Quantum Well LED (InGaN-GaN and AlGaInP)

1. Theory

Definition: A quantum-well LED is a double-heterostructure LED whose active layer is one or more thin quantum wells, with InGaN/GaN (blue and green) and AlGaInP (red to amber) as the two main materials.
Quantum confinement: Wells only a few nanometres thick form discrete subbands ( $E_{n} \propto n^{2} / L_{w}^{2}$ ) and a step-like 2D density of states.
Polarization field (InGaN): Spontaneous and piezoelectric polarization in c-plane nitrides creates MV/cm fields in the wells.
Quantum-confined Stark effect: The field separates electrons and holes, lowering their overlap and radiative rate and red-shifting the emission (by about $q F_{w} L_{w}$ only in wide wells; in 2 to 3 nm wells the shift must be computed).
Alloy bandgap: InGaN’s gap follows a bowing formula between GaN (3.43 eV) and InN (0.64 to 0.78 eV), so indium content sets the colour.
AlGaInP limits: Lattice-matched to GaAs, its direct gap ends near the direct-indirect crossover (~2.26 eV) and its conduction-band offset is small.

2. Working Principle

Carrier capture: Injected carriers fall into the wells and recombine across the confined gap.
Field screening: Rising current screens the polarization field, blue-shifting the emission.
Multiple wells: Spreading carriers over several wells lowers density and Auger loss, but holes pile up in the wells nearest the p-side.
Electron blocking layer: A wide-gap layer on the p-side reduces electron overflow and helps hole injection.
Thermionic escape: Carriers leak over shallow barriers, with escape time falling exponentially with barrier height and rising temperature (worst for AlGaInP electrons).
Green gap: More indium for green brings strain, stronger fields, lower overlap and more defects, so green LEDs are less efficient than blue or red.

3. Applications

General lighting: Blue InGaN LEDs with yellow phosphor make white LEDs for homes, streets and buildings.
Display backlights and full-colour displays: LCD backlights, video walls and signage using RGB LEDs.
Automotive and signal lighting: Headlamps (InGaN) and red/amber brake, turn and traffic lights (AlGaInP).
Horticulture and specialty lighting: Tunable blue and red spectra for plant growth and medical lighting.

