Light Emitting Diode (Homojunction and Heterojunction LED)

1. Theory

Definition: A light-emitting diode (LED) is a forward-biased PN diode in which injected electrons and holes recombine radiatively and the photons escape through the surface.
Scope: This page holds quantities common to all LEDs; quantum-well, UV, organic, quantum-dot, perovskite, micro-, superluminescent and resonant-cavity LEDs have their own pages.
Direct bandgap: Efficient emission needs the conduction-band minimum and valence-band maximum at the same momentum, so LEDs use direct-gap III-V alloys (AlGaAs, AlGaInP, InGaN) rather than Si or Ge.
Homojunction vs double heterostructure: A double heterostructure sandwiches a narrow-gap active layer between wider-gap layers, confining carriers, avoiding re-absorption and improving injection efficiency.
ABC model: Recombination rate $R = A n + B n^{2} + C n^{3}$ (Shockley-Read-Hall, radiative and Auger terms); internal efficiency is the radiative share.
Photon energy: Emission is near the bandgap, peaking at $E_{g} + k_{B} T /2$ with a width of about $1.8 k_{B} T$ .

2. Working Principle

Injection: Forward bias injects electrons into the p-side and holes into the n-side (or both into the active layer of a heterostructure).
Recombination: Carriers recombine radiatively ( $B n^{2}$ ) or non-radiatively through defects ( $A n$ ) and Auger processes ( $C n^{3}$ ).
Efficiency droop: Internal efficiency peaks at $n = A / C$ and falls at higher current as Auger loss grows faster than emission.
Light extraction: Total internal reflection traps most light; a planar GaAs chip extracts only about 1–2% through one face, so roughening, shaping and thin-film designs are used.
Efficiencies: External quantum efficiency is the product of injection, radiative and extraction efficiencies, $η_{E QE} = η_{inj} η_{r a d} η_{e x t r}$ ; wall-plug efficiency also includes the voltage.
Temperature: The bandgap narrows (Varshni), red-shifting the emission, and non-radiative losses reduce output.
Modulation speed: Bandwidth $f \approx 1/2 π τ$ , limited by the carrier lifetime (tens of MHz typical), slower than a laser.
Current spreading: A finite spreading length in the top layer sets contact and chip design.

3. Applications

Infrared emitters: Remote controls, optocouplers, proximity and smoke sensors (GaAs and AlGaAs).
Indicators and signalling: Status lamps, displays and traffic or automotive signals.
Short-reach optical links: Plastic optical fibre and low-cost free-space data links where laser coherence is not needed.

