This page covers the degradation and the burnout of microwave diodes: IMPATT oscillators, Gunn (transferred-electron) diodes, PIN limiter and switch diodes, Schottky mixer and detector diodes, tunnel (Esaki) diodes, and varactor and step-recovery multiplier diodes. TRAPATT diodes share the avalanche, heating and contact mechanisms of IMPATT diodes, and no separate source was read for them. The device physics is on the device pages of this vault, the Wunsch-Bell and overstress models on the Electrical Stress page, the thermal models on the Thermal Stress page, and the low-frequency PN, PIN and Schottky mechanisms on the Diodes (PN, PIN, Schottky, Zener) page; this page covers what is specific to microwave operation.

1. Why It Happens

Trigger: A high junction temperature in an avalanche or transferred-electron oscillator, which turns most of its input into heat (RADC; Hughes; Microsemi); a circuit mismatch that sends an IMPATT diode out of its intended operating point (Hughes); a high-power RF pulse on a limiter or on a receiver mixer (HDL; MACOM; HP); a forward current through a tunnel diode (Ono et al.); an RF swing beyond the breakdown voltage of a varactor or multiplier (SemiGen; background knowledge).
Mechanism, contact and metal penetration: Contact metal moves into the semiconductor: Pt Schottky metal and Au through thin Pt into the GaAs active region of IMPATT diodes, where gold forms acceptors and lowers the net doping (RADC), and Ni through the AuGe grains into the buffer of Gunn diodes, where it forms compensated regions (Gulati et al.).
Mechanism, surface and edge: Mesa-surface failures and edge burnout at mesa irregularities, where the field is highest, in overstressed GaAs IMPATT diodes, and a dendrite along a surface channel in a silicon one (RADC); shorting is the main failure mode of 60 $GHz$ IMPATT diodes (Hughes).
Mechanism, RF burnout: An RF pulse heats the I region of a PIN limiter or the junction of a mixer diode; for short pulses the heat cannot leave the volume, so a nearly constant energy destroys the diode (HDL).
Mechanism, tunnel diodes: Dopant (carbon) diffuses out of the degenerate junction and lowers the peak current, and the forward current speeds up the diffusion through recombination energy (Ono et al.).
What sets the rate: The junction temperature through the efficiency and the thermal resistance (RADC; Hughes), the bias polarity for contact metal migration (Gulati et al.), the pulse energy and the I-region width for limiters (HDL; MACOM), and the current density for tunnel diodes (Ono et al.).
Failure criterion: A change above 10 per cent in breakdown voltage, forward voltage, leakage or zero-bias capacitance, or more than 2 $dB$ loss of RF power (RADC); a fall of the reverse breakdown voltage after a pulse (HDL); a loss of peak current in a tunnel diode (Ono et al.).

2. Where to Look

Location: The epi-side metallization and the mesa edge of IMPATT diodes (RADC; Hughes); the AuGeNi cathode contact and the buffer of Gunn diodes (Gulati et al.); the I region of limiter diodes (HDL); the tunnel junction itself (Ono et al.).
Signature: A shift of the doping peak or a lower net doping seen as a breakdown-voltage change (RADC); a falling Gunn current at all biases after 50 to 60 $h$ at 250 $^{\circ} C$ (Gulati et al.); higher leakage and a rounded breakdown after high-temperature storage (RADC); a lower peak current with an unchanged valley current in tunnel diodes (Ono et al.).
Confirm with: The junction temperature from the breakdown-voltage shift (RADC); SEM of shorted and opened diodes (Hughes); I-V curves after anneals at 200 to 300 $^{\circ} C$ (Gulati et al.); spike and flat leakage with a fast-rise RF pulse (HDL).
Look-alikes: Process-related failures spread over a wide temperature range, unlike wear-out (Hughes); a spike measured with a slow rise time can disappear, since HDL used a 1.5 $ns$ rise time and notes that it might not be seen with a microsecond rise; noise degradation after a mismatch rather than slow wear (Hughes).

3. Prevention

Design: Keep the oscillator junction at or below the design temperature, 250 $^{\circ} C$ for the Hughes diodes, by heat sinking, since the RF output is limited by $(T_{M} - T_{A}) / θ_{T H}$ (Hughes); use a diamond heat sink, which can halve the thermal resistance (Hughes); match the circuit, since a mismatch burned out IMPATT diodes even at low bias (Hughes); choose the limiter I-region width for the threat power and the allowed spike leakage (HDL; MACOM).
Material and process: Control the AuGe grain size, on which the contact resistance strongly depends (Gulati et al.), and use a diffusion barrier in the contact stack thick enough to stop gold, which penetrated thin Pt in the RADC diodes (my reading of both).
Operation: Bias Gunn diodes with the epi side negative, as intended, since the opposite polarity speeds up Ni migration (Gulati et al.); handle Gunn and mixer diodes as ESD sensitive (Microsemi); keep the reverse bias of a PIN switch above the peak RF voltage (Skyworks).
Screening and test: Use an oscillating RF burn-in for single-drift IMPATT diodes, since DC and pulsed DC burn-in missed many weak devices (RADC); step-stress tests show the process-related failure temperatures (Hughes).

