HgCdTe Photodiode

1. Theory

Definition: A HgCdTe photodiode is a p-n junction in the alloy $H g_{1 - x} C d_{x} Te$ of the semimetal HgTe and the wide-gap CdTe; the gap is set by the cadmium fraction $x$ and the temperature, $E_{g} (x, T)$ , so that the cutoff wavelength can be chosen between about 1 μm and 30 μm.
Composition: A fraction $x \approx 0.30$ gives the mid-wave band (3 μm to 5 μm), $x \approx 0.20$ the long-wave band (8 μm to 12 μm), and the lattice mismatch of the alloy to CdTe is only about 0.2%, which allows graded and heterojunction layers.
Structures: The n-on-p homojunction has an acceptor-doped base of $N_{a} \approx 5 \times 1 0^{15} c m^{- 3}$ , and the p-on-n double-layer heterojunction (DLHJ) an indium-doped n-type base of $1 0^{15} c m^{- 3}$ or less with an arsenic-doped p-type cap of $(1 to 4) \times 1 0^{16} c m^{- 3}$ and a gap wider by $Δ x \approx 0.04$ .
Illumination: The diode is illuminated through the transparent CdZnTe substrate, the base is about 10 μm to 15 μm thick, i.e. of the order of the absorption length $1/ α$ , and the surface is passivated by a CdTe layer of about 500 Å.
Fundamental limit: The dark current is limited by thermal generation in the base. Auger recombination dominates (Auger 1 in n-type, Auger 7 in p-type material) and the radiative process is weaker because photon recycling lengthens the radiative lifetime.
Rule 07: The dark current density of the best planar p-on-n diodes grown by molecular-beam epitaxy follows the empirical law $J = J_{0} exp [C 1.24 q / (k_{B} λ_{e} T)]$ of the cutoff wavelength and the temperature, which predicts 67 layers within 0.4 to 2.5 times over about 13 orders of magnitude and, at mid-wave and longer wavelengths, agrees with the Auger 1 limit of a base of $4 \times 1 0^{14}$ to $1 0^{15} c m^{- 3}$ .
Non-fundamental currents: Generation in the depletion region, band-to-band and trap-assisted tunnelling, surface leakage and dislocations add to the fundamental current, and a base doping below about $1 0^{16} c m^{- 3}$ is needed to keep the tunnelling current low.
Cooling: The long-wave detectors operate near 77 K (and near 40 K for cutoffs of 17 μm to 20 μm), and the mid-wave detectors reach the background limit with thermoelectric cooling.
Avalanche variant: The electron-initiated avalanche photodiode (e-APD) uses a mid-wave absorber with a multiplication layer in which only the electrons ionize.
Figures of merit: The resistance-area product $R_{0} A$ , the quantum efficiency, $D^{*}$ , the dark current density compared with the empirical benchmark called Rule 07, and for the e-APD the gain and the excess noise factor.

2. Working Principle

Absorption: A photon above the gap creates a pair in the base; the thickness of 10 μm to 15 μm gives a quantum efficiency of 70% to 80%, because the base is thin compared with the minority diffusion length (25 μm to 50 μm in starting material).
Collection: The minority carriers diffuse to the junction, the wider-gap cap keeps the Auger generation of the cap and the surface recombination low, and the diode is operated at zero bias or at a few tens of mV (-10 mV to -40 mV in a direct-injection readout).
Dark current: At small bias and moderate temperature the diffusion current of the base dominates and sets $R_{0} A$ ; at a reverse bias above about 0.3 V the tunnelling current dominates in long-wave diodes, and a low base doping suppresses it.
Resistance: The $R_{0} A$ of an Auger-limited base falls with the intrinsic density $n_{i}^{2} \propto exp (- E_{g} / k_{B} T)$ , so a longer cutoff needs a lower temperature to keep the same resistance.
Avalanche: At a reverse bias of the order of 10 V the electrons ionize with an exponentially growing gain, the holes do not ionize, and the multiplication is nearly deterministic, so that the excess noise factor stays between 1 and 1.25.

3. Applications

Cooled long-wave and mid-wave focal plane arrays: Backside-illuminated p-on-n arrays, such as the 10.6 μm bilinear arrays operated at 72 K, and dual-band arrays with quantum efficiencies of 79% (MW) and 67% (LW).
Astronomy and space: Hybrid arrays with a dark current of at most 0.05 $e^{-}$ /s per pixel at 77 K for a 2.5 μm cutoff and about 0.003 $e^{-}$ /s per pixel at 38.5 K for a 5 μm cutoff.
Photon counting and wavefront sensing: Electron-initiated avalanche arrays with a readout noise below 1 $e^{-}$ and gains above 500, which resolve single photons.
Heterodyne detection of $C O_{2}$ lasers: Photodiodes with a bandwidth above 1 GHz at 10.6 μm.