Name	Formula	Related Formulae	Variables/Constants
Recombination rate and internal quantum efficiency (ABC model)	$R (n) = A n + B n^{2} + C n^{3}$	$η_{r a d} = \frac{B n ^{2}}{A n + B n ^{2} + C n ^{3}}$ $n_{p e ak} = \frac{A}{C}$ $η_{r a d, ma x} = \frac{B}{B + 2 A C}$ $τ_{d} = \frac{1}{A + 2 B n + 3 C n ^{2}}$	$R : t o t a l r eco mbina t i o n r a t e p er u ni t v o l u m e o f t h e a c t i v e r e g i o n (u ni t c m^{- 3} s^{- 1})$ $n : c a r r i er d e n s i t y in t h e a c t i v e r e g i o n, t ak e n e q u a l f or e l ec t r o n s an d h o l es (t h e a c t i v e r e g i o n i s u n d o p e d or t h e inj ec t i o n i s hi g h)$ $A : S h oc k l ey - R e a d - H a l l (d e f ec t) coe f f i c i e n t, n o n - r a d ia t i v e an d l in e a r in n$ $B : bim o l ec u l a r r a d ia t i v e coe f f i c i e n t$ $C : A ug er coe f f i c i e n t, n o n - r a d ia t i v e an d c u bi c in n (i t in c l u d es t h e c a r r i er l e ak a g e t ha t i s m o d e l l e d w i t h a c u bi c t er m in so m e p a p er s)$ $η_{r a d} : r a d ia t i v e e f f i c i e n cy o f t h e a c t i v e r e g i o n, t h e f r a c t i o n o f t h e r eco mbina t i o n e v e n t s t ha t e mi t a p h o t o n$ $n_{p e ak} : c a r r i er d e n s i t y a t w hi c h η_{r a d} i s ma x im u m$ $η_{r a d, ma x} : t h e ma x im u m v a l u e o f η_{r a d} (e f f i c i e n cy f a l l s a t hi g h er n b ec a u se o f t h e A ug er t er m, w hi c h i s t h e e f f i c i e n cy d r oo p)$ $τ_{d} : d i f f er e n t ia l c a r r i er l i f e t im e a t t h e o p er a t in g p o in t, t h e in v er se o f d R / d n, u se d f or t h e s ma l l - s i g na l m o d u l a t i o n r es p o n se$
Steady-state carrier density and drive current	$J = \frac{q d}{η _{inj}} (A n + B n^{2} + C n^{3})$	$n \approx \frac{η _{inj} J τ}{q d}$ $τ = \frac{1}{A + B n + C n ^{2}}$	$J : c u r r e n t d e n s i t y t h r o ug h t h e d i o d e$ $q : e l e m e n t a r y c ha r g e$ $d : t hi c k n ess o f t h e a c t i v e r e g i o n (a t hinn er d o u b l e h e t er os t r u c t u r e g i v es a hi g h er c a r r i er d e n s i t y f or t h e s am e c u r r e n t, b ec a u se t h e c a r r i er s a r e co n f in e d b y t h e ba r r i er s o f t h e w i d er - g a p l a y er s)$ $η_{inj} : inj ec t i o n e f f i c i e n cy, t h e f r a c t i o n o f t h e t er mina l c u r r e n t t ha t i s inj ec t e d in t o t h e a c t i v e r e g i o n a s t h e u se f u l c a r r i er t y p e$ $n : c a r r i er d e n s i t y in t h e a c t i v e r e g i o n$ $A, B, C : r eco mbina t i o n coe f f i c i e n t s o f t h e A B C m o d e l$ $τ : e f f ec t i v e c a r r i er l i f e t im e a t d e n s i t y n (t h e a pp r o x ima t i o n f or n h o l d s w h e n τ i s t r e a t e d a s co n s t an t, t ha t i s a t l o w inj ec t i o n)$
Injection efficiency: homojunction and heterojunction	$η_{inj} = \frac{J _{n}}{J _{n} + J _{p}}$	$(\frac{J _{n}}{J _{p}})_{h o m o} = \frac{D _{n} L _{p} N _{D}}{D _{p} L _{n} N _{A}}$ $(\frac{J _{p}}{J _{n}})_{h e t er o} = \frac{D _{p} L _{n} N _{A}}{D _{n} L _{p} N _{D}} exp (- \frac{Δ E _{g}}{k _{B} T})$	$η_{inj} : inj ec t i o n e f f i c i e n cy o f e l ec t r o n s in t o t h e p - t y p e a c t i v e r e g i o n (t h e l i g h t i s p r o d u ce d o n t h e p s i d e in t hi s c a se)$ $J_{n}, J_{p} : e l ec t r o n an d h o l e co m p o n e n t s o f t h e d i f f u s i o n c u r r e n t a t t h e j u n c t i o n (J_{p} i s t h e u