1. Theory

Definition: A CCD pixel is a MOS capacitor (a gate on a thin oxide on silicon) that stores the photogenerated charge in a potential well under the gate and passes it to the next gate when the voltages of the gates are changed; a CCD is a row of such gates, and the charge packet is read as a voltage only at the output node.
Full-frame device: The Hamamatsu full-frame transfer (FFT) CCD has two clock phases, a vertical register with a full well of about 300000 electrons, a horizontal register and a summing well that hold more, a read noise of a few to about 10 electrons and a dynamic range of 37500 to 50000.
Charge capacity: The charge that a well holds is the oxide capacitance times the change of the gate voltage, $N_{f u l l} = C_{e f f} Δ V_{G} / q$ ; the numbers of the 1970s are 6×10 $^{11}$ cm $^{- 2}$ for a surface channel and 1.4×10 $^{11}$ cm $^{- 2}$ for a deep buried channel at 5 V.
Surface and buried channel: In a surface-channel CCD the charge is held at the interface and meets the interface states; in a buried-channel CCD an implanted or epitaxial n layer holds the charge away from the interface, which makes it free of interface trapping and faster.
Transfer efficiency: Good devices have a charge transfer efficiency of 0.99999 per transfer, so that the charge is carried over thousands of transfers with a loss of a few percent.
Dark current: It is generated at the interface and in the bulk and halves for every 5 to 7 °C of cooling; multi-pinned-phase operation (MPP) suppresses the surface generation, at the cost of about half the full well in the low-dark-current mode of Hamamatsu.
Quantum efficiency: Light that enters through the gates of the front face gives up to about 40% at 600 nm to 700 nm; back-thinned devices of 10 μm to 20 μm give 50% to 90% from the ultraviolet to the near infrared.
Variants: The frame-transfer and interline formats, the time-delay-and-integrate operation (TDI), and the electron-multiplying CCD (EMCCD), which adds a gain register before the output node.
Figures of merit: The full well, the sensitivity in volts per electron, the read noise, the dynamic range, the charge transfer efficiency, the dark current at a given temperature, the quantum efficiency, the non-uniformity and the readout rate.

2. Working Principle

Integration: A gate held at a high voltage (the integrating phase) makes a deep well under it; photoelectrons collect there while the neighbouring gates form the barriers, and the pixel is isolated from its neighbours in the other direction by channel stops.
Transfer: The voltages of the gates are changed in sequence (two phases in the devices of Hamamatsu, three phases in those of the early work); the charge is moved by the field of its own space charge at first (self-induced drift), by thermal diffusion at the end, and by the fringing fields of the neighbouring gates, which work in the whole time.
Readout: The last charge packet is moved through a horizontal register to the floating node, where its charge is converted to a voltage by the capacitance of the node and read by a MOS transistor; the node is reset after each pixel.
Binning and gain: Several pixels can be summed in the register before the read-out, and in an EMCCD a register of several hundred stages multiplies the packet by impact ionization.
Cooling: Thermoelectric coolers reach about -10 °C (one stage), -20 °C to -30 °C (two) and -50 °C to -70 °C (three) from a room at 25 °C, with a halving of the dark current for every 5 to 7 °C.

3. Applications

Astronomy and spectroscopy: Cooled full-frame and back-thinned CCDs of the large formats, read slowly (below 100 kHz) for a noise of a few electrons.
X-ray spectroscopy: A thick or deep-depleted CCD counts single photons and gives their energy from the charge (3.65 eV per electron), with 140 eV measured at 5.9 keV with cooling.
Low-light and fast imaging: The EMCCD for photon-limited imaging at high frame rates, and the time-delay-and-integrate CCD for the scanning of moving objects.

