o f p h o t o n r ecy c l in g)$
Layered (Ruddlesden-Popper) perovskite emitters	$(A^{'})_{2} A_{n - 1} B_{n} X_{3 n + 1}$	$E_{b}^{2 D} \approx 4 E_{b}^{3 D}$ $k_{F} = \frac{1}{τ _{D}} (\frac{R _{0}}{R})^{6}$	$(A^{'})_{2} A_{n - 1} B_{n} X_{3 n + 1} : co m p os i t i o n o f t h e l a y er e d p er o v s k i t e in w hi c h n l a y er s o f oc t ah e d r a a r e se p a r a t e d b y l a y er s o f a l a r g e or g ani c c a t i o n A^{'}$ $n : n u mb er o f oc t ah e d r a l l a y er s in o n e q u an t u m w e l l, n = 1 b e in g t h e na r r o w es t w e l l w i t h t h e l a r g es t g a p an d t h e l a r g es t e x c i t o n bin d in g e n er g y$ $E_{b}^{2 D}, E_{b}^{3 D} : e x c i t o n bin d in g e n er g i es in t h e i d e a l tw o - d im e n s i o na l an d t h e t h r ee - d im e n s i o na l l imi t s$ $k_{F}, τ_{D}, R_{0}, R : r a t e, d o n or l i f e t im e, F oer s t er r a d i u s an d d i s t an ce o f t h e e n er g y t r an s f er (t h e e x p r ess i o n i s e x pl ain e d o n t h e O L E D p a g e; in a f i l m w i t h a mi x t u r e o f w e l l s o f d i f f er e n t n t h e e x c i t o n s a r e t r an s f er r e d t o t h e w e l l s o f t h e l o w es t g a p, t h e e n er g y f u nn e l, w hi c h co n ce n t r a t es t h e e x c i t o n s an d r ai ses t h e r a d ia t i v e y i e l d)$

Important Questions

Ques: What makes the halide perovskite a good LED emitter, and how efficient have the devices become?

The halide perovskites combine a sharp absorption edge, a high photoluminescence yield in thin films, a bandgap tunable over the visible range by the halide, and solution processing at a low temperature. Room-temperature perovskite LEDs were first reported in 2014, with an EQE of 0.76% for the infrared device and 0.1% for the green device. In 2018 two groups reported EQEs above 20% (20.3% by Lin et al. and 20.7% at 18 mA/cm $^{2}$ by Cao et al.), which, in the words of the Cao et al. abstract, approach those of the best-performing OLEDs. Reference: Z.-K. Tan et al., “Bright light-emitting diodes based on organometal halide perovskite,” Nat. Nanotechnol. 9, 687 (2014); K. Lin et al., “Perovskite light-emitting diodes with external quantum efficiency exceeding 20 per cent,” Nature 562, 245 (2018); Y. Cao et al., “Perovskite light-emitting diodes based on spontaneously formed submicrometre-scale structures,” Nature 562, 249 (2018).

Ques: Why is the bandgap of a mixed-halide perovskite hard to predict?

The reported bowing parameter of MAPb(I $_{1 - x}$ Br $_{x}$ ) $_{3}$ ranges from almost zero to a significant value, and the gaps of the end compounds also differ between reports (1.56 to 1.65 eV for the iodide and 2.23 to 2.37 eV for the bromide, in a table of a recent paper). With $E_{g}^{I} = 1.56$ eV and $E_{g}^{B r} = 2.23$ eV, $x = 0.5$ gives 1.895 eV (654 nm) for $b = 0$ and 1.77 eV (699 nm) for $b = 0.49$ eV (the value of a calculation in that paper), a difference of 45 nm, so a composition cannot be read from a wavelength with confidence. The mixed halides can also separate into iodide-rich and bromide-rich regions during operation, which is a stability issue covered on the reliability pages. Reference: The band gaps and the calculated bowing are those of arXiv:2409.16775 (“On the band gap variation in CH $_{3}$ NH $_{3}$ Pb(I $_{1 - x}$ Br $_{x}$ ) $_{3}$ ”); the experimental values attributed there to earlier work (including J.H. Noh et al., “Chemical management for colorful, efficient, and stable inorganic-organic hybrid nanostructured solar cells,” Nano Lett. 13, 1764 (2013)) were not re-checked against the originals.

Ques: Are the electrons and holes in a lead halide perovskite bound as excitons at room temperature?

The exciton binding energy of the 3D lead halide perovskites is of the order of the thermal energy: a review gives 19 to 32 meV for MAPbI $_{3}$ , 53 meV for MAPbBr $_{3}$ and 22 meV for FAPbBr $_{3}$ , while $k_{B} T$ is 26 meV, and states that these values are more an upper bound than an exact value. The hydrogenic formula with $μ = 0.104 m_{0}$ gives 2.3 meV for the static permittivity of 25, which is far too low, and 57 meV for a permittivity of 5 (computed), because the lattice screening is frequency dependent. With the Saha expression of the table and $μ = 0.104 m_{0}$ at 300 K, the free fraction of the pairs is 0.94 for $E_{b} = 25$ meV at $n_{t o t} = 1 0^{16}$ cm $^{- 3}$ and 0.70 at $1 0^{17}$ cm $^{- 3}$ , and 0.86 for $E_{b} = 53$ meV at $1 0^{16}$ cm $^{- 3}$ (computed), so the emission of a 3D film at the density of an LED is a mixture of the free-carrier and the excitonic processes. The layered perovskites have a much higher binding energy (about four times in the ideal 2D limit), so they emit from excitons. Reference: The binding energies are those quoted in the review arXiv:2007.15882 (“Shining light on photoluminescence properties of metal halide perovskites”); the Saha form is the standard one for the equilibrium of excitons and free carriers and was not checked against a source for perovskites.

Ques: How does photon recycling raise the efficiency of a perovskite LED?

The film has a high index, so for a planar film the probability that a photon escapes at one pass is only about $1/2 n^{2}$ , 10% for $n = 2.2$ . A photon that does not escape can be reabsorbed by the perovskite, which is a strong absorber at the emission wavelength, and be re-emitted with the probability $η_{in t}$ , and the sum over all re-emission events gives the efficiency of the first row. With $η_{esc} = 0.103$ the external efficiency is 0.094 for $η_{in t} = 0.5$ , 0.19 for 0.7, 0.48 for 0.9 and 0.91 for 0.99, so the efficiency is a steep function of the internal yield. This is the upper limit, because the parasitic absorption in the electrodes and in the layers (the last term in the third expression) is not included in it. A review of the photoluminescence of the perovskites gives the same relation between the internal and the external yield and a value of 12.7% for the escape probability of a glass/perovskite/air sample. Reference: arXiv:2007.15882, “Shining light on photoluminescence properties of metal halide perovskites” (the equations and the 12.7% are those of that review); photon recycling in a thin-film LED was used by I. Schnitzer, E. Yablonovitch, C. Caneau and T.J. Gmitter, “Ultrahigh spontaneous emission quantum efficiency, 99.7% internally and 72% externally, from AlGaAs/GaAs/AlGaAs double heterostructures,” Appl. Phys. Lett. 62, 131 (1993), and I. Schnitzer et al., “30% external quantum efficiency from surface textured, thin-film light-emitting diodes,” Appl. Phys. Lett. 63, 2174 (1993).

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