Name	Formula	Related Formulae	Variables/Constants
Alloy gap and cutoff wavelength	$E_{g} (x, T) = - 0.302 + 1.93 x - 0.81 x^{2} + 0.832 x^{3} + 5.35 \times 1 0^{- 4} T (1 - 2 x)$	$λ_{c} (μ m) = \frac{1.2398}{E _{g} ( eV )}$ $\frac{m _{e}^{}}{m _{0}} \approx 0.071 E_{g} (eV), m_{h}^{} = 0.55 m_{0}$	$E_{g} : g a p in e V$ $x : c a d mi u m f r a c t i o n o f H g_{1 - x} C d_{x} Te$ $T : t e m p er a t u r e in K$ $λ_{c} : c u t o f f w a v e l e n g t h$ $m_{e}^{}, m_{h}^{} : e f f ec t i v e ma sses o f t h e e l ec t r o n an d t h e h o l e$ $m_{0} : f r ee e l ec t r o n ma ss, t h e f or m u l a i s t h e H an se n - S c hmi t - C a sse l man r e l a t i o n$ $(x = 0.228 a t 77 K g i v es 0.1282 e V an d 9.67 μ m, an d x = 0.3 a t 77 K g i v es 0.243 e V an d 5.10 μ m, co m p u t e d, a g ain s t t h e s t a t e d 9.7 μ m an d 5.1 μ m)$
Intrinsic carrier concentration	$n_{i} = (5.585 - 3.82 x + 1.753 \times 1 0^{- 3} T - 1.364 \times 1 0^{- 3} x T) \times 1 0^{14} E_{g}^{3/4} T^{3/2} exp (- \frac{E _{g}}{2 k _{B} T})$	$n p = n_{i}^{2}, n_{i} \propto exp (- \frac{E _{g}}{2 k _{B} T})$ $\frac{n _{i}^{2} ( T _{2} )}{n _{i}^{2} ( T _{1} )} \approx (\frac{T _{2}}{T _{1}})^{3} exp [\frac{E _{g}}{k _{B}} (\frac{1}{T _{1}} - \frac{1}{T _{2}})]$	$n_{i} : in t r in s i c co n ce n t r a t i o n in c m^{- 3}$ $E_{g} : g a p in e V f r o m t h e f i r s t r o w$ $k_{B} : B o l t z mann co n s t an t in e V / K, t h e p r e f a c t or i s t h e H an se n - S c hmi t f or m (r ec a l l e d, see t h e n o t e o n t h e so u r ces), t h e r a t i o i s f or a co n s t an t g a p$ $(x = 0.228 a t 77 K g i v es 4.5 \times 1 0^{12} c m^{- 3} an d x = 0.3 a t 77 K g i v es 1.2 \times 1 0^{9} c m^{- 3}, co m p u t e d)$
Diffusion-limited resistance-area product	$(R_{0} A)_{D} = \frac{k _{B} T}{q ^{2}} \frac{N}{n _{i}^{2}} \frac{L}{D} coth (\frac{d}{L})$	$(R_{0} A)_{D} \approx \frac{k _{B} T}{q ^{2}} \frac{N}{n _{i}^{2}} \frac{τ}{d} (d ≪ L)$ $(R_{0} A)_{D} = \frac{k _{B} T}{q ^{2}} \frac{N}{n _{i}^{2}} \frac{L}{D} tanh (\frac{d}{L}) (ohmic back contact)$	$N : d o p in g o f t h e ba se (d o n or s f or t h e n - t y p e ba se o f a p - o n - n d i o d e)$ $n_{i} : in t r in s i c co n ce n t r a t i o n$ $τ, D : l i f e t im e an d d i f f u s i v i t y o f t h e min or i t y c a r r i er s, L = D τ$ $d : t hi c k n ess o f t h e ba se, t h e co t h f or m h o l d s f or a r e f l ec t in g ba c k co n t a c t (t hin - ba se f or m : k_{B} T τ N / (q^{2} n_{i}^{2} d))$ $k_{B} : B o l t z mann co n s t an t$ $(x = 0.228, 77 K, N = 1 0^{15} c m^{- 3}, d = 15 μ m an d τ = 100 n s (a ss u m e d) g i v e 1.4 \times 1 0^{2} Ω c m^{2}, co m p u t e d; t h e m e a s u r e d v a l u e o f an n - o n - p h o m o j u n c t i o n a t 10 μ m w a s 655 Ω c m^{2} a g ain s t 680 Ω c m^{2} f or t h e A ug er l imi t)$
Auger and Shockley-Read-Hall generation	$(G - R)_{A U G} = (C_{n} n + C_{p} p) (n p - n_{i}^{2})$	$(G - R)_{S R H} = \frac{n p - n _{i}^{2}}{τ _{p 0} ( n + n _{1} ) + τ _{n 0} ( p + p _{1} )}$ $γ = \frac{C _{p}}{C _{n}}, n_{1} = n_{i} exp (\frac{E _{T} - E _{F i}}{k _{B} T})$	$C_{n}, C_{p} : A ug er coe f f i c i e n t s o f t h e p r ocesses w i t h tw o e l ec t r o n s an d w i t h tw o h o l es$ $γ : r a t i o o f t h e coe f f i c i e n t s (3 t o 60 d e p e n d in g o n t h e t e m p er a t u r e)$ $τ_{n 0}, τ_{p 0} : S R H l i f e t im es (1/ (c_{n} N_{T}) an d 1/ (c_{p} N_{T}))$ $E_{T} : t r a p l e v e l$ $E_{F i} : in t r in s i c F er mi l e v e l$ $n_{1} : t r a p occ u p a t i o n d e n s i t y$ $k_{B} : B o l t z mann co n s t an t$ $(f or an n - t y p e ba se w i t h n = N t h e min or i t y h o l es ha v e τ_{A 1} = 1/ (C_{n} N^{2}) an d t h e t h er ma l g e n er a t i o n r a t e i s g_{t h} = C_{n} N n_{i}^{2}, d er i v e d, so t ha t (R_{0} A)_{D} \approx k_{B} T / (q^{2} C_{n} N n_{i}^{2} d) f or d ≪ L)$