Name	Formula	Related Formulae	Variables/Constants
Subband energies of a quantum well	$E_{n} = \frac{ℏ ^{2} π ^{2} n ^{2}}{2 m ^{*} L _{w}^{2}}$	$\frac{k}{m _{w}} tan (\frac{k L _{w}}{2}) = \frac{κ}{m _{b}}$ $- \frac{k}{m _{w}} cot (\frac{k L _{w}}{2}) = \frac{κ}{m _{b}}$ $k = \frac{2 m _{w} E}{ℏ}$ $κ = \frac{2 m _{b} ( V _{0} - E )}{ℏ}$	$E_{n} : e n er g y o f t h e n - t h s u bban d ab o v e t h e ban d e d g e o f t h e w e l l f or an in f ini t e ba r r i er (an u pp er l imi t o f t h e co n f in e m e n t e n er g y)$ $ℏ : r e d u ce d P l an c k co n s t an t$ $m^{*} : e f f ec t i v e ma ss o f t h e c a r r i er in t h e w e l l$ $L_{w} : w e l l w i d t h$ $n : s u bban d in d e x, n = 1, 2, 3$ $k, κ : w a v e n u mb er in t h e w e l l an d d ec a y co n s t an t in t h e ba r r i er o f a f ini t e w e l l$ $m_{w}, m_{b} : e f f ec t i v e ma sses in t h e w e l l an d in t h e ba r r i er (t h e f a c t or s 1/ m in t h e co n d i t i o n s co m e f r o m t h e co n t in u i t y o f t h e d er i v a t i v e o f t h e e n v e l o p e f u n c t i o n d i v i d e d b y t h e ma ss, t h e B e n D ani e l - D u k e b o u n d a r y co n d i t i o n)$ $E : e n er g y o f t h e s t a t e m e a s u r e d f r o m t h e ban d e d g e o f t h e w e l l$ $V_{0} : ba r r i er h e i g h t, t h e ban d o f f se t o f t h e c a r r i er t y p e (t h e f i r s t co n d i t i o n g i v es t h e e v e n s t a t es an d t h e seco n d t h e o dd s t a t es, an d t h e so l u t i o n s a r e f o u n d n u m er i c a l l y)$
Two-dimensional density of states and sheet carrier density	$g_{2 D} = \frac{m ^{*}}{π ℏ ^{2}}$	$n_{s} = \frac{m ^{} k _{B} T}{π ℏ ^{2}} ln [1 + exp (\frac{E _{F} - E _{1}}{k _{B} T})]$ $n_{s} \approx N_{2 D} exp (\frac{E _{F} - E _{1}}{k _{B} T}), N_{2 D} = \frac{m ^{} k _{B} T}{π ℏ ^{2}}$	$g_{2 D} : d e n s i t y o f s t a t es p er u ni t a r e a an d e n er g y o f o n e s u bban d, in c l u d in g t h e s p in d e g e n er a cy (a co n s t an t, so t h e t o t a l d e n s i t y o f s t a t es i s a s t ai r c a se)$ $m^{*} : in - pl an e e f f ec t i v e ma ss$ $n_{s} : s h ee t c a r r i er d e n s i t y in t h e l o w es t s u bban d$ $E_{F} : q u a s i - F er mi l e v e l o f t h e c a r r i er t y p e$ $E_{1} : e n er g y o f t h e f i r s t s u bban d$ $N_{2 D} : e f f ec t i v e d e n s i t y o f s t a t es p er u ni t a r e a, u se d f or t h e n o n - d e g e n er a t e c a se E_{F} w e l l b e l o w E_{1}$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e$
Polarization-induced electric field in a c-plane quantum well	$F_{w} = \frac{L _{b} ( P _{b} - P _{w} )}{ε _{0} ( ε _{w} L _{b} + ε _{b} L _{w} )}$	$F_{b} = - \frac{L _{w}}{L _{b}} F_{w}$ $P = P_{s p} + P_{p z}$	$F_{w} : e l ec t r i c f i e l d in t h e w e l l f or a p er i o d i c m u l t i pl e q u an t u m w e l l w i t h n o a ppl i e d v o l t a g e (t h e v o l t a g e o v er o n e p er i o d i s z er o)$ $F_{b} : f i e l d in t h e ba r r i er, o pp os i t e in s i g n t o F_{w}$ $P_{w}, P_{b} : t o t a l p o l a r i z a t i o n o f t h e w e l l an d o f t h e ba r r i er ma t er ia l a l o n g t h e g r o w t h a x i s$ $P_{s p}, P_{p z} : s p o n t an eo u s an d p i ez oe l ec t r i c (s t r ain - in d u ce d) p a r t s o f t h e p o l a r i z a t i o n$ $L_{w}, L_{b} : w e l l an d ba r r i er t hi c k n esses$ $ε_{w}, ε_{b} : r e l a t i v e p er mi tt i v i t i es o f t h e w e l l an d t h e ba r r i er$ $ε_{0} : v a c uu m p er mi tt i v i t y$ $(t h e e x p r ess i o n f o l l o w s f r o m t h e co n t in u i t y o f t h e d i s pl a ce m e n t a t t h e in t er f a ces an d t h e z er o v o l t a g e o v er a p er i o d; t h e s i g n d e p e n d s o n t h e p o l a r i t y o f t h e cr y s t a l an d o n t h e d i r ec t i o n t ak e n a s p os i t i v e f or P an d F . F or a s in g l e w e l l b e tw ee n t hi c k ba r r i er s, L_{b} ≫ L_{w}, t h e e x p r ess i o n t e n d s t o F_{w} = (P_{b} - P_{w}) / ε_{0} ε_{w} w i t h F_{b} \to 0, t h e l a r g es t f i e l d; an a ppl i e d or b u i l t - in j u n c t i o n f i e l d a dd s t o i t)$