se l ess h o l e c u r r e n t inj ec t e d in t o t h e n s i d e)$ $D_{n}, D_{p} : d i f f u s i o n coe f f i c i e n t s o f e l ec t r o n s an d h o l es$ $L_{n}, L_{p} : d i f f u s i o n l e n g t h s o f e l ec t r o n s in t h e p s i d e an d o f h o l es in t h e n s i d e$ $N_{D}, N_{A} : d o n or d e n s i t y o f t h e n s i d e an d a cce pt or d e n s i t y o f t h e p s i d e$ $Δ E_{g} : ban d g a p o f t h e n - t y p e e mi tt er min u s t h e ban d g a p o f t h e p - t y p e a c t i v e l a y er, p os i t i v e f or a w i d e - g a p e mi tt er$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e$ $(t h e h e t er o j u n c t i o n e x p r ess i o n n e g l ec t s t h e d i f f er e n ce o f t h e e f f ec t i v e d e n s i t i es o f s t a t es o f t h e tw o ma t er ia l s an d a ss u m es t h e s h or t - d i o d e or l o n g - d i o d e f or m i s t h e s am e o n b o t h s i d es)$
Diode current-voltage characteristic and forward voltage	$J = J_{s} [exp (\frac{q V _{j}}{n _{i d} k _{B} T}) - 1]$	$J_{s} = q (\frac{D _{p}}{L _{p} N _{D}} + \frac{D _{n}}{L _{n} N _{A}}) n_{i}^{2}$ $V \approx V_{j} + I R_{s}$ $V_{j} \approx \frac{E _{g}}{q}$	$J : c u r r e n t d e n s i t y$ $J_{s} : s a t u r a t i o n c u r r e n t d e n s i t y o f t h e d i f f u s i o n c u r r e n t in t h e l o n g - d i o d e c a se$ $V_{j} : v o l t a g e a cr oss t h e j u n c t i o n$ $n_{i d} : i d e a l i t y f a c t or, ab o u t o n e f or d i f f u s i o n c u r r e n t an d ab o u t tw o f or r eco mbina t i o n in t h e d e pl e t i o n r e g i o n$ $n_{i} : in t r in s i c c a r r i er d e n s i t y, w hi c h f a l l s e x p o n e n t ia l l y w i t h t h e ban d g a p so J_{s} i s v er y s ma l l f or a w i d e - g a p L E D$ $V : t er mina l v o l t a g e$ $I, R_{s} : c u r r e n t an d ser i es r es i s t an ce o f t h e co n t a c t s an d t h e n e u t r a l r e g i o n s$ $E_{g} : ban d g a p o f t h e a c t i v e r e g i o n (t h e f or w a r d v o l t a g e a t t h e o p er a t in g c u r r e n t i s c l ose t o E_{g} / q pl u s t h e d r o p a cr oss R_{s}, an d i t i s hi g h er in a h e t er os t r u c t u r e w i t h ban d o f f se t s; V_{j} \approx E_{g} / q i s a r u l e o f t h u mb f or t h e o p er a t in g p o in t an d d oes n o t h o l d a t l o w c u r r e n t)$ $k_{B} : B o l t z mann co n s t an t$
Emission spectrum of spontaneous emission	$I (E) \propto E - E_{g} exp (- \frac{E - E _{g}}{k _{B} T})$	$E_{p e ak} = E_{g} + \frac{k _{B} T}{2}$ $Δ E_{F W H M} \approx 1.8 k_{B} T$ $Δ λ_{F W H M} \approx \frac{1.8 k _{B} T λ ^{2}}{h c}$ $λ (μ m) = \frac{1.2398}{E ( e V )}$	$I (E) : e mi tt e d in t e n s i t y p er u ni t p h o t o n e n er g y f r o m ban d - t o - ban d r eco mbina t i o n o f a n o n - d e g e n er a t e se mi co n d u c t or (t h e p r o d u c t o f t h e j o in t d e n s i t y o f s t a t es an d t h e B o l t z mann d i s t r ib u t i o n)$ $E : p h o t o n e n er g y$ $E_{g} : ban d g a p e n er g y o f t h e a c t i v e r e g i o n$ $E_{p e ak} : p h o t o n e n er g y a t t h e ma x im u m o f t h e s p ec t r u m$ $Δ E_{F W H M} : f u l l w i d t h a t ha l f ma x im u m in e n er g y (1.795 k_{B} T w h e n so l v e d n u m er i c a l l y f or t hi s l in es ha p e, q u o t e d a s 1.8 k_{B} T)$ $Δ λ_{F W H M} : f u l l w i d t h a t ha l f ma x im u m in w a v e l e n g t h, o b t ain e d f r o m Δ E w i t h Δ λ = λ^{2} Δ E / h c$ $λ : p e ak w a v e l e n g t h$ $h, c : P l an c k co n s t an t an d s p ee d o f l i g h t$ $k_{B} : B o l t z mann co n s t an t$ $(a l l oy d i sor d er, ban d f i l l in g a t hi g h inj ec t i o n an d t h e q u an t u m co n f in e m e n t b r o a d e n t h e r e a l s p ec t r u m b ey o n d t hi s i d e a l l in es ha p e)$