Rule 07 dark current benchmark	$J = J_{0} exp (C \frac{1.24 q}{k λ _{e} T})$	$λ_{e} = λ_{c u t o f f} (λ_{c u t o f f} \geq λ_{t h})$ $λ_{e} = \frac{λ _{c u t o f f}}{1 - ( λ _{sc a l e} / λ _{c u t o f f} - λ _{sc a l e} / λ _{t h} ) ^{P w r}} (λ_{c u t o f f} < λ_{t h})$ $J_{d} = \frac{k _{B} T}{q R _{0} A}$	$J : d a r k c u r r e n t d e n s i t y o f a p - o n - n d i o d e in A/c m^{2}, m e a s u r e d se v er a l k_{B} T / q in r e v er se bia s$ $λ_{c u t o f f} : c u t o f f w a v e l e n g t h in μ m a t w hi c h t h e q u an t u m e f f i c i e n cy f a l l s t o ha l f o f i t s pl a t e a u$ $λ_{e} : e f f ec t i v e w a v e l e n g t h$ $T : t e m p er a t u r e, 1.24 : p r o d u c t o f p h o t o n e n er g y an d w a v e l e n g t h in e V μ m$ $R_{0} A : z er o - bia s r es i s t an ce - a r e a p r o d u c t, t h e or i g ina l f i t i s J_{0} = 8367 A/c m^{2}, C = - 1.1624, P w r = 0.5441, λ_{sc a l e} = 0.2008 μ m, λ_{t h} = 4.635 μ m, t h e r e f i t o f 2010 i s 5315 A/c m^{2}, - 1.1403, 0.5696, 0.2151 μ m an d 4.843 μ m$ $k_{B} : B o l t z mann co n s t an t$ $(λ_{c u t o f f} = 9.67 μ m a t 77 K g i v es 1.5 \times 1 0^{- 6} A/c m^{2} an d R_{0} A = 4.5 \times 1 0^{3} Ω c m^{2}, 12 μ m a t 77 K g i v es 1.2 \times 1 0^{- 4}, 5 μ m a t 120 K g i v es 6.5 \times 1 0^{- 9}; t h e e x p o n e n t e q u a l s C E_{e} / k_{B} T w i t h E_{e} = 1.24/ λ_{e} in e V, a l l co m p u t e d)$
Auger 1 lifetime of the n-type base	$τ_{A 1} = \frac{2 n _{i}^{2}}{( N _{D} + p _{0} ) ( N _{D} + 2 p _{0} )} τ_{A i 1}$	$τ_{A i 1} = \frac{3.8 \times 1 0 ^{- 18} ϵ _{\infty}^{2} 1 + μ ( 1 + 2 μ ) e x p [ \frac{1 + 2 μ}{1 + μ} \frac{E _{g}}{k _{B} T} ]}{m _{c} ∣ F _{1} F _{2} ∣ ^{2} ( k _{B} T / E _{g} ) ^{3/2}}$ $J_{A 1} \approx \frac{q N _{D} d}{2 τ _{A i 1}} (d ≪ L)$	$τ_{A 1} : A ug er 1 l i f e t im e o f t h e min or i t y h o l es$ $τ_{A i 1} : in t r in s i c A ug er 1 l i f e t im e in s$ $n_{i} : in t r in s i c co n ce n t r a t i o n$ $N_{D} : d o n or d e n s i t y$ $p_{0} : min or i t y h o l e d e n s i t y (se t t o z er o)$ $ϵ_{\infty} : hi g h - f r e q u e n cy d i e l ec t r i c co n s t an t$ $μ : r a t i o o f t h e e l ec t r o n t o t h e h o l e e f f ec t i v e ma ss$ $m_{c} : e l ec t r o n e f f ec t i v e ma ss in u ni t s o f m_{0} (a ss u m e d)$ $∣ F_{1} F_{2} ∣ : o v er l a p in t e g r a l (0.2)$ $J_{A 1} : t hin - ba se A ug er c u r r e n t$ $q : e l ec t r o n c ha r g e$ $d : ba se t hi c k n ess$ $L : d i f f u s i o n l e n g t h (t h e l a s t f or m i s d er i v e d)$ $k_{B} : B o l t z mann co n s t an t$ $(x = 0.228 a t 77 K, N_{D} = 1 0^{15} c m^{- 3}, ϵ_{\infty} = 12.07 (r ec a l l e d, a ss u m e d), m_{c} = 0.071 E_{g} an d m_{h} = 0.55 g i v e τ_{A i 1} = 45 m s an d τ_{A 1} = 1.8 μ s; d = 10 μ m g i v es J_{A 1} = 1.8 \times 1 0^{- 6} A/c m^{2}, co m p u t e d)$