IMPATT Thermal Limits and Life Tests

Name	Formula	Related Formulae	Variables/Constants
Junction temperature of an oscillating diode	$T_{J} = T_{C} + θ_{T H} P_{D}, P_{D} = P_{D C} - P_{R F} = P_{R F} \frac{1 - η}{η}$	$θ_{T H} = \frac{V _{B} ( T _{J} ) - V _{B 0}}{K _{T} P _{D}} (V_{B} = K_{T} (T_{J} - T_{C}) + V_{B 0})$	$T_{J} : j u n c t i o n t e m p er a t u r e$ $T_{C} : c a se or h e a t - s ink t e m p er a t u r e$ $θ_{T H} : t h er ma l r es i s t an ce$ $P_{D} : d i ss i p a t e d p o w er$ $P_{D C} : D C in p u t p o w er$ $P_{R F} : R F o u tp u t p o w er$ $η : D C - t o - R F e f f i c i e n cy$ $V_{B} : b r e ak d o w n v o l t a g e a t t h e j u n c t i o n t e m p er a t u r e$ $V_{B 0} : b r e ak d o w n v o l t a g e a t t h e c a se t e m p er a t u r e$ $K_{T} : t e m p er a t u r e coe f f i c i e n t o f t h e b r e ak d o w n v o l t a g e (R A D C E q . 2.15, u se d t o m e a s u r e θ_{T H}); R A D C : s in g l e - d r i f t G a A s I M P A T T b u r n - in a t 16.9 W an d 95^{\circ} C c a se w i t h 7.7 K/W g i v es ab o u t 224^{\circ} C, s i l i co n d o u b l e - d r i f t a t 14 W an d 85^{\circ} C ab o u t 180^{\circ} C w i t h 6.5 t o 7.0 K/W, an d t h e h e a t in g d u r in g a 1 μ s p u l se w a s n e g l i g ib l e so t h e a v er a g e t e m p er a t u r e w a s u se d; H ug h es : 1 W a t 52.75 GHz an d 14.7 p er ce n t e f f i c i e n cy a t 228^{\circ} C j u n c t i o n$ $(co m p u t e d : 95 + 7.7 x 16.9 = 225^{\circ} C a g ain s t R A D C^{'} s 224; 176 t o 183^{\circ} C f or t h e S i d i o d e; H ug h e s^{'} s 1 W a t 14.7 p er ce n t d i ss i p a t es 5.80 W, so 228^{\circ} C f r o m 25^{\circ} C m e an s θ_{T H} = 35.0 K/W, an d i t s d e l i v er e d d i o d es, a 236 K r i se a t 1 W an d 13 p er ce n t, g i v e 35.3 K/W)$
Thermal limit of the RF output	$P_{R F, ma x} = \frac{T _{M} - T _{A}}{θ _{T H}} \frac{η}{1 - η}$	$θ_{D} = \frac{l}{k _{D} A}$	$P_{R F, ma x} : hi g h es t R F o u tp u t a t w hi c h t h e j u n c t i o n s t a y s a t T_{M}$ $T_{M} : ma x im u m a l l o w e d j u n c t i o n t e m p er a t u r e$ $T_{A} : ambi e n t or h e a t - s ink t e m p er a t u r e$ $θ_{T H} : t o t a l t h er ma l r es i s t an ce$ $η : e f f i c i e n cy$ $θ_{D} : r es i s t an ce o f t h e G a A s l a y er b e tw ee n j u n c t i o n an d h e a t s ink$ $l : i t s t hi c k n ess$ $k_{D} : t h er ma l co n d u c t i v i t y o f G a A s$ $A : d e v i ce a r e a (H ug h es E q . 34 an d 35, E q . 34 r eco n s t r u c t e d f r o m a g a r b l e d sc an); H ug h es : d iam o n d h e a t s ink s l o w er e d t h e t h er ma l r es i s t an ce b y u p t o 50 p er ce n t a g ain s t pl a t e d h e a t s ink s, an d t h e ma x im u m s a f e j u n c t i o n t e m p er a t u r e se t s t h e a v er a g e l i f e$ $(co m p u t e d : w i t h θ_{T H} = 35.0 K/W an d T_{M} = 200^{\circ} C (a ss u m e d), 0.86 W a t 14.7 p er ce n t an d 0.56 W a t 10 p er ce n t e f f i c i e n cy; ha l v in g θ_{T H} w i t h d iam o n d d o u b l es P_{R F, ma x})$
Arrhenius extrapolation of an IMPATT life test	$t_{f} = e^{Δ H / (k_{B} T) + C}, A F = exp [\frac{E _{a}}{k _{B}} (\frac{1}{T _{u se}} - \frac{1}{T _{t es t}})]$	$t_{u se} = A F t_{t es t}$	$t_{f} : t im e t o f ai l u r e$ $Δ H : a c t i v a t i o n e n t ha l p y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T : ab so l u t e j u n c t i o n t e m p er a t u r e$ $C : co n s t an t (R A D C E q . 1.5, E y r in g - A r r h e ni u s f or m)$ $A F : a cce l er a t i o n f a c t or$ $E_{a} : a c t i v a t i o n e n er g y$ $T_{u se} an d T_{t es t} : j u n c t i o n t e m p er a t u r es in u se an d in t h e t es t$ $t_{u se} an d t_{t es t} : l i v es in u se an d in t h e t es t; R A D C : t h r ee f u l l - R e a d G a A s d i o d es f ai l e d a f t er 809, 764 an d 768 h a t 265^{\circ} C j u n c t i o n, a h y b r i d d o u b l e - d r i f t d i o d e a f t er 723 h a t 220^{\circ} C, an d s t or a g e a t 335^{\circ} C f or 168 h r ai se d t h e l e ak a g e an d r o u n d e d t h e b r e ak d o w n; t h e e x cer pt r e a d g i v es n o n u m er i c a l a c t i v a t i o n e n er g y$ $(co m p u t e d : m e an o f t h e t h r ee l i v es 780 h; f r o m 265 t o 200^{\circ} C, A F = 19.3, 85 an d 207 f or E_{a} = 1.0, 1.5 an d 1.8 eV (a ss u m e d), g i v in g 1.5 \times 1 0^{4}, 6.6 \times 1 0^{4} an d 1.6 \times 1 0^{5} h, t ha t i s 1.7, 7.6 an d 18.4 y e a r s)$
MTBF demonstrated by a life test	$M T B F = \frac{T _{t o t}}{r}, M T B F_{L} = \frac{2 T _{t o t}}{χ _{γ; 2 r + 2}^{2}}$	$M T B F = \frac{1}{λ}, P_{s} = e^{- λ t}$	$M T B F : p o in t es t ima t e$ $T_{t o t} : t o t a l d e v i ce h o u r s$ $r : n u mb er o f f ai l u r es$ $M T B F_{L} : o n e - s i d e d l o w er l imi t a t co n f i d e n ce γ$ $χ^{2} : c hi - s q u a r e q u an t i l e w i t h 2 r + 2 d e g r ees o f f r ee d o m f or a t im e - t er mina t e d t es t (2 r f or a f ai l u r e - t er mina t e d t es t)$ $λ : co n s t an t f ai l u r e r a t e$ $P_{s} : s u r v i v a l p r o babi l i t y f or t im e t (R A D C E q . 1.1 t o 1.4); R A D C : a s w i t c hin g t es t t ha t t o g g l e d t h e bia s e v er y tw o min u t es o n t e n d e v i ces s h o w e d n o s i g ni f i c an t c han g e in f ai l u r e r a t e; t h e g e n er a l ba t h t u b m o d e l i s o n t h e F ai l u r e R a t e p a g e$ $(co m p u t e d : t e n d i o d es f or 168 h w i t h o u t f ai l u r e g i v e T_{t o t} = 1680 h an d M T B F_{L} = 1830 h a t 60 p er ce n t an d 730 h a t 90 p er ce n t co n f i d e n ce, w i t h χ^{2} = 1.833 an d 4.605 f or 2 d e g r ees o f f r ee d o m)$
Fallout of a two-stage burn-in screen	$F = 1 - \prod_{i} \frac{N _{i} - n _{i}}{N _{i}}$	$F_{i} = \frac{n _{i}}{N _{i}}$	$F : t o t a l f r a c t i o n r e m o v e d b y t h e scr ee n$ $N_{i} : d e v i ces e n t er in g s t a g e i$ $n_{i} : f ai l u r es in s t a g e i$ $F_{i} : f a l l o u t o f o n e s t a g e; R A D C, s in g l e - d r i f t G a A s : 5 o f 25 f ai l e d a 48 h D C b u r n - in an d 2 o f 30 a p u l se d D C b u r n - in, t h e n 4 o f 19 an d 7 o f 28 s u r v i v or s f ai l e d t h e R F t es t; s i l i co n d o u b l e - d r i f t 0 o f 10 u n d er D C an d 1 o f 10 u n d er p u l se d D C; f ai l u r e cr i t er i o n a c han g e ab o v e 10 p er ce n t in V_{B}, V_{F}, l e ak a g e or z er o - bia s c a p a c i t an ce, or m or e t han 2 dB l oss o f R F p o w er$ $(co m p u t e d : 20 an d 6.7 p er ce n t in t h e D C s t a g es, 21 an d 25 p er ce n t in t h e R F s t a g e, t o t a l 36.8 p er ce n t o n t h e D C p a t h an d 30.0 p er ce n t o n t h e p u l se d p a t h, so t h e R F t es t r e m o v e d m or e d i o d es t han e i t h er D C b u r n - in)$