Temperature dependence of the wavelength and of the output	$E_{g} (T) = E_{g} (0) - \frac{α T ^{2}}{T + β}$	$\frac{d λ}{d T} = - \frac{λ ^{2}}{h c} \frac{d E _{g}}{d T}$ $P (T) = P (T_{0}) exp (- \frac{T - T _{0}}{T _{1}})$	$E_{g} (T) : ban d g a p a t t e m p er a t u r e T (t h e V a r s hni e q u a t i o n)$ $E_{g} (0) : ban d g a p a t z er o t e m p er a t u r e$ $α, β : V a r s hni p a r am e t er s, f i tt e d f or e a c h ma t er ia l$ $\frac{d λ}{d T} : s hi f t o f t h e p e ak w a v e l e n g t h w i t h t h e j u n c t i o n t e m p er a t u r e, p os i t i v e b ec a u se t h e ban d g a p f a l l s w i t h t e m p er a t u r e$ $P (T) : o pt i c a l o u tp u t p o w er a t a co n s t an t c u r r e n t an d j u n c t i o n t e m p er a t u r e T$ $T_{0} : r e f er e n ce t e m p er a t u r e$ $T_{1} : c ha r a c t er i s t i c t e m p er a t u r e o f t h e L E D, an e m p i r i c a l p a r am e t er (a l a r g er T_{1} m e an s a w e ak er l oss o f o u tp u t w i t h h e a t in g)$
Light extraction from a planar chip	$θ_{c} = arcsin (\frac{n _{s}}{n _{se mi}})$	$η_{co n e} = \frac{1 - c o s θ _{c}}{2} \approx \frac{1}{4} (\frac{n _{s}}{n _{se mi}})^{2}$ $T_{F} = \frac{4 n _{se mi} n _{s}}{( n _{se mi} + n _{s} ) ^{2}}$ $η_{e x t r} \approx η_{co n e} T_{F}$	$θ_{c} : cr i t i c a l an g l e f or t o t a l in t er na l r e f l ec t i o n a t t h e se mi co n d u c t or s u r f a ce$ $n_{s} : r e f r a c t i v e in d e x o f t h e s u r r o u n d in g m e d i u m (ai r or t h e e n c a p s u l an t)$ $n_{se mi} : r e f r a c t i v e in d e x o f t h e se mi co n d u c t or$ $η_{co n e} : f r a c t i o n o f i so t r o p i c a l l y e mi tt e d p h o t o n s t ha t f a l l in s i d e t h e esc a p e co n e o f o n e s u r f a ce$ $T_{F} : F r es n e l t r an s mi ss i o n a t n or ma l in c i d e n ce (t h e v a l u e a t o t h er an g l es in s i d e t h e co n e i s d i f f er e n t)$ $η_{e x t r} : e x t r a c t i o n e f f i c i e n cy t h r o ug h o n e s u r f a ce w i t h o u t r e ab sor pt i o n in t h e c hi p an d w i t h o u t r e f l ec t i o n f r o m t h e ba c k co n t a c t (a mi r r or or a seco n d s u r f a ce c an a dd t o i t, an d r o ug h e nin g or s ha p in g in cr e a ses i t b y r an d o mi z in g t h e p h o t o n d i r ec t i o n)$
External quantum efficiency, wall-plug efficiency and luminous flux	$η_{E QE} = η_{inj} η_{r a d} η_{e x t r}$	$P_{o pt} = η_{E QE} \frac{h ν}{q} I$ $η_{w p} = \frac{P _{o pt}}{I V} = η_{E QE} \frac{h ν}{q V}$ $Φ_{v} = 683 lm/W \cdot V (λ) P_{o pt}$	$η_{E QE} : e x t er na l q u an t u m e f f i c i e n cy, t h e n u mb er o f p h o t o n s e mi tt e d in t o f r ee s p a ce p er inj ec t e d e l ec t r o n$ $η_{inj}, η_{r a d}, η_{e x t r} : inj ec t i o n, r a d ia t i v e an d e x t r a c t i o n e f f i c i e n c i es (t h e p r o d u c t o f t h e f i r s t tw o i s t h e in t er na l q u an t u m e f f i c i e n cy)$ $P_{o pt} : e mi tt e d o pt i c a l p o w er$ $h ν : p h o t o n e n er g y$ $I, V : d r i v e c u r r e n t an d t er mina l v o l t a g e$ $η_{w p} : w a l l - pl ug e f f i c i e n cy, t h e o pt i c a l p o w er o u t d i v i d e d b y t h e e l ec t r i c a l p o w er in$ $Φ_{v} : l u min o u s f l ux o f a m o n oc h r o ma t i c so u r ce$ $V (λ) : p h o t o p i c l u min o u s e f f i c i e n cy f u n c t i o n o f t h e ey e, e q u a l t o o n e a t 555 nm (f or a b r o a d s p ec t r u m t h e p r o d u c t i s in t e g r a t e d o v er t h e w a v e l e n g t h)$