Band-to-band and trap-assisted tunnelling	$G_{T} (F) = \frac{q ^{2} m ^{} F ^{2}}{4 2 π ^{2} ℏ ^{2} E _{g}} exp [- \frac{4 2 m ^{} E _{g}^{3/2}}{3 q ℏ F}]$	$(G - R)_{T A T} = \frac{n p - n _{i}^{2}}{τ _{p 0} δ _{p} ( n + n _{1} ) + τ _{n 0} δ _{n} ( p + p _{1} )}$ $F_{ma x} = \frac{2 q N ( V _{bi} + V )}{ϵ}$	$G_{T} : ban d - t o - ban d g e n er a t i o n r a t e f or a t r ian g u l a r ba r r i er$ $F : e l ec t r i c f i e l d, m^{*} = m_{c} m_{v} / (m_{c} + m_{v}) : r e d u ce d ma ss$ $δ_{n}, δ_{p} : f i e l d - d e p e n d e n t e nhan ce m e n t f a c t or s o f t h e t r a p - a ss i s t e d p r ocess (H u r k x m o d e l)$ $F_{ma x} : p e ak f i e l d o f an ab r u pt o n e - s i d e d j u n c t i o n (d er i v e d)$ $ϵ : p er mi tt i v i t y (ab o u t 18 ϵ_{0})$ $V_{bi} : b u i l t - in v o l t a g e$ $(E_{g} = 0.128 e V g i v es an e x p o n e n t o f 33 a t N = 1 0^{15} c m^{- 3} an d 10.5 a t N = 1 0^{16} c m^{- 3} f or V = 0.3 V an d V_{bi} = 0.1 V (a ss u m e d), co m p u t e d)$
Detectivity and background limit	$D^{*} = R \frac{R _{0} A}{4 k _{B} T}$	$D_{B L I P}^{*} = \frac{λ}{h c} \frac{η}{2 Φ _{B}}$ $R_{0} A ≫ \frac{2 k _{B} T}{q ^{2} η Φ _{B}}$	$R : c u r r e n t r es p o n s i v i t y$ $R_{0} A : r es i s t an ce - a r e a p r o d u c t$ $Φ_{B} : ba c k g r o u n d p h o t o n f l ux p er u ni t a r e a$ $η : q u an t u m e f f i c i e n cy$ $λ : w a v e l e n g t h, t h e f i r s t f or m i s t h e J o hn so n - n o i se l imi t a t z er o bia s, t h e seco n d t h e l imi t o f t h e s h o t n o i se o f t h e ba c k g r o u n d p h o t oc u r r e n t (d er i v e d)$ $k_{B} : B o l t z mann co n s t an t$ $(a 300 K h e mi s p h er i c a l ba c k g r o u n d an d a c u t o f f o f 10 μ m w i t h η = 0.8 (a ss u m e d) g i v e Φ_{B} = 4.9 \times 1 0^{17} c m^{- 2} s^{- 1} an d D_{B L I P}^{*} = 4.5 \times 1 0^{10} cm H z^{1/2} /W, an d 1.4 \times 1 0^{11} f or 5 μ m, co m p u t e d)$
Quantum efficiency and optimum base thickness	$η = (1 - R) (1 - e^{- α d}) η_{c}, R = \frac{η q λ}{h c}$	$D^{*} \propto \frac{1 - e ^{- α d}}{d} \Rightarrow α d_{o pt} = 1.256$ $η = (1 - R) (1 - e^{- 2 α d}) η_{c} (retroreflector)$	$R : r e f l ec t an ce$ $α : ab sor pt i o n coe f f i c i e n t$ $d : t hi c k n ess o f t h e ba se$ $η_{c} : co l l ec t i o n e f f i c i e n cy (n e a r 1 f or d ≪ L), α d_{o pt} : r oo t o f 2 u = e^{u} - 1 t ha t ma x imi ses t h e d e t ec t i v i t y o f a t hin ba se (d er i v e d w i t h R_{0} A \propto 1/ d), t h e r e t r or e f l ec t or ha l v es t h e o pt im u m t hi c k n ess$ $(η = 79.7% a t 10.6 μ m an d 77 K f or a 13 μ m c u t o f f d e v i ce, an d 79% (M W) an d 67% (L W) f or a d u a l - ban d a r r a y; 1/ α = 10 μ m i s q u o t e d f or s in g l e p a ss)$
Avalanche multiplication and excess noise	$F = κ M + (2 - \frac{1}{M}) (1 - κ), κ = \frac{β}{α _{e}}$	$α_{e} (F) = a_{e} exp (- \frac{b _{e}}{F ^{m_{e}}})$ $F (κ \to 0) = 2 - \frac{1}{M}$	$F : e x cess n o i se f a c t or$ $M : g ain$ $κ : r a t i o o f t h e h o l e t o t h e e l ec t r o n i o ni z a t i o n coe f f i c i e n t$ $α_{e}, β : i o ni z a t i o n coe f f i c i e n t s o f t h e e l ec t r o n an d t h e h o l e$ $a_{e}, b_{e}, m_{e} : f i tt in g co n s t an t s (a_{e} = 3 \times 1 0^{5} c m^{- 1} a f t er K in c h), t h e f i r s t f or m i s t h e M c I n t y r e e x p r ess i o n f or e l ec t r o n inj ec t i o n$ $(t h e M c I n t y r e l imi t f or κ = 0 i s 1.8 a t M = 5 an d 2.0 f or M ≫ 1, co m p u t e d, w h er e a s F = 1 t o 1.25 w a s m e a s u r e d a t g ain s u p t o 23 in an L W I R d e v i ce an d c l ose t o 1 b ey o n d 1000 in an M W I R d e v i ce)$