MOS Well and Charge Capacity

Name	Formula	Related Formulae	Variables/Constants
Charge held under a photogate	$Q_{in v} = A_{e l ec} C_{o x} V_{G}$	$C_{o x} = \frac{ε _{o x}}{t _{o x}}$ $N_{e} = \frac{Q _{in v}}{q}$	$Q_{in v} : in v er s i o n c ha r g e u n d er t h e e l ec t r o d e$ $A_{e l ec} : e l ec t r o d e a r e a$ $C_{o x} : o x i d e c a p a c i t an ce p er u ni t a r e a$ $V_{G} : g a t e v o l t a g e$ $ε_{o x} : p er mi tt i v i t y o f t h e o x i d e$ $t_{o x} : o x i d e t hi c k n ess$ $N_{e} : n u mb er o f e l ec t r o n s, t h e e x p r ess i o n i s an u pp er b o u n d an d n o t t h e u s ab l e f u l l w e l l$ $(10 \times 20 μ m^{2}, t_{o x} = 0.1 μ m an d V_{G} = 10 V g i v e 0.69 pC or 4.3 \times 1 0^{6} e^{-} w i t h ε_{r} = 3.9 (a ss u m e d); t h e so u r ce p r in t s 0.6 pC an d 3.6 \times 1 0^{6} e^{-}, w hi c h im pl i es ε_{r} = 3.39, a l l co m p u t e d)$
Signal-handling capacity of a well	$N_{f u l l} = \frac{C _{e f f} Δ V _{G}}{q}$	$Δ V_{G} = V_{G, hi g h} - V_{G, l o w}$ $Q_{f u l l} = q N_{f u l l} A_{e l ec}$	$N_{f u l l} : f u l l w e l l p er u ni t a r e a$ $C_{e f f} : e f f ec t i v e c a p a c i t an ce p er u ni t a r e a (e q u a l t o C_{o x} f or a s u r f a ce c hann e l)$ $Δ V_{G} : d i f f er e n ce o f t h e g a t e v o l t a g es o f t h e f u l l an d t h e e m pt y e l ec t r o d e$ $A_{e l ec} : e l ec t r o d e a r e a$ $(a s u r f a ce c hann e l w i t h 2 \times 1 0^{- 8} F/c m^{2} an d 5 V g i v es 6.2 \times 1 0^{11} c m^{- 2}, a s ha l l o w b u r i e d c hann e l w i t h 1.5 \times 1 0^{- 8} g i v es 4.7 \times 1 0^{11} an d a d ee p o n e w i t h 0.46 \times 1 0^{- 8} g i v es 1.4 \times 1 0^{11}, a g ain s t 6, 5 an d 1.4 \times 1 0^{11} in t h e so u r ce, co m p u t e d)$
Gate voltage, surface potential and depletion depth	$V_{G} = V_{F B} + ψ_{s} + \frac{q N _{s} + Q _{B}}{C _{o x}}$	$Q_{B} = 2 ε_{s} q N_{A} ψ_{s}$ $W = \frac{2 ε _{s} ψ _{s}}{q N _{A}}$	$V_{F B} : f l a t - ban d v o l t a g e$ $ψ_{s} : s u r f a ce p o t e n t ia l$ $N_{s} : d e n s i t y o f m o bi l e e l ec t r o n s in t h e w e l l$ $Q_{B} : c ha r g e o f t h e d e pl e t i o n r e g i o n$ $ε_{s} : p er mi tt i v i t y o f s i l i co n$ $N_{A} : a cce pt or d e n s i t y$ $W : d e pt h o f t h e d e pl e t i o n r e g i o n, t h e t h r ee f or m s a r e t h e s t an d a r d d e pl e t i o n a pp r o x ima t i o n o f a M O S c a p a c i t or an d w er e n o t co p i e d f r o m a so u r ce$ $(N_{A} = 1 0^{15} c m^{- 3} (a ss u m e d) an d ψ_{s} = 10 V g i v e W = 3.6 μ m, an d 1 0^{14} c m^{- 3} g i v es 11.4 μ m, co m p u t e d)$