Gunn Diode Heating and Contacts

Name	Formula	Related Formulae	Variables/Constants
Transit length and threshold voltage of a Gunn diode	$f \approx \frac{v _{d}}{L}, V_{t h} = E_{t h} L$	$n_{0} L > 1 0^{12} c m^{- 2}$	$f : t r an s i t - t im e f r e q u e n cy$ $v_{d} : d o main v e l oc i t y (ab o u t 1 \times 1 0^{7} cm/s in G a A s)$ $L : a c t i v e l e n g t h$ $V_{t h} : t h r es h o l d v o l t a g e$ $E_{t h} : t h r es h o l d f i e l d (ab o u t 3.2 kV/cm in G a A s)$ $n_{0} : d o p in g (ba c k g r o u n d k n o w l e d g e, an d o n t h e G u nn D i o d e p a g e); M i cr ose mi M G 1001 ser i es : t y p i c a l bia s 12 V in C ban d, 10 V in X ban d an d 4 t o 5.5 V ab o v e 26 GHz, p o l a r i t y an o d e o n t h e c a p an d c a t h o d e o n t h e h e a t s ink, an d t h e p a r t s a r e E S D se n s i t i v e$ $(co m p u t e d : 10 GHz n ee d s L = 10 μ m an d V_{t h} = 3.2 V, so t h e 10 V X - ban d bia s i s ab o u t t h r ee t im es t h e t h r es h o l d)$
Self-heating of a Gunn oscillator	$P_{D} = V_{0} I_{0} - P_{R F}, T_{a c t} = T_{H S} + θ_{T H} P_{D}$	$η = \frac{P _{R F}}{V _{0} I _{0}}$	$P_{D} : d i ss i p a t e d p o w er$ $V_{0} : bia s v o l t a g e$ $I_{0} : bia s c u r r e n t$ $P_{R F} : R F o u tp u t$ $T_{a c t} : a c t i v e - r e g i o n t e m p er a t u r e$ $T_{H S} : h e a t - s ink t e m p er a t u r e$ $θ_{T H} : t h er ma l r es i s t an ce$ $η : e f f i c i e n cy; M i cr ose mi : X ban d 200 t o 1600 mA a t 10 V f or 50 t o 500 mW, an d t h e c a t h o d e i s t h e h e a t - s ink s i d e; G u l a t i e t a l . : C W G u nn d i o d es r u n w i t h a c t i v e r e g i o n s a t 250 t o 300^{\circ} C$ $(co m p u t e d : 10 V a t 1.6 A f or 0.5 W i s 3.1 p er ce n t e f f i c i e n cy an d 15.5 W o f h e a t, a r i se o f 78 an d 155 K f or θ_{T H} = 5 an d 10 K/W (a ss u m e d))$
Penetration of contact metal into the buffer	$x (t) = 2 D (T) t, D (T) = D_{0} e^{- E_{a} / (k_{B} T)}$	$J = - D \frac{\partial c}{\partial x} + c μ_{i} E$	$x : p e n e t r a t i o n d e pt h$ $D : d i f f u s i o n coe f f i c i e n t o f t h e co n t a c t m e t a l (N i or A u) in G a A s$ $t : t im e$ $D_{0} : p r e f a c t or$ $E_{a} : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T : ab so l u t e t e m p er a t u r e$ $J : f l ux$ $c : co n ce n t r a t i o n$ $μ_{i} : d r i f t m o bi l i t y o f t h e m e t a l s p ec i es$ $E : f i e l d (t h e d r i f t t er m i s m y r e a d in g o f t h e bia s e f f ec t, ba c k g r o u n d k n o w l e d g e); G u l a t i e t a l . : A u G e N i co n t a c t s a t 200 t o 300^{\circ} C, t h e c u r r e n t f e l l a f t er 50 t o 60 h a t 250^{\circ} C an d k e pt f a l l in g t o ab o u t 230 h t h r o ug h a s ma l l er e f f ec t i v e co n t a c t a r e a, N i d i f f u se d t h r o ug h t h e A u G e g r ain s in t o t h e b u f f er an d co m p e n s a t e d i t, a n e g a t i v e e p i - s i d e bia s o f a t l e a s t ab o u t 2.5 V a r r es t e d t h e mi g r a t i o n f or ab o u t 500 h, a p os i t i v e bia s e nhan ce d i t t o c a t a s t r o p hi c f ai l u r e, an d t h e c i t e d s p ec i f i c co n t a c t r es i s t an ce r ose b y 20 t o 25 p er ce n t a f t er ab o u t 150 h a t 250^{\circ} C; R A D C a l so nam es g o l d p e n e t r a t i o n t h r o ug h an n + b u f f er f or I M P A T T an d G u nn d i o d es$ $(co m p u t e d : cr oss in g an a ss u m e d 1 μ m b u f f er in 150 h n ee d s D = 4.6 \times 1 0^{- 15} c m^{2} /s; t h e co n t a c t - r es i s t an ce r i se i s 0.13 t o 0.17 p er ce n t p er h o u r o n a v er a g e)$