Quantum-confined Stark effect: emission energy and radiative rate	$E_{e m} \approx E_{g, w} + E_{e 1} + E_{h 1} - E_{x} - q F_{w} L_{w}$	$Γ_{e h} = [\int ψ_{e} (z) ψ_{h} (z) d z]^{2}$ $B \approx B_{0} Γ_{e h}$ $τ_{r a d} \propto \frac{1}{Γ _{e h}}$	$E_{e m} : p h o t o n e n er g y o f t h e e mi ss i o n, r e d u ce d b e l o w t h e f i e l d - f r ee v a l u e b y t h e f i e l d (t h e f i r s t - or d er f or m h o l d s o n l y f or a s t r o n g f i e l d in a w i d e w e l l, w h er e E_{e 1} an d E_{h 1} a r e t h e t r ian g u l a r - w e l l e n er g i es 2.338 (ℏ^{2} q^{2} F_{w}^{2} /2 m^{*})^{1/3} an d n o t t h e f l a t - w e l l o n es . F or b l u e w e l l s o f 2 t o 3 nm, q F_{w} L_{w} i s co m p a r ab l e t o t h e co n f in e m e n t e n er g i es an d t h e f or m u l a i s n o t r e l iab l e : t h e s hi f t m u s t b e f o u n d b y so l v in g t h e S c h r o d in g er e q u a t i o n in t h e t i l t e d w e l l)$ $E_{g, w} : ban d g a p o f t h e w e l l ma t er ia l$ $E_{e 1}, E_{h 1} : co n f in e m e n t e n er g i es o f t h e f i r s t e l ec t r o n an d h o l e s u bban d s$ $E_{x} : e x c i t o n bin d in g e n er g y, s ma l l f or a s t r o n g f i e l d$ $q : e l e m e n t a r y c ha r g e$ $F_{w} : e l ec t r i c f i e l d in t h e w e l l$ $L_{w} : w e l l w i d t h$ $Γ_{e h} : o v er l a p o f t h e e l ec t r o n an d h o l e e n v e l o p e f u n c t i o n s (i t f a l l s a s t h e f i e l d p u l l s t h e c a r r i er s t o o pp os i t e s i d es o f t h e w e l l)$ $ψ_{e}, ψ_{h} : e n v e l o p e f u n c t i o n s o f t h e e l ec t r o n an d t h e h o l e$ $B, B_{0} : r a d ia t i v e coe f f i c i e n t w i t h t h e f i e l d an d w i t h o u t t h e f i e l d$ $τ_{r a d} : r a d ia t i v e l i f e t im e, l o n g er a t a l o w er o v er l a p$
Screening of the polarization field by the injected carriers	$F_{e f f} = F_{w} - \frac{q n _{s}}{ε _{0} ε _{w}}$	$Δ E_{b l u e} \approx q L_{w} \frac{q n _{s}}{ε _{0} ε _{w}}$	$F_{e f f} : n e t f i e l d in t h e w e l l a f t er t h e scr ee nin g b y t h e e l ec t r o n an d h o l e s h ee t s$ $F_{w} : p o l a r i z a t i o n f i e l d o f t h e u n scr ee n e d w e l l$ $n_{s} : s h ee t d e n s i t y o f t h e e l ec t r o n s an d o f t h e h o l es, t ak e n e q u a l$ $ε_{w} : r e l a t i v e p er mi tt i v i t y o f t h e w e l l$ $Δ E_{b l u e} : u pp er es t ima t e o f t h e in cr e a se o f t h e e mi ss i o n e n er g y (t h e b l u e s hi f t) w i t h t h e c u r r e n t, f or t h e l in e a r s hi f t w i t h t h e f i e l d o f t h e p r e v i o u s r o w$ $(a cr u d e es t ima t e : i t t r e a t s t h e c a r r i er s a s tw o c ha r g e d s h ee t s a t t h e o pp os i t e in t er f a ces o f t h e w e l l, so i t g i v es t h e l a r g es t scr ee nin g f or t h e d e n s i t y n_{s}; ban d f i l l in g a l so co n t r ib u t es t o t h e s hi f t)$