Modulation bandwidth	$f_{τ} = \frac{1}{2 π τ _{d}}$	$f_{R C} = \frac{1}{2 π R _{s} C _{d}}$ $\frac{P ( ω )}{P ( 0 )} = \frac{1}{1 + ( ω τ _{d} ) ^{2}}$	$f_{τ} : m o d u l a t i o n f r e q u e n cy a t w hi c h t h e m o d u l a t e d o pt i c a l p o w er f a l l s t o 1/ 2 o f i t s l o w - f r e q u e n cy v a l u e, se t b y t h e c a r r i er l i f e t im e . T hi s i s t h e u s u a l e l ec t r i c a l 3 d B ban d w i d t h, s in ce t h e d e t ec t e d e l ec t r i c a l p o w er g oes a s t h e s q u a r e o f t h e o pt i c a l p o w er; t h e o pt i c a l 3 d B p o in t (o pt i c a l p o w er ha l v e d) i s 3 f_{τ}$ $τ_{d} : d i f f er e n t ia l c a r r i er l i f e t im e a t t h e o p er a t in g p o in t (i t f a l l s w i t h t h e c u r r e n t, so t h e ban d w i d t h r i ses w i t h t h e bia s)$ $f_{R C} : l imi t se t b y t h e ser i es r es i s t an ce an d t h e j u n c t i o n an d p a d c a p a c i t an ce$ $R_{s} : ser i es r es i s t an ce$ $C_{d} : t o t a l c a p a c i t an ce o f t h e d i o d e, t h e s u m o f t h e d e pl e t i o n an d d i f f u s i o n c a p a c i t an ce an d t h e p a d c a p a c i t an ce$ $\frac{P ( ω )}{P ( 0 )} : s ma l l - s i g na l r es p o n se o f t h e o pt i c a l p o w er f or a s in g l e t im e co n s t an t$ $ω : an g u l a r m o d u l a t i o n f r e q u e n cy$ $(t h e tw o l imi t s co mbin e a pp r o x ima t e l y a s t h e l o w er o f t h e tw o, an d t h e L E D ha s n o r eso nan ce p e ak u n l ik e t h e l a ser d i o d e)$
Current spreading in the top layer	$L_{s} = \frac{n _{i d} k _{B} T t _{p}}{q ρ _{p} J _{e}}$	$ρ_{s h} = \frac{ρ _{p}}{t _{p}}$ $L_{s} = \frac{n _{i d} k _{B} T}{q ρ _{s h} J _{e}}$	$L_{s} : c u r r e n t s p r e a d in g l e n g t h, t h e d i s t an ce f r o m t h e e d g e o f t h e t o p co n t a c t o v er w hi c h t h e c u r r e n t d e n s i t y f a l l s b y a f a c t or o f or d er e (t h e r es u l t o f t h e ba l an ce b e tw ee n t h e l a t er a l v o l t a g e d r o p an d t h e e x p o n e n t ia l d i o d e l a w)$ $n_{i d} : i d e a l i t y f a c t or o f t h e d i o d e$ $t_{p}, ρ_{p} : t hi c k n ess an d r es i s t i v i t y o f t h e l a y er in w hi c h t h e c u r r e n t s p r e a d s$ $J_{e} : c u r r e n t d e n s i t y t h r o ug h t h e j u n c t i o n a t t h e e d g e o f t h e co n t a c t$ $(t h e o n e - d im e n s i o na l m o d e l o f G . H . B . T h o m p so n, a s g i v e n in S c h u b er t, sec t i o n 8.2, f or a r es i s t i v e t o p l a y er o n a l o w - r es i s t an ce s u b s t r a t e; G a N c hi p s o n in s u l a t in g s u b s t r a t es n ee d a m o d e l t ha t in c l u d es t h e n - l a y er an d t h e co n t a c t r es i s t an ce)$ $ρ_{s h} : s h ee t r es i s t an ce o f t h e s p r e a d in g l a y er$ $k_{B} : B o l t z mann co n s t an t$ $(a s p r e a d in g l e n g t h s h or t er t han t h e c hi p w i d t h mak es t h e l i g h t co m e f r o m t h e e d g e o f t h e co n t a c t, an d i t i s in cr e a se d b y a c u r r e n t s p r e a d in g l a y er or a f in g er co n t a c t)$