Limiter and Mixer Burnout

Name	Formula	Related Formulae	Variables/Constants
Junction temperature under pulsed RF	$T_{J} = T_{A} + P_{D} (D F θ_{A V} + θ_{T P})$	$P_{D, ma x} = \frac{T _{J, ma x} - T _{A}}{θ _{A V}} (CW)$	$T_{J} : p e ak j u n c t i o n t e m p er a t u r e$ $T_{A} : ambi e n t or h e a t - s ink t e m p er a t u r e$ $P_{D} : p e ak d i ss i p a t e d p o w er$ $D F : d u t y f a c t or (b e l o w 10 p er ce n t)$ $θ_{A V} : a v er a g e (C W) t h er ma l r es i s t an ce$ $θ_{T P} : p u l se t h er ma l im p e d an ce f or t h e p u l se l e n g t h, w hi c h m u s t b e s h or t er t han t h e t h er ma l t im e co n s t an t (S k y w or k s E q . 7 an d 8)$ $T_{J, ma x} : ma x im u m j u n c t i o n t e m p er a t u r e (u s u a l l y 175^{\circ} C, S k y w or k s); C L A 4609 : θ_{A V} = 25 K/W, θ_{T P} = 0.1 K/W f or a 0.1 μ s p u l se, 2.5 W C W an d 25 W p e ak a t 10 p er ce n t d u t y, b o t h a t 85^{\circ} C, 16 W C W in p u t a t 25^{\circ} C d er a t e d l in e a r l y t o z er o a t 175^{\circ} C, + 72 dBm p u l se d in p u t a t 0.1 μ s an d 0.1 p er ce n t d u t y; S K Y 16603 m o d u l e : 52 K/W, 0.4 W d i ss i p a t i o n a t 120^{\circ} C c a se; M A C O M l imi t er s : T_{J} a t m os t 175^{\circ} C an d 25 t o 175 K/W$ $(co m p u t e d : C L A 4609 a t 25 W p e ak an d 10 p er ce n t d u t y, 85 + 25 x (0.1 x 25 + 0.1) = 150^{\circ} C, a t 2.5 W C W 147.5^{\circ} C, b o t h 25 t o 28 K b e l o w 175^{\circ} C i f θ_{T P} = 0.1 K/W a l so h o l d s f or t h e l o n g er p u l se (a ss u m e d); C W in p u t r a t in g a t 85^{\circ} C 9.6 W; S K Y 16603 r i ses 21 K ab o v e t h e c a se)$
Power dissipated in a shunt limiter or switch diode	$P_{D} = \frac{4 R _{S} Z _{0}}{( Z _{0} + 2 R _{S} ) ^{2}} P_{A V} \approx \frac{4 R _{S}}{Z _{0}} P_{A V}$	$P_{D, ser i es} \approx \frac{R _{S}}{Z _{0}} P_{A V}, k_{S W R} = (\frac{2 σ}{σ + 1})^{2}$	$P_{D} : p o w er d i ss i p a t e d in a co n d u c t in g s h u n t d i o d e$ $R_{S} : i t s f or w a r d r es i s t an ce$ $Z_{0} : l in e im p e d an ce (50 Ω)$ $P_{A V} : a v ai l ab l e (in c i d e n t) p o w er$ $P_{D, ser i es} : d i ss i p a t i o n o f a co n d u c t in g ser i es d i o d e$ $k_{S W R} : f a c t or f or a l o a d w i t h s t an d in g - w a v e r a t i o σ (S k y w or k s E q . 12 an d 21 an d t h e mi s ma t c h f a c t or; t h e s h u n t f or m i s m y d er i v a t i o n, s in ce t h e E q . 21 r e a d w a s g a r b l e d); C L A 4609 : R_{S} a t m os t 1.5 Ω a t 10 mA an d 500 MHz$ $(co m p u t e d : 16 W in c i d e n t o n 1.5 Ω in 50 Ω d i ss i p a t es 1.71 W e x a c tl y an d 1.92 W b y t h e a pp r o x ima t i o n, an d a t o t a l mi s ma t c h, σ t o in f ini t y, m u l t i pl i es t h e d i ss i p a t i o n b y u p t o 4)$
Peak RF voltage and I-region breakdown	$V_{p k} = 2 P_{A V} Z_{0}, V_{B} \approx E_{B} W$	$∣ V_{R} ∣ > V_{p k} (reverse-biased switch)$	$V_{p k} : p e ak R F v o l t a g e o n a ma t c h e d l in e$ $P_{A V} : a v ai l ab l e p o w er$ $Z_{0} : l in e im p e d an ce$ $V_{B} : b u l k a v a l an c h e v o l t a g e$ $E_{B} : a v er a g e b r e ak d o w n f i e l d o f t h e I r e g i o n$ $W : I - r e g i o n w i d t h$ $V_{R} : D C r e v er se bia s, w hi c h m u s t e x cee d t h e p e ak R F s w in g so t ha t t h e d i o d e d oes n o t co n d u c t (S k y w or k s); S k y w or k s : ab o u t 10 V p er mi cr o n (p r in t e d a s V / mm in t h e co p y r e a d); M A C O M : 15 t o 35 V a t 2 μ m, 60 t o 75 V a t 4 μ m, a t l e a s t 100 V a t 13 μ m an d 250 V a t 25 μ m, a l l a t 10 μ A; H D L : ab o u t 35, 60 an d 122 V f or 1, 2 an d 5 μ m, an d a v a l an c hin g a t 1.5 GHz n ee d e d ab o u t 300 V f or t h e 5 μ m d i o d e$ $(co m p u t e d : 100 W in 50 Ω g i v es V_{p k} = 100 V; M A C O M g i v es 7.5 t o 19 V/ μ m an d H D L 24 t o 35 V/ μ m, so t h e t hinn es t d i o d es b r e ak d o w n a t a hi g h er a v er a g e f i e l d)$