Charge Transfer

Name	Formula	Related Formulae	Variables/Constants
Self-induced drift	$\frac{N _{t}}{N _{0}} = \frac{1}{1 + t / τ _{s i}}$	$τ_{s i} = \frac{2 l ^{2} C _{e f f}}{π μ _{n} q N _{0}}$ $t_{ϵ} = τ_{s i} (\frac{1}{ϵ} - 1)$	$N_{t} / N_{0} : f r a c t i o n o f t h e c ha r g e l e f t u n d er t h e e l ec t r o d e a f t er t h e t im e t$ $τ_{s i} : t im e co n s t an t o f t h e d r i f t c a u se d b y t h e f i e l d o f t h e c ha r g e i t se l f$ $l : d i s t an ce o f t h e e l ec t r o d e ce n t r es$ $C_{e f f} : e f f ec t i v e c a p a c i t an ce p er u ni t a r e a$ $μ_{n} : e l ec t r o n m o bi l i t y$ $N_{0} : ini t ia l d e n s i t y o f t h e c ha r g e$ $t_{ϵ} : t im e t o l e a v e a f r a c t i o n ϵ, t h e e x p r ess i o n o f τ_{s i} w a s r eco n s t r u c t e d f r o m an u n c l e a r sc an$ $(a s u r f a ce c hann e l w i t h 2 \times 1 0^{- 8} F/c m^{2}, 500 c m^{2} /Vs an d 6 \times 1 0^{11} c m^{- 2} g i v es 0.27 n s f or l = 0.4 mi l (a ss u m e d, o n e t hi r d o f t h e ce l l o f 1.2 mi l), a g ain s t 0.26 n s in t h e t ab l e; t h e s ha l l o w an d d ee p b u r i e d c hann e l s g i v e 0.12 an d 0.09 n s a g ain s t 0.13 an d 0.09; a r es i d u a l o f 1 0^{- 5} n ee d s 26 μ s, a l l co m p u t e d)$
Thermal diffusion	$τ_{t h} = \frac{4 L ^{2}}{π ^{2} D _{n}}$	$D_{n} = \frac{k _{B} T}{q} μ_{n}$ $\frac{N _{t}}{N _{0}} \propto exp (- \frac{t}{τ _{t h}})$	$τ_{t h} : t im e co n s t an t o f t h e l a s t s t a g e o f t h e d i sc ha r g e o f an e l ec t r o d e w i t h o u t f r in g in g f i e l d$ $L : l e n g t h o f t h e e l ec t r o d e$ $D_{n} : d i f f u s i o n coe f f i c i e n t o f t h e e l ec t r o n s$ $μ_{n} : m o bi l i t y$ $k_{B} : B o l t z mann co n s t an t$ $(L = 25 μ m an d D_{n} = 10 c m^{2} /s g i v e 0.25 μ s, t h e v a l u e o f t h e s l i d es, an d D_{n} = 10 m e an s μ_{n} = 387 c m^{2} /Vs; a r es i d u a l o f 1 0^{- 5} n ee d s ln (1 0^{5}) = 11.5 t im e co n s t an t s, i . e . 2.9 μ s, a l l co m p u t e d)$
Capacity and speed of transfer	$N_{f u l l} \cdot \frac{1}{τ} \propto \frac{μ _{n} ε _{s} W Δ V _{G}^{2}}{l ^{2}}$	$N_{f u l l} \cdot \frac{1}{τ _{s i}} \propto \frac{μ _{n} W C _{e f f} Δ V _{G}^{2}}{l ^{2}}$ $\frac{( N _{f u l l} / τ ) _{2}}{( N _{f u l l} / τ ) _{1}} = (\frac{l _{1}}{l _{2}})^{2}$	$W : w i d t h o f t h e c hann e l$ $l : l e n g t h o f t h e e l ec t r o d e$ $Δ V_{G} : v o l t a g e d i f f er e n ce o f t h e e l ec t r o d es$ $ε_{s} : p er mi tt i v i t y o f s i l i co n, t h e f i r s t f or m i s f or a t r an s f er l imi t e d b y t h e f r in g in g f i e l d, t h e seco n d f or t h e se l f - in d u ce d d r i f t, b o t h w er e r eco n s t r u c t e d a s p r o p or t i o na l i t i es f r o m an u n c l e a r sc an$ $(ha l v in g t h e l e n g t h o f t h e e l ec t r o d e g i v es f o u r t im es t h e p r o d u c t o f c a p a c i t y an d s p ee d in b o t h r e g im es, co m p u t e d; t h e so u r ce f in d s t h e s ha l l o w b u r i e d c hann e l b es t f or se l f - in d u ce d d r i f t an d t h e d ee p b u r i e d c hann e l b es t f or t h e f r in g in g f i e l d)$
Transfer efficiency and loss	$T T E = C T E^{p}$	$C T I = 1 - C T E$ $\frac{Δ Q}{Q} \approx pC T I$	$T T E : t o t a l t r an s f er e f f i c i e n cy$ $C T E : c ha r g e t r an s f er e f f i c i e n cy o f o n e t r an s f er$ $C T I : c ha r g e t r an s f er in e f f i c i e n cy$ $p : n u mb er o f t r an s f er s, Δ Q / Q : f r a c t i o n o f t h e c ha r g e t ha t i s l os t or l e f t b e hin d, t h e so u r ce w r i t es t h e l oss a s n C T I$ $(C T E = 0.99999 an d p = 2048 g i v e 0.980, i . e . a l oss o f 2.03% a g ain s t 2.05% f or p C T I; 0.9999 g i v es 0.815; t h e l a g o f a l in e - binn e d r e g i s t er w i t h C T E = 0.99999 i s 0.00064 an d 0.00128, i . e . n = 64 an d 128 (in f er r e d), a l l co m p u t e d)$