Important Questions

Ques: Why are LEDs made from direct-bandgap materials, and why is a double heterostructure used?

A photon carries almost no momentum, so band-to-band recombination needs the conduction band minimum and the valence band maximum at the same crystal momentum. In an indirect-gap semiconductor such as Si or Ge the radiative coefficient $B$ is several orders of magnitude lower (about $1 0^{- 14}$ cm $^{3}$ s $^{- 1}$ for Si against about $1 0^{- 10}$ cm $^{3}$ s $^{- 1}$ for GaAs) and the non-radiative terms dominate, so the efficient visible and infrared LEDs use direct-gap III-V alloys (AlGaAs, AlGaInP, InGaN). A double heterostructure puts a thin narrow-gap active layer between two wider-gap layers: the barriers confine the carriers (a higher $n$ for the same current, as in the row of the carrier density), the wide-gap layers do not absorb the emission, and the injection ratio is improved by the factor $exp (- Δ E_{g} / k_{B} T)$ of the heterojunction row. Reference: E.F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge University Press, 2006), chapters 2, 4 and 7.

Ques: Why does the efficiency of an LED fall at high current (efficiency droop)?

In the ABC model $η_{r a d} = B n / (A + B n + C n^{2})$ is maximum at $n_{p e ak} = A / C$ and falls at a higher density, because the Auger term grows as $n^{3}$ while the radiative term grows as $n^{2}$ . As an illustration (assumed values, not measured ones), $A = 1 0^{7}$ s $^{- 1}$ , $B = 2 \times 1 0^{- 11}$ cm $^{3}$ s $^{- 1}$ and $C = 1 0^{- 30}$ cm $^{6}$ s $^{- 1}$ give $n_{p e ak} = 3.2 \times 1 0^{18}$ cm $^{- 3}$ and $η_{r a d, ma x} = 0.76$ . The physical origin of the cubic loss in InGaN LEDs (direct Auger recombination, carrier leakage out of the wells, or the poor hole injection) has been debated, and the coefficients reported in the literature differ by orders of magnitude. Shen et al. measured $C = 1.4$ to $2.0 \times 1 0^{- 30}$ cm $^{6}$ s $^{- 1}$ in InGaN layers by photoluminescence, which is the order of the assumed value above. The identification of the mechanism by the electron-emission experiment of Iveland et al. has been disputed (see the comment of Bertazzi et al., arXiv:1305.2512), so the model is used as a fit to the measured efficiency curve. Reference: Y.C. Shen et al., “Auger recombination in InGaN measured by photoluminescence,” Appl. Phys. Lett. 91, 141101 (2007); J. Iveland et al., “Direct measurement of Auger electrons emitted from a semiconductor light-emitting diode under electrical injection: identification of the dominant mechanism for efficiency droop,” Phys. Rev. Lett. 110, 177406 (2013).