Burnout power against pulse length	$P_{f} \propto ⎩ ⎨ ⎧ t_{p}^{- 1} t_{p}^{- 1/2} const t_{p} ≪ τ_{t h} intermediate t_{p} ≫ τ_{t h}$	$τ_{t h} \approx \frac{W ^{2}}{D _{t h}}, D_{t h} = \frac{k}{ρ c}$	$P_{f} : ab sor b e d p o w er a t f ai l u r e$ $t_{p} : p u l se l e n g t h$ $τ_{t h} : t h er ma l d i f f u s i o n t im e a cr oss t h e h e a t e d r e g i o n$ $W : i t s w i d t h (t h e I r e g i o n o f a P I N d i o d e)$ $D_{t h} : t h er ma l d i f f u s i v i t y$ $k : t h er ma l co n d u c t i v i t y$ $ρ : d e n s i t y$ $c : s p ec i f i c h e a t; t h e t h r ee r e g im es a r e a d iaba t i c, W u n sc h - B e l l an d s t e a d y s t a t e (ba c k g r o u n d k n o w l e d g e; t h e W u n sc h - B e l l l a w i s o n t h e E l ec t r i c a l S t r ess p a g e); H D L : t h e c a l c u l a t e d d ama g e p o w er f a l l s ab o u t l in e a r l y w i t h p u l se l e n g t h, so a r o ug h l y co n s t an t e n er g y c a u ses d ama g e, an d f or 0.1 μ s i t r i ses w i t h t h e w i d t h w i t h a l o g - l o g s l o p e o f 2.2; M A C O M : 80 t o 90 W p e ak a t 2 μ m, 125 W a t 3 μ m, 200 W a t 4 μ m, 250 W a t 13 μ m, 500 W a t 20 μ m an d 1000 W a t 25 μ m, a l l f or 1 μ s a t 9.4 GHz$ $(co m p u t e d : f or S i, k = 1.48 W/ (cm K), ρ c = 1.63 J/ (c m^{3} K) (ba c k g r o u n d k n o w l e d g e), D_{t h} = 0.91 c m^{2} /s, so τ_{t h} = 0.044, 1.1 an d 6.9 μ s f or 2, 10 an d 25 μ m; a 0.1 μ s p u l se i s a d iaba t i c f or t h e 10 μ m d i o d e; M A C O M p e ak p o w er g r o w s ab o u t a s W^{1.0} f r o m 2 t o 25 μ m)$
Adiabatic burnout energy	$E_{f} = ρ c A W Δ T_{f}$	$P_{f} = \frac{E _{f}}{t _{p}} (t_{p} ≪ τ_{t h})$	$E_{f} : ab sor b e d e n er g y a t f ai l u r e$ $ρ c : v o l u m e t r i c h e a t c a p a c i t y$ $A : j u n c t i o n a r e a$ $W : h e a t e d t hi c k n ess$ $Δ T_{f} : t e m p er a t u r e r i se a t f ai l u r e$ $P_{f} : ab sor b e d p o w er a t f ai l u r e$ $t_{p} : p u l se l e n g t h; H D L : d ama g e w a s t ak e n a s a 500 K a v er a g e r i se ab o v e 300 K (e a r l i er 350 K, f r o m D C seco n d b r e ak d o w n o f t h e 1 N 4148), t h e 10 μ m d i o d e a r e a i s g i v e n a s 3 \times 1 0^{- 4} c m^{2} in t h e t e x t an d 3 \times 1 0^{- 5} c m^{2} in t h e c a pt i o n, m e a s u r e d t h r es h o l d s w er e ab o u t t e n t im es hi g h er in p o w er t han c a l c u l a t e d w hi l e t h e e n er g i es a g r ee d r e a so nab l y, an d d ama g e m e an t a d e f ini t e f a l l o f t h e r e v er se b r e ak d o w n v o l t a g e; H P 5082 mi x er an d d e t ec t or d i o d es : 1 W p e ak ab sor b e d f or 1 μ s a t 0.001 d u t y, 100 t o 250 mW C W, T_{J} a t m os t 100, 150 or 200^{\circ} C b y t y p e, an d 15 W R F p u l se p o w er in c i d e n t in t h e f e a t u r e l i s t$ $(co m p u t e d : 10 μ m x 3 \times 1 0^{- 5} c m^{2} h e a t e d b y 500 K n ee d s 24 μ J, 245 W f or 0.1 μ s; w i t h 3 \times 1 0^{- 4} c m^{2}, 245 μ J an d 2450 W; t h e H P r a t in g i s 1 μ J ab sor b e d p er p u l se, an d 1 W i s 6.7 p er ce n t o f t h e 15 W in c i d e n t)$
Spike leakage energy of a limiter	$E_{s p} \approx P_{s p} t_{s p}$	$P_{f l a t} = P_{in} 1 0^{- L_{i so} /10}$	$E_{s p} : e n er g y t ha t p a sses t h e l imi t er b e f or e i t t u r n s o n$ $P_{s p} : p e ak p o w er o f t h e s p ik e$ $t_{s p} : s p ik e d u r a t i o n (e x t r a p o l a t e d f r o m t h e f a l l in g e d g e, w hi c h H D L c a l l s n ecess a r i l y s u bj ec t i v e)$ $P_{f l a t} : f l a t l e ak a g e a f t er t u r n - o n$ $P_{in} : in c i d e n t p o w er$ $L_{i so} : i so l a t i o n in dB; H D L : f or a 10 μ m d i o d e a t 8.79 GHz, s p ik es o f 8.7 W f or 3.5 μ s a t 23.1 W in p u t, 16.1 W f or 750 ns a t 49.3 W, 59.1 W f or 60 ns a t 257 W an d 106 W f or 40 ns a t 498 W, t h e e n er g y g r o w s ab o u t o n e d ec a d e f or t h r ee d ec a d es o f p o w er an d m u c h m or e w i t h t h e I - r e g i o n w i d t h, t h e d a t a u se d a 1.5 ns r i se t im e, an d a 5 μ m d i o d e l imi t e d f r o m ab o u t 0.1 W w i t h 10 t o 13 dB i so l a t i o n; C L A 4609 : r eco v er y 5 ns an d t h r es h o l d + 36 dBm; M A C O M : r eco v er y 10 ns a t 2 μ m t o 250 ns a t 25 μ m; S K Y 16603 : f l a t l e ak a g e 5.5 dBm t y p i c a l an d 9 dBm ma x im u m a t + 10 dBm in p u t$ $(co m p u t e d : 30, 12, 3.5 an d 4.2 μ J f or t h e f o u r H D L s p ik es, e n er g i es es t ima t e d f r o m t h e f i g u r es r e a d an d n o t g i v e n b y t h e a u t h or s; 10 dB i so l a t i o n o n 1 W l e a v es 100 mW)$