Thermionic escape from a quantum well	$τ_{esc} = \frac{2 π m ^{*} L _{w}^{2}}{k _{B} T} exp (\frac{Δ E _{b}}{k _{B} T})$	$\frac{1}{τ _{esc}} = \frac{1}{L _{w}} \frac{k _{B} T}{2 π m ^{*}} exp (- \frac{Δ E _{b}}{k _{B} T})$	$τ_{esc} : esc a p e t im e o f a c a r r i er f r o m t h e w e l l b y t h er mi o ni c e mi ss i o n o v er t h e ba r r i er$ $m^{*} : e f f ec t i v e ma ss o f t h e c a r r i er in t h e w e l l$ $L_{w} : w e l l w i d t h$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e$ $Δ E_{b} : e n er g y f r o m t h e q u a s i - F er mi l e v e l (or f r o m t h e b o tt o m o f t h e l o w es t s u bban d in t h e n o n - d e g e n er a t e c a se) t o t h e t o p o f t h e ba r r i er$ $(t h e e x p r ess i o n i s t h e f l ux o f c a r r i er s w i t h t h e v e l oc i t y co m p o n e n t a l o n g t h e g r o w t h a x i s d i v i d e d b y t h e w e l l w i d t h; i t g i v es t h e or d er o f t h e esc a p e t im e an d i t n e g l ec t s t h e r e f l ec t i o n a t t h e ba r r i er, t h e p o l a r i z a t i o n f i e l d t ha t l o w er s t h e ba r r i er o n o n e s i d e an d t h e p h o n o n - a ss i s t e d esc a p e; t h e s am e f or m i s g i v e n b y H . S c hn e i d er an d K . v . K l i t z in g, P h y s . R e v . B 38, 6160 (1988))$
Carrier density in a multiple-quantum-well active region	$n \approx \frac{η _{inj} J τ}{q N _{w} L _{w}}$	$\frac{R _{A ug er}}{R _{r a d}} = \frac{C n}{B}$ $R_{A ug er} \propto \frac{J ^{3}}{( N _{w} L _{w} ) ^{3}}$	$n : a v er a g e c a r r i er d e n s i t y in t h e w e l l s, a ss u min g t ha t a l l t h e w e l l s c a r r y t h e s am e d e n s i t y$ $η_{inj} : inj ec t i o n e f f i c i e n cy$ $J : c u r r e n t d e n s i t y$ $τ : c a r r i er l i f e t im e$ $q : e l e m e n t a r y c ha r g e$ $N_{w} : n u mb er o f q u an t u m w e l l s$ $L_{w} : w i d t h o f o n e w e l l$ $C, B : A ug er an d r a d ia t i v e coe f f i c i e n t s o f t h e A B C m o d e l o n t h e L E D p a g e$ $R_{A ug er}, R_{r a d} : A ug er an d r a d ia t i v e r eco mbina t i o n r a t es p er u ni t v o l u m e (t h e c u bi c d e p e n d e n ce h o l d s a t a co n s t an t l i f e t im e an d i t i s o n l y a sc a l in g, b ec a u se t h e l i f e t im e i t se l f f a l l s a s t h e d e n s i t y r i ses)$ $(in a r e a l d e v i ce t h e h o l es a r e n o t s ha r e d e q u a l l y am o n g t h e w e l l s b ec a u se o f t h e i r l o w m o bi l i t y an d t h e ba r r i er s, so t h e w e l l n e x t t o t h e p - s i d e c a r r i es m os t o f t h e r eco mbina t i o n an d t h e b e n e f i t o f t h e e x t r a w e l l s i s s ma l l er)$
Alloy bandgap of the active layer	$E_{g} (I n_{x} G a_{1 - x} N) = x E_{g} (I n N) + (1 - x) E_{g} (G a N) - b x (1 - x)$	$a (I n_{x} G a_{1 - x} N) = x a_{I n N} + (1 - x) a_{G a N}$ $ε_{∥} = \frac{a _{G a N} - a _{l a y er}}{a _{l a y er}}$	$E_{g} (I n_{x} G a_{1 - x} N) : ban d g a p o f t h e in d i u m g a l l i u m ni t r i d e a l l oy, ab o u t 3.43 e V f or G a N, ab o u t 0.64 e V f or I n N an d a b o w in g p a r am e t er b o f ab o u t 1.4 e V (t h e r e p or t e d v a l u es o f t h e I n N g a p an d o f b d i f f er b e tw ee n p a p er s)$ $x : in d i u m f r a c t i o n$ $b : b o w in g p a r am e t er$ $a : in - pl an e l a tt i ce co n s t an t b y V e g a r d l a w, ab o u t 0.3189 nm f or G a N an d 0.3533 nm f or I n N$ $ε_{∥} : in - pl an e s t r ain o f t h e l a y er g r o w n o n G a N, n e g a t i v e (co m p r ess i v e) f or I n G a N$ $(t h e r e d - t o - y e l l o w A l G a I n P a l l oy l a tt i ce - ma t c h e d t o G a A s, (A l_{y} G a_{1 - y})_{0.5} I n_{0.5} P, ha s a d i r ec t g a p f r o m ab o u t 1.9 e V (y = 0) t o ab o u t 2.26 e V n e a r t h e d i r ec t - in d i r ec t cr osso v er a t y o f ab o u t 0.53, an d t h e e f f i c i e n cy f a l l s a s t h e cr osso v er i s a pp r o a c h e d)$