Important Questions

Ques: Which base gives the higher resistance-area product, the p-type or the n-type, and why are the best photodiodes p-on-n?

The review states that a p-type base gives a higher $R_{0} A$ than an n-type base of the same doping because $γ = C_{p} / C_{n} > 1$ (the n-type base is limited by Auger 1, which is stronger), and an acceptor-doped n-on-p diode reached 655 $Ω c m^{2}$ at 10 μm and 77 K against an Auger limit of 680 $Ω c m^{2}$ . The p-on-n structure nevertheless gives the lower dark current in practice, with an indium-doped base of $5 \times 1 0^{14}$ to $1 0^{15} c m^{- 3}$ , whereas the vacancy-doped p-type base has a short SRH lifetime and a higher leakage, and the p-type doping is hard to lower below $5 \times 1 0^{14} c m^{- 3}$ . The theoretical advantage of the p-type base is therefore not reached, and the n-type base is the more likely choice. Reference: A. Rogalski, Mercury Cadmium Telluride Photodiodes at the Beginning of the Next Millennium, Defence Sci. J. 51, 5 (2001); Jóźwikowski, Kopytko and Rogalski, Enhanced numerical analysis of current-voltage characteristics of long wavelength infrared p-on-n HgCdTe photodiodes, Bull. Pol. Acad. Sci. Tech. Sci. 58, 523 (2010); M. B. Reine, Status of LWIR HgCdTe Infrared Detector Technology, NASA N91-14383 (1990).