Tunnel, Varactor and Step-Recovery Diodes

Name	Formula	Related Formulae	Variables/Constants
Current-enhanced dopant diffusion in a tunnel diode	$D = D_{t h}^{0} e^{- E_{a} / (k_{B} T)} + K J^{2} e^{- (E_{a} - Δ E) / (k_{B} T)}$	$\frac{D _{e nh}}{D _{t h er m}} \propto J^{2} e^{Δ E / (k_{B} T)}$	$D : d i f f u s i o n coe f f i c i e n t o f t h e d o p an t (c a r b o n in I n G a A s)$ $D_{t h}^{0} : t h er ma l p r e f a c t or$ $E_{a} : t h er ma l a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t (8.617 \times 1 0^{- 5} eV / K)$ $T : ab so l u t e t e m p er a t u r e$ $K : co n s t an t o f t h e c u r r e n t - e nhan ce d t er m$ $J : f or w a r d c u r r e n t d e n s i t y$ $Δ E : l o w er in g o f t h e a c t i v a t i o n e n er g y b y r eco mbina t i o n$ $D_{e nh} an d D_{t h er m} : e nhan ce d an d t h er ma l p a r t s; O n o e t a l . : ann e a l in g an d f or w a r d - c u r r e n t s t r ess l o w er e d t h e p e ak c u r r e n t t h r o ug h c a r b o n d i f f u s i o n w hi l e t h e v a l l ey c u r r e n t d i d n o t c han g e, t h e e nhan ce d p a r t sc a l es ab o u t a s J^{2} an d i s in d e p e n d e n t o f t h e d e f ec t d e n s i t y, an d Δ E = 0.8 eV, c l ose t o t h e I n G a A s ban d g a p, f r o m n o n r a d ia t i v e ban d - t o - ban d r eco mbina t i o n in t h e d e pl e t i o n l a y er$ $(co m p u t e d : e^{Δ E / (k_{B} T)} = 2.8 \times 1 0^{13} a t 300 K an d 1.2 \times 1 0^{10} a t 400 K, so t h e s ha r e o f t h e c u r r e n t - d r i v e n p a r t g r o w s 2300 t im es o n coo l in g f r o m 400 t o 300 K a t t h e s am e c u r r e n t)$
Temperature drift of the tunnel-diode operating points	$V_{P} (T) = V_{P 0} + α_{P} (T - T_{0}), V_{V} (T) = V_{V 0} + α_{V} (T - T_{0})$	$I_{V} (T) = I_{V 0} e^{β_{V} (T - T_{0})}$	$V_{P} an d V_{V} : p e ak an d v a l l ey v o l t a g es$ $α_{P} an d α_{V} : t h e i r t e m p er a t u r e coe f f i c i e n t s$ $T_{0} : r e f er e n ce t e m p er a t u r e$ $I_{V} : v a l l ey c u r r e n t$ $β_{V} : i t s e x p o n e n t ia l t e m p er a t u r e coe f f i c i e n t; E l ec t r o ni cs W or l d : α_{P} = - 0.1 mV/K an d α_{V} = - 0.8 mV/K, I_{P} an d V_{P} v a r y ab o u t 10 p er ce n t f r o m - 55 t o 150^{\circ} C, I_{V} a t 150^{\circ} C ma y b e t h r ee t im es i t s v a l u e a t - 55^{\circ} C, G e d i o d es a r e t h e m os t s t ab l e w i t h t e m p er a t u r e, S i i s n ee d e d u p t o 150^{\circ} C, an d t h e v a l l ey c u r r e n t i s t h e p a r am e t er m os t se n s i t i v e t o r a d ia t i o n, w i t h n o t ab l e s hi f t s n e a r 1 \times 1 0^{16} n/c m^{2}$ $(co m p u t e d : β_{V} = 0.0054 K^{- 1}, a d o u b l in g e v er y 129 K; t h e 1 \times 1 0^{16} n/c m^{2} l e v e l i s f a r ab o v e t h e n e u t r o n f l u e n ces t ha t d e g r a d e min or i t y - c a r r i er d e v i ces, see t h e R a d ia t i o n p a g e (m y r e a d in g))$
RF voltage window of a varactor or step-recovery diode	$V_{0} = - \frac{V _{B} - ϕ}{2}, V_{1, ma x} = \frac{V _{B} + ϕ}{2}$	$- V_{B} \leq V_{0} \pm V_{1} \leq ϕ$	$V_{0} : D C bia s$ $V_{1, ma x} : l a r g es t R F am pl i t u d e t ha t n e i t h er f or w a r d - bia ses t h e j u n c t i o n n or r e a c h es a v a l an c h e$ $V_{B} : b r e ak d o w n v o l t a g e$ $ϕ : f or w a r d t u r n - o n v o l t a g e (ab o u t 0.7 V f or S i, a ss u m e d); ba c k g r o u n d k n o w l e d g e; S e mi G e n S S R s t e p - r eco v er y d i o d es : minim u m V_{B} o f 15, 20, 30 an d 40 V a t 10 mA, c a r r i er l i f e t im e a t l e a s t 8 t o 21 ns, t r an s i t i o n t im e a t m os t 60 t o 150 ps, an d s i l i co n d i o x i d e p a ss i v a t i o n f or s t ab l e l o w l e ak a g e o v er t e m p er a t u r e$ $(co m p u t e d : f or V_{B} = 15 V t h e bia s i s - 7.15 V an d t h e am pl i t u d e a t m os t 7.85 V)$
Thermal limit of a multiplier diode	$P_{D, ma x} = \frac{T _{ma x} - T _{A}}{θ _{T H}}$	$P_{D} = I_{r f}^{2} R_{S} /2$	$P_{D, ma x} : hi g h es t d i ss i p a t i o n$ $T_{ma x} : ma x im u m o p er a t in g t e m p er a t u r e$ $T_{A} : ambi e n t or h e a t - s ink t e m p er a t u r e$ $θ_{T H} : t h er ma l r es i s t an ce$ $P_{D} : d i ss i p a t i o n o f a s in u so i d a l R F c u r r e n t o f am pl i t u d e I_{r f} in t h e ser i es r es i s t an ce R_{S}; S e mi G e n S S R : o p er a t i o n - 55 t o + 150^{\circ} C, θ_{T H} 40 t o 125 K/W (p u l se d, in f ini t e h e a t s ink, t h e l o w er - c a p a c i t an ce p a r t s a t t h e hi g h er v a l u es), R_{S} 0.20 t o 1.20 Ω$ $(co m p u t e d : f r o m 25^{\circ} C t o 150^{\circ} C, 3.1 W a t 40 K/W an d 1.0 W a t 125 K/W)$