Important Questions

Ques: Why does the polarization field of the InGaN/GaN quantum well reduce the efficiency?

The wurtzite nitrides in the c-plane orientation have a large spontaneous and piezoelectric (strain-induced) polarization, and the difference between the well and the barrier produces a field of the order of MV/cm in the well (the field row). The field pulls the electrons and holes to opposite sides of the well, which lowers the overlap $Γ_{e h}$ of the envelope functions, lengthens the radiative lifetime and reduces $B$ , and it red-shifts the emission (by $q F_{w} L_{w}$ to first order in a wide well; in the 2 to 3 nm wells of blue LEDs the shift has to be calculated numerically). A wider well is shifted more but has a smaller overlap, so the well widths of blue LEDs are kept at about 2 to 3 nm. The injected carriers screen the field, so the emission shifts to the blue as the current rises, as in the screening row. Non-polar and semi-polar growth orientations remove or reduce the field. Reference: T. Takeuchi et al., “Quantum-confined Stark effect due to piezoelectric fields in GaInN strained quantum wells,” Jpn. J. Appl. Phys. 36, L382 (1997); F. Bernardini, V. Fiorentini and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56, R10024 (1997).

Ques: How does the number and width of the wells affect the efficiency droop?

For a fixed current density, a larger total active thickness $N_{w} L_{w}$ lowers the carrier density $n$ , and the Auger rate, which scales as $n^{3}$ , falls faster than the radiative rate, which scales as $n^{2}$ (the carrier-density row). A thicker well is limited by the polarization field, which lowers the overlap, and more wells are limited by the uneven injection of holes, because the holes are concentrated in the wells nearest to the p-side. The design is therefore a compromise between the density, the overlap and the hole distribution, and the choices of electron-blocking layers and of the barrier composition aim to improve the hole injection. Reference: M.-H. Kim et al., “Origin of efficiency droop in GaN-based light-emitting diodes,” Appl. Phys. Lett. 91, 183507 (2007); A. David, M.J. Grundmann et al., “Carrier distribution in (0001)InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 92, 053502 (2008).

Ques: What are typical InGaN compositions for blue and green emission, and why is green harder?

With the bowing form of the alloy row ( $E_{g} (GaN) = 3.43$ eV, $E_{g} (InN) = 0.64$ eV, $b = 1.4$ eV) the bulk gap is 2.83 eV (438 nm) at $x = 0.15$ , 2.47 eV (502 nm) at $x = 0.25$ and 2.14 eV (581 nm) at $x = 0.35$ . The quantum-well emission is at a longer wavelength than the bulk gap because of the field shift, so the indium needed for a given colour is lower than these numbers suggest, and the numbers are given as an illustration of the trend and not as growth recipes. More indium means a larger lattice mismatch (the strain row), a larger piezoelectric field, a lower overlap, a lower growth temperature that increases the defect density and the alloy fluctuations, and these effects together lower the efficiency in the green, which is called the green gap. AlGaInP faces the same trend from the other side, as the direct gap closes at the crossover. Reference: J. Wu, “When group-III nitrides go infrared: new properties and perspectives,” J. Appl. Phys. 106, 011101 (2009); M.R. Krames et al., “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Display Technol. 3, 160 (2007).

Ques: How fast can carriers escape a quantum well by thermionic emission?

With the escape-time expression of the thermionic row, an electron of mass $0.2 m_{0}$ in a 3 nm well takes $τ_{esc} = 5.5$ ns at 300 K when the barrier is 0.3 eV above the Fermi level, 0.11 ns for 0.2 eV and 2.4 ps for 0.1 eV, with a prefactor of about 50 fs. The escape time falls exponentially with the barrier height and with the temperature, so a small band offset (the electrons in the AlGaInP red LED) leads to a leakage that rises with the temperature and with the current. These numbers are computed from the formula for illustration, and the barrier heights are assumed. Reference: H. Schneider and K. v. Klitzing, “Thermionic emission and Gaussian transport of holes in a GaAs/Al $_{x}$ Ga $_{1 - x}$ As multiple-quantum-well structure,” Phys. Rev. B 38, 6160 (1988), for the expression (the numbers are computed from it). The documentation of the Ansys Lumerical TWLM solver prints the same formula without the square root, which would give the prefactor the unit of time squared, so the form with the root is used here.