Ques: Why does the dark current of a long-wave diode rise so steeply with the cutoff wavelength and the temperature?

In the Auger-limited n-type base $τ_{A 1} = 1/ (C_{n} N^{2})$ , so the thermal generation rate is $g_{t h} = C_{n} N n_{i}^{2}$ and $(R_{0} A)_{D} \approx k_{B} T / (q^{2} C_{n} N n_{i}^{2} d)$ (derived), i.e. the resistance falls with $n_{i}^{2} \propto T^{3} exp (- E_{g} / k_{B} T)$ . The Hansen relation gives $n_{i} = 4.5 \times 1 0^{12} c m^{- 3}$ for $x = 0.228$ (9.7 μm) and $1.2 \times 1 0^{9} c m^{- 3}$ for $x = 0.3$ (5.1 μm) at 77 K (computed), so $n_{i}^{2}$ differs by a factor of $1.4 \times 1 0^{7}$ between the two cutoffs. A longer cutoff therefore needs a lower temperature for the same resistance, which is why the mid-wave diodes reach the background limit with thermoelectric cooling and the long-wave diodes at 77 K or below. Reference: Jóźwikowski, Kopytko and Rogalski, Enhanced numerical analysis of current-voltage characteristics of long wavelength infrared p-on-n HgCdTe photodiodes, Bull. Pol. Acad. Sci. Tech. Sci. 58, 523 (2010); Kopytko and Martyniuk, InTechOpen chapter 51331 on HgCdTe barrier detectors (2016); A. Rogalski, Mercury Cadmium Telluride Photodiodes at the Beginning of the Next Millennium, Defence Sci. J. 51, 5 (2001).

Ques: Why would a thinner base raise the resistance-area product, and why was a test of this inconclusive?

For a base with a reflecting back contact the diffusion current is $q g_{t h} L tanh (d / L)$ , so $R_{0} A$ equals the value for a thick base times $coth (d / L)$ , which rises as $L / d$ when $d ≪ L$ ; with an ohmic back contact it is multiplied by $tanh (d / L)$ and falls as the base is thinned. The 1979 study fabricated tapered $n^{+}$ -p- $p^{+}$ arrays to test the rise, but the data were inconclusive because of excess junction current and surface inversion, and the measured $R_{0} A$ was 2 to 3 times below the model since the lifetime was below the radiative limit. A thinner base also absorbs less; with $R_{0} A \propto 1/ d$ the detectivity is largest at $α d = 1.256$ (computed), i.e. 12.6 μm for $1/ α = 10$ μm. Reference: Sood, Marciniec and Reine, Moderate Temperature Detector Development, NASA CR-160306 (1979); A. Rogalski, Mercury Cadmium Telluride Photodiodes at the Beginning of the Next Millennium, Defence Sci. J. 51, 5 (2001).

Ques: Why is the Rule 07 benchmark quoted for p-on-n diodes only, and where does it fail?

Rule 07 is a fit to the dark current of planar, ion-implanted p-on-n diodes grown by molecular-beam epitaxy, with an n-type base 4 μm to 10 μm thick and a donor doping the paper gives as 0.5 to $2.0 \times 1 0^{15} c m^{- 3}$ in one place and 0.4 to $1.0 \times 1 0^{15} c m^{- 3}$ in another. It was derived for temperatures above 77 K and describes the best devices; the paper does not say why p-on-n only. Tennant finds that very long-wave diodes down to 60 K still agree, whereas Baier et al. find that below about 78 K the measured current departs from the rule. The later, dedicated study is the more likely for long-wave diodes at those temperatures, but only its abstract was read; above 78 K the rule is a heuristic good to a factor of 0.4 to 2.5. Reference: W. E. Tennant, “Rule 07” Revisited: Still a Good Heuristic Predictor of p/n HgCdTe Photodiode Performance?, J. Electron. Mater. 39, 1030 (2010); Tennant et al., MBE HgCdTe Technology: A Very General Solution to IR Detection, Described by Rule 07, J. Electron. Mater. 37, 1406 (2008) (abstract); Baier et al., HgCdTe Diode Dark Current Modeling: Rule 07 Revisited for LW and VLW, J. Electron. Mater. (2019), doi:10.1007/s11664-019-07299-z (abstract).