Ques: Why does a planar LED extract only a few percent of the light?

The light is emitted in all directions inside a semiconductor of high index, and only the photons inside the escape cone ( $θ < θ_{c}$ ) leave. For GaAs ( $n_{se mi} = 3.5$ ) in air $θ_{c} = 16. 6^{\circ}$ , the cone holds $(1 - cos θ_{c}) /2 = 2.1%$ of the isotropic emission, the Fresnel transmission at normal incidence is $T_{F} = 0.69$ , and $η_{e x t r} \approx 1.4%$ through one surface. For GaN ( $n \approx 2.5$ ) the same numbers are $θ_{c} = 23. 6^{\circ}$ , 4.2%, 0.82 and 3.4%. An epoxy dome of index 1.5 on GaAs raises the cone to a half-angle of $25. 4^{\circ}$ and $η_{e x t r}$ to about 4.1% for a single escape surface in this simple model. Surface roughening, photonic crystals, shaped chips (for example the truncated inverted pyramid) and thin-film (flip-chip) designs raise the extraction further by changing the direction of the trapped photons or by letting them be re-emitted after absorption. Reference: E.F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge University Press, 2006), chapter 9.

Ques: What sets the width of the LED spectrum, and how does the width in wavelength depend on the colour?

The line shape $E - E_{g} exp (- (E - E_{g}) / k_{B} T)$ peaks at $E_{g} + k_{B} T /2$ and has a full width of about $1.8 k_{B} T$ (1.795 $k_{B} T$ exactly for this lineshape), which is 46.4 meV at 300 K. The width in wavelength follows from $Δ λ = Δ E λ^{2} / (h c)$ and grows as $λ^{2}$ : about 7.6 nm at 450 nm and 15.8 nm at 650 nm for the same energy width. The measured width is larger than this ideal value in alloys because of the alloy disorder and in quantum wells because of the thickness fluctuations, so the ideal width is a lower limit. Reference: E.F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge University Press, 2006), chapter 5.

Ques: Why does the wavelength shift to the red, and the output fall, when the LED heats up?

The bandgap narrows with temperature, which is described by the Varshni equation, so the peak shifts to a longer wavelength. For GaAs with $E_{g} (0) = 1.519$ eV, $α = 5.405 \times 1 0^{- 4}$ eV/K and $β = 204$ K, the gap is 1.422 eV at 300 K and 1.399 eV at 350 K, which moves the band edge from 872 nm to 886 nm (the values are recalled for GaAs and are an illustration of the size of the effect). The output falls because the non-radiative rates rise and the injection and the carrier confinement worsen, and the fall is summarized by the empirical $T_{1}$ of the output-power equation. In a continuously driven LED the electrical power $I V$ not emitted as light heats the junction, so the shift and the loss also follow the drive current. Reference: Y.P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149 (1967); E.F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge University Press, 2006), sections 5.7 and chapter 6.

Ques: What limits the modulation speed of an LED, and why is it slower than a laser diode?

The LED responds as a single-pole system with the carrier lifetime, so the optical bandwidth is $1/ (2 π τ_{d})$ , and the RC time of the diode adds a second limit. With the assumed ABC values above, $τ_{d} = 1/ (A + 2 B n + 3 C n^{2}) = 6$ ns at $n_{p e ak}$ , which gives about 26 MHz (the electrical 3 dB bandwidth, as defined in the bandwidth row). A higher current shortens the lifetime and raises the bandwidth, at the cost of the efficiency. A laser diode is faster because the stimulated recombination shortens the effective lifetime above threshold, and its response has a relaxation-oscillation peak, which the LED does not have. Reference: E.F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge University Press, 2006), chapter 24.

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