Important Questions

Ques: Why does an IMPATT diode that gives 1 W of RF need about 6 W of cooling?

Because its efficiency is low, and all the DC input that does not leave as RF heats the junction. The Hughes 60 $GHz$ diodes gave 1 $W$ at 14.7 per cent efficiency, so they dissipated $P_{R F} (1 - η) / η = 5.80$ $W$ , and the reported 228 $^{\circ} C$ junction implies a thermal resistance of 35.0 $K/W$ from a 25 $^{\circ} C$ heat sink (computed). The delivered diodes, at 13 per cent and a 236 $K$ rise, give 35.3 $K/W$ , a consistent value. The usable RF output is then $(T_{M} - T_{A}) / θ_{T H} \times η / (1 - η)$ , which is why Hughes moved to diamond heat sinks, which lowered the thermal resistance by up to half. At an assumed 200 $^{\circ} C$ limit, the same diode could give only 0.86 $W$ (computed). Reference: Y. E. Ma, J. Chen, E. Benko, M. J. Barger, H. Nghiem, T. Q. Trinh and J. Kung, Hughes Aircraft Company, ‘60 GHz IMPATT Diode Development’, Final Report NASA CR-174969, May 1985 (NTRS 19850024980; Eq. 34 partly garbled, fetch-tool summary); Thermal Stress page of this vault.

Ques: Did the Au-Zn metallized 60 GHz IMPATT diodes fail at 350 or at 375 degrees Celsius?

The Hughes report gives both values. Its abstract states that most diodes with the Au-Zn epi-side metallization failed around 350 $^{\circ} C$ in the DC step-stress test, while its introduction gives 375 $^{\circ} C$ as the maximum failure temperature of that group. The two do not contradict each other if 350 $^{\circ} C$ is the typical value and 375 $^{\circ} C$ the highest one, which fits the stated wide spread of failure temperatures (my reading). For design, 350 $^{\circ} C$ is the more likely and the safer value. The Pt-Ti-Pt-Au diodes failed between 400 and 425 $^{\circ} C$ , and the authors attribute the difference of about 50 $K$ to a higher thermal resistance, an inference, not a measurement. Since the failures were process related, neither value is an intrinsic limit. Reference: Y. E. Ma, J. Chen, E. Benko, M. J. Barger, H. Nghiem, T. Q. Trinh and J. Kung, Hughes Aircraft Company, ‘60 GHz IMPATT Diode Development’, Final Report NASA CR-174969, May 1985 (NTRS 19850024980; Eq. 34 partly garbled, fetch-tool summary).

Ques: How much longer does an IMPATT diode live at 200 instead of 265 degrees Celsius?

It depends on the activation energy, which the RADC report read does not give as a number. Three full-Read GaAs diodes failed after 809, 764 and 768 $h$ at 265 $^{\circ} C$ , a mean of 780 $h$ (computed). With the Arrhenius form of RADC Eq. 1.5, the acceleration factor from 265 to 200 $^{\circ} C$ is 19 for 1.0 $eV$ , 85 for 1.5 $eV$ and 207 for 1.8 $eV$ (assumed), so the expected life at 200 $^{\circ} C$ ranges from 1.7 to 18 years (computed). The range shows why the activation energy must come from tests at several temperatures, which RADC plotted as reciprocal failure temperature against cumulative failures. Three failures at one temperature fix only the scale, not the slope. Reference: M. S. Ayyagari, J. L. Heaton and N. Jansen, Microwave Associates, ‘Reliability of High-Power Pulsed IMPATT Diodes’, Final Technical Report RADC-TR-81-315, November 1981 (DTIC ADA110798; OCR text partly garbled, fetch-tool summary); Failure Rate and the Bathtub Curve page of this vault.

Ques: Is a DC burn-in enough to screen IMPATT diodes?

Not for single-drift GaAs diodes in the RADC study. A 48 $h$ DC burn-in removed 5 of 25 diodes and a pulsed DC burn-in 2 of 30, but the RF test that followed removed 4 of 19 and 7 of 28 of the survivors (computed: 21 and 25 per cent). The total fallout was 36.8 per cent on the DC path and 30.0 per cent on the pulsed path, so most weak diodes passed the DC stages and failed only when oscillating. The authors therefore conclude that single-drift diodes need an oscillating burn-in, while for GaAs double-drift diodes pulsed burn-in with or without oscillation was equally effective. A DC stress does not reproduce the large RF voltage swing and the different current distribution of an oscillating diode (my reading). Reference: M. S. Ayyagari, J. L. Heaton and N. Jansen, Microwave Associates, ‘Reliability of High-Power Pulsed IMPATT Diodes’, Final Technical Report RADC-TR-81-315, November 1981 (DTIC ADA110798; OCR text partly garbled, fetch-tool summary).

Ques: Why does a Gunn diode lose current at 250 degrees Celsius, and why does the bias polarity matter?

Gulati et al. annealed AuGeNi-contacted GaAs Gunn diodes at 200 to 300 $^{\circ} C$ and found that the current fell after 50 to 60 $h$ at 250 $^{\circ} C$ and kept falling to about 230 $h$ . They attribute this to a smaller effective contact area: Ni penetrates between the AuGe eutectic grains, diffuses into the buffer and forms compensated regions. With the epi side biased negative by at least about 2.5 $V$ , the migration was largely arrested for about 500 $h$ , while a positive bias enhanced it to catastrophic failure. This suggests a field-driven drift of ionized Ni added to diffusion (my reading). It explains why Gunn diodes run for thousands of hours at 250 to 300 $^{\circ} C$ in the intended polarity, and why a reversed connection is harmful. Reference: R. Gulati, I. Chandra and B. L. Sharma, ‘Proposed Model for Degradation of Gunn Diodes as Observed from Study of the I-V Characteristics’, Defence Science Journal 33(2), 113-117 (1983) (fetch-tool summary); Microsemi, ‘MG1001-MG1060 Series GaAs Gunn Diodes’ data sheet, Rev. 2011-06-20 (fetch-tool summary).