Ques: Does an Auger 1 estimate reproduce the Rule 07 current?

For $x = 0.228$ at 77 K the gap is 0.1282 eV and the cutoff 9.67 μm, where Rule 07 gives $1.5 \times 1 0^{- 6} A/c m^{2}$ (computed). With the Auger 1 lifetime of Eq. 2, $∣ F_{1} F_{2} ∣ = 0.2$ , $ϵ_{\infty} = 12.07$ (recalled, assumed) and the masses of the first table, $τ_{A i 1} = 45$ ms, and a thin base of 10 μm gives $J = q N_{D} d / (2 τ_{A i 1}) = 1.8 \times 1 0^{- 6} A/c m^{2}$ for $1 0^{15} c m^{- 3}$ and $7 \times 1 0^{- 7}$ for $4 \times 1 0^{14}$ (derived, computed), i.e. 1.2 and 0.48 times the rule. Taking $ϵ_{\infty}$ from 11 to 13 moves the first value from 2.1 to $1.5 \times 1 0^{- 6}$ . The paper shows this agreement in a figure for the two dopings but gives neither the thickness nor the numbers, so the check is of the order of magnitude. Reference: W. E. Tennant, “Rule 07” Revisited: Still a Good Heuristic Predictor of p/n HgCdTe Photodiode Performance?, J. Electron. Mater. 39, 1030 (2010).

Ques: Do the original and the refitted Rule 07 parameters give different currents?

Every parameter differs (the 2010 refit is $J_{0} = 5315 A/c m^{2}$ , $C = - 1.1403$ , Pwr = 0.5696, $λ_{sc a l e} = 0.2151$ μm, $λ_{t h} = 4.843$ μm), but the currents differ by 0.7 to 2.0 times for cutoff-temperature products of 400 μm K to 2700 μm K between 60 K and 300 K (computed), by 0.96 at 10 μm and 77 K and by 1.45 at 5 μm and 77 K, where the current is $1 0^{- 15} A/c m^{2}$ . That is inside the scatter of 0.4 to 2.5 for the data and 0.434 to 2.3 for the refit. The author states that the 15 digits have no physical meaning: 0.1% changes give almost as good a fit, $C$ and $λ_{t h}$ are the most sensitive, and the parameters compensate each other. The original set is the one named Rule 07. Reference: W. E. Tennant, “Rule 07” Revisited: Still a Good Heuristic Predictor of p/n HgCdTe Photodiode Performance?, J. Electron. Mater. 39, 1030 (2010).

Ques: Is the gain from a p-type base and Auger 7 stated correctly?

The paper says that a p-type absorber, with its sixfold lower Auger 7 recombination, would lower the dark current of a diffusion-limited device by six, equal to a rise of the operating temperature of about 11% at a cutoff-temperature product of 800 μm K and 22% at 1500 μm K, and calls this incremental. Rule 07 behaves as $exp (- 16 726/ λ T)$ in $μ m K$ (computed from $C$ and $1.24 q / k_{B}$ ), and a factor of six then means a rise of 9.4% at 800 μm K and 19.1% at 1500 μm K (computed); the stated percentages correspond to products of 925 and 1683 μm K. The paper’s figures are therefore approximate by about 15%, which does not change the conclusion that the gain in temperature is modest. Reference: W. E. Tennant, “Rule 07” Revisited: Still a Good Heuristic Predictor of p/n HgCdTe Photodiode Performance?, J. Electron. Mater. 39, 1030 (2010).

Ques: Why does the background limit require so low a dark current, and what resistance does it demand?

The shot noise of the background photocurrent exceeds the Johnson noise when $R_{0} A ≫ 2 k_{B} T / (q^{2} η Φ_{B})$ . For a 300 K hemispherical background below a cutoff of 10 μm, $Φ_{B} = 4.9 \times 1 0^{17} c m^{- 2} s^{- 1}$ (computed), so the condition at 77 K is $R_{0} A ≫ 0.21 Ω c m^{2}$ for $η = 0.8$ (assumed), and $D_{B L I P}^{*} = 4.5 \times 1 0^{10} cm H z^{1/2} /W$ . For a field of view of 30° (taken as a half angle of 15°, assumed) the flux is smaller by $sin^{2} 15° = 0.067$ and the condition becomes $R_{0} A ≫ 3 Ω c m^{2}$ (computed); the review asks for an impedance 10 to 20 times that of the background, so that the measured 655 $Ω c m^{2}$ leaves a wide margin. Reference: M. B. Reine, Status of LWIR HgCdTe Infrared Detector Technology, NASA N91-14383 (1990); A. Rogalski, Mercury Cadmium Telluride Photodiodes at the Beginning of the Next Millennium, Defence Sci. J. 51, 5 (2001).