Ques: Why does the AlGaInP LED lose efficiency at the shorter wavelengths and at high temperature?

The active layer is lattice-matched to GaAs and its direct gap rises with the aluminium fraction only up to the direct-indirect crossover near an aluminium fraction of 0.53 (about 2.26 eV), so the yellow-green LED is close to the crossover and a part of the electrons occupies the indirect minima, which lowers the radiative efficiency. The conduction band offset to the confining layers is small, and the electrons leak over the barrier, especially at a high temperature and a high current, so the confining layers use a high aluminium fraction (above 0.7) and the active layer is made thick enough to limit the leakage, at the cost of the absorption. Commercial AlGaInP LEDs are therefore mostly red, orange and amber. Reference: “AlGaInP LEDs break performance barriers,” Compound Semiconductor (cover story), compoundsemiconductor.net, article 82202 (a trade article); F.A. Kish and R.M. Fletcher, “AlGaInP light-emitting diodes,” in High Brightness Light Emitting Diodes, Semiconductors and Semimetals vol. 48 (Academic Press, 1997); E.F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge University Press, 2006), chapter 12.

Ques: What is the difference between the InN band gap and lattice constant of Wu and those of Vurgaftman, and which is more likely to be right?

The InN gap is the endpoint of the InGaN bowing formula, so a different InN value moves the whole green and red range. Wu (2009) recommends $E_{g} (InN) = 0.64$ eV at 300 K (0.69 eV at 0 K) with $a = 0.3533$ nm and $c = 0.5693$ nm, and lists the older values 0.77 eV and 0.79 eV for comparison. The compilation of Vurgaftman and Meyer (2003) is recalled by me to give about 0.78 eV at 0 K and $a$ near 0.3545 nm; these two numbers could not be read in the sources available and are not verified. The independent theory paper arXiv:cond-mat/0401306 gives 0.78 eV for the wurtzite gap and $a = 0.3546$ nm, which agrees with that recollection. With $E_{g} (GaN) = 3.43$ eV and $b = 1.4$ eV, the gap of $In_{x} Ga_{1 - x} N$ is 2.65 eV (468 nm) for $x = 0.2$ with 0.64 eV and 2.68 eV (463 nm) with 0.78 eV, and for $x = 0.3$ it is 2.30 eV (539 nm) against 2.34 eV (530 nm) (computed), so the difference is about 5 to 10 nm in the blue and green. The lattice difference of 0.3 percent changes the compressive strain of $In_{0.2} Ga_{0.8} N$ on GaN from 2.11 to 2.18 percent (computed), which is negligible. Of the two gap values, 0.64 to 0.7 eV is the more likely: Davydov et al. concluded from photoluminescence that the true gap is about 0.7 eV, and the reflectance study of Feneberg et al. confirms a low value of about 0.7 eV at low temperature (both as summarised on the Ioffe page), and Wu’s own review adopts 0.64 eV at 300 K. The value of 0.78 to 0.8 eV is close to this range and is supported by some structural and theoretical work (EXAFS and EELS studies adopt about 0.8 eV, the theory paper gives 0.78 eV), so the two are not far apart; the much older value of 1.9 eV is attributed to films with a high electron density and the Burstein-Moss shift, where the filled conduction band moves the absorption edge to a higher energy. The interpretation of the photoluminescence is still discussed (Wu and Davydov report opposite temperature shifts). One caution applies: the bowing parameter of $b = 1.4$ eV is fitted together with the endpoints, so the endpoints and the bowing should be taken from the same source, and I could not read the Vurgaftman and Meyer bowing value to check which endpoint it goes with. Reference: J. Wu, J. Appl. Phys. 106, 011101 (2009), Table I and text; Ioffe Institute, “InN - Indium Nitride”, www.ioffe.ru/SVA/NSM/Semicond/InN/dvdv.html (Davydov, Feneberg and others); arXiv:cond-mat/0401306 (authors not retrieved); I. Vurgaftman and J.R. Meyer, J. Appl. Phys. 94, 3675 (2003) (values recalled, not checked).

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