Ques: Does the peak power of a PIN limiter rise linearly with the I-region width or with its 2.2 power?

The two sources answer different questions. The MACOM data sheet rates diodes from 2 to 25 $μ m$ at 80 to 1000 $W$ for 1 $μ s$ pulses, a growth about as $W^{1.0}$ (computed), but these parts also differ in area, capacitance and thermal resistance. HDL calculated the damage power of diodes whose area grew with the width to keep the capacitance constant, so the heated volume grew as $W^{2}$ , and found a log-log slope of 2.2 for 0.1 $μ s$ pulses. For a given product line, the data-sheet value is the more reliable guide, and the HDL slope applies only when the capacitance is held fixed (my reading). Both agree that a thicker I region survives more power but leaks a larger spike and recovers more slowly, 250 $ns$ at 25 $μ m$ against 10 $ns$ at 2 $μ m$ . Reference: MACOM, ‘Silicon PIN Limiter Diodes, MA4L and MADL Series’ data sheet, Rev. V20 (year not stated) (fetch-tool summary); A. L. Ward, R. J. Tan and R. Kaul, ‘Spike Leakage and Burnout of Silicon PIN Diode Microwave Limiters’, Harry Diamond Laboratories HDL-TR-2195, June 1991 (DTIC ADA238392; values partly read from plots, fetch-tool summary).

Ques: Why does a short RF pulse destroy a limiter at a fixed energy rather than a fixed power?

Because the heat cannot leave the I region during the pulse. The thermal diffusion time across a width $W$ is about $W^{2} / D_{t h}$ , and with $D_{t h} = 0.91$ $c m^{2} /s$ for silicon (background knowledge) it is 1.1 $μ s$ for 10 $μ m$ (computed). A 0.1 $μ s$ pulse is therefore adiabatic, and the energy to heat the volume by 500 $K$ , the HDL damage criterion, is $ρ c A W Δ T$ : 24 $μ J$ for $3 \times 1 0^{- 5}$ $c m^{2}$ and 245 $μ J$ for $3 \times 1 0^{- 4}$ $c m^{2}$ , the two areas the report gives (computed). HDL indeed found that the calculated damage power falls about linearly with pulse length, a constant energy. For pulses longer than the diffusion time the Wunsch-Bell square-root law and finally a steady-state power limit take over. Reference: A. L. Ward, R. J. Tan and R. Kaul, ‘Spike Leakage and Burnout of Silicon PIN Diode Microwave Limiters’, Harry Diamond Laboratories HDL-TR-2195, June 1991 (DTIC ADA238392; values partly read from plots, fetch-tool summary); Electrical Stress page of this vault.

Ques: Is a Schottky mixer diode rated for 15 W or for 1 W of pulse power?

The HP 5082 data sheet uses both numbers for different quantities. The feature list claims 15 $W$ of RF pulse power incident, while the maximum rating is 1 $W$ of peak power absorbed for a 1 $μ s$ pulse at 0.001 duty and 25 $^{\circ} C$ case. The difference is the mismatch: a diode driven far above its normal level reflects most of the incident power, and 1 $W$ absorbed is 6.7 per cent of 15 $W$ (computed). The absorbed rating, 1 $μ J$ per pulse, is the more reliable limit for a design, since the reflected fraction depends on the mount and the frequency (my reading). The CW limits of 100 to 250 $mW$ and the junction limits of 100 to 200 $^{\circ} C$ show that low-barrier detector types are the most fragile. Reference: Hewlett-Packard, ‘Schottky Barrier Diodes for Mixers and Detectors, 5082 series’ data sheet (year not stated) (fetch-tool summary).

Ques: Why does the peak current of an InGaAs tunnel diode fall under forward current?

Ono et al. stressed InGaAs Esaki diodes grown on GaAs and on InP with different defect densities. Both thermal annealing and forward-current stress lowered the peak current, while the valley current stayed the same, so the defect density did not grow. The cause is carbon diffusing out of the degenerate junction, which widens it and reduces tunnelling. The current enhances the diffusion: the enhanced part scales about as the square of the current density, does not depend on the defect density, and has an activation energy 0.8 $eV$ lower, close to the band gap, which points to energy released by nonradiative band-to-band recombination in the depletion layer. The enhancement is much weaker than in GaAs, so InGaAs tunnel junctions are the more stable choice (my reading). Reference: H. Ono, S. Taniguchi and T. Suzuki, ‘Impurity Diffusion in InGaAs Esaki Tunnel Diodes of Varied Defect Densities’, IEICE Transactions on Electronics E89-C(7), 1020-1024 (2006) (abstract only).

Ques: How much RF swing can a varactor or step-recovery diode take?

The junction must stay between forward conduction and avalanche over the whole RF cycle, $- V_{B} \leq V_{0} \pm V_{1} \leq ϕ$ . The largest amplitude is $(V_{B} + ϕ) /2$ , with the bias at the centre of the window (background knowledge). For a SemiGen SSR70x with a minimum breakdown of 15 $V$ and an assumed 0.7 $V$ turn-on, this gives a bias of -7.15 $V$ and at most 7.85 $V$ amplitude (computed). A larger drive avalanches the junction on the negative peak, which adds noise and heat, or forward-biases it, which a step-recovery diode does intentionally but a varactor should not. The heat must also stay within $(T_{ma x} - T_{A}) / θ_{T H}$ , 1.0 to 3.1 $W$ for the 40 to 125 $K/W$ of the SSR range up to 150 $^{\circ} C$ (computed). Reference: SemiGen, ‘SSR Series Step Recovery Diodes’ data sheet (year not stated) (fetch-tool summary); Skyworks Solutions, ‘Design With PIN Diodes’, Application Note 200312D, 22 October 2012 (fetch-tool summary; some equations garbled).

Quartz 5

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Microwave Diodes (IMPATT, Gunn, Limiter, Mixer, Tunnel, Varactor)