Ques: Why does a low base doping suppress the tunnelling current, and by how much?

The peak field of an abrupt junction is $F_{ma x} = 2 q N (V_{bi} + V) / ϵ$ , and the band-to-band generation contains $exp [- 4 2 m^{*} E_{g}^{3/2} / (3 q ℏ F)]$ (derived). For $E_{g} = 0.128$ eV, $m^{*} = 0.0089 m_{0}$ (computed from the effective masses of the chapter), $V = 0.3$ V and $V_{bi} = 0.1$ V (assumed) the field is $9.0 \times 1 0^{3}$ V/cm at $N = 1 0^{15} c m^{- 3}$ , with an exponent of 33, and $2.8 \times 1 0^{4}$ V/cm at $1 0^{16} c m^{- 3}$ , with an exponent of 10.5 (computed). A tenfold lower doping thus lowers the tunnelling factor by about $1 0^{9}$ , which agrees with the requirement of a base doping below about $1 0^{16} c m^{- 3}$ and with the tunnelling-dominated current above about 0.3 V. Reference: Jóźwikowski, Kopytko and Rogalski, Enhanced numerical analysis of current-voltage characteristics of long wavelength infrared p-on-n HgCdTe photodiodes, Bull. Pol. Acad. Sci. Tech. Sci. 58, 523 (2010); Kopytko and Martyniuk, InTechOpen chapter 51331 on HgCdTe barrier detectors (2016); A. Rogalski, Mercury Cadmium Telluride Photodiodes at the Beginning of the Next Millennium, Defence Sci. J. 51, 5 (2001).

Ques: Is the excess noise of the HgCdTe avalanche photodiode close to 1 or to the McIntyre value of 2?

The McIntyre expression with $κ = 0$ gives $F = 2 - 1/ M$ , i.e. 1.8 at a gain of 5 and 2.0 at large gain (computed), whereas the measurements give $F = 1$ to 1.25 at gains up to 23 in an LWIR device and close to unity beyond 1000 in an MWIR device. The array paper explains the low value by the immobile holes, which does not follow from the McIntyre expression, since $κ = 0$ already assumes that only the electrons ionize; the magazine article attributes it to an electron ionization localized in space and energy, so that the multiplication is nearly deterministic, and the Monte Carlo studies find that alloy scattering raises the gain and lowers the noise. The measured values are the more likely, because the memoryless ionization assumed by McIntyre does not hold. Reference: A Monte Carlo study of multiplication and noise in HgCdTe avalanche photodiodes (IP Paris research portal, abstract); Experimental performances and Monte Carlo modelling of LWIR HgCdTe avalanche photodiodes (IP Paris research portal, abstract); Laser Focus World, Advances in Imaging: Electron-initiated APDs improve high-speed SWIR imaging; HgCdTe APD Arrays for Astronomy: Natural Guide Star Wavefront Sensing and Space Astronomy, arXiv:1807.00772.

Ques: Which cutoff is right, the value from the gap formula or the value quoted for the device?

The Hansen relation gives $λ_{c} = 1.2398/ E_{g}$ with $E_{g} = 0.1282$ eV for $x = 0.228$ at 77 K, i.e. 9.67 μm (computed), against 9.7 μm stated for the device, and 5.10 μm for $x = 0.3$ against 5.1 μm. For $x = 0.235$ at 80 K the formula gives 8.84 μm (computed), against 9.0 μm quoted for a modelled LWIR avalanche diode, a difference of 1.8%. The device cutoff is defined as the wavelength at which the response falls to 50% of the peak, which lies at a slightly longer wavelength than the gap wavelength, so the two values are compatible, and the formula is the more reliable for design while the 50% value describes the measured spectrum. Reference: Kopytko and Martyniuk, InTechOpen chapter 51331 on HgCdTe barrier detectors (2016); Experimental performances and Monte Carlo modelling of LWIR HgCdTe avalanche photodiodes (IP Paris research portal, abstract); Jóźwikowski, Kopytko and Rogalski, Enhanced numerical analysis of current-voltage characteristics of long wavelength infrared p-on-n HgCdTe photodiodes, Bull. Pol. Acad. Sci. Tech. Sci. 58, 523 (2010).

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