Read-out, Noise and Detection

Name	Formula	Related Formulae	Variables/Constants
Charge-to-voltage conversion	$Δ V_{o u t} = \frac{A _{v} Q}{C _{f d}}$	$S_{v} = \frac{q Δ V _{o u t}}{Q} = \frac{q A _{v}}{C _{f d}}$ $C_{f d} = \frac{q A _{v}}{S _{v}}$	$Δ V_{o u t} : c han g e o f t h e o u tp u t v o l t a g e$ $A_{v} : v o l t a g e g ain o f t h e o u tp u t t r an s i s t or$ $Q : c ha r g e in e l ec t r o n s$ $C_{f d} : c a p a c i t an ce o f t h e f l o a t in g n o d e$ $S_{v} : se n s i t i v i t y in v o l t s p er e l ec t r o n$ $(S_{v} = 2.0 μ V / e^{-} an d C_{f d} = 44 f F im pl y A_{v} = 0.55, co m p u t e d, w hi c h t h e n o t e d oes n o t p r in t; q / C_{f d} a t t h e n o d e i s 3.6 μ V / e^{-}, co m p u t e d)$
Saturation voltage and dynamic range	$V_{s a t} = N_{F W} S_{v}$	$D R = \frac{N _{F W}}{σ _{r}}$ $D R_{d B} = 20 lo g_{10} (\frac{N _{F W}}{σ _{r}})$	$V_{s a t} : o u tp u t v o l t a g e a t s a t u r a t i o n$ $N_{F W} : f u l l w e l l in e l ec t r o n s$ $S_{v} : se n s i t i v i t y$ $D R : d y nami c r an g e$ $σ_{r} : r e a d n o i se in e l ec t r o n s r m s$ $(300 k e^{-} w i t h 6 an d 8 e^{-} r m s g i v e 50000 an d 37500, i . e . 94.0 an d 91.5 d B; 300 k e^{-} a t 1.7 an d 2.2 μ V / e^{-} g i v e 0.51 an d 0.66 V, co m p u t e d; t h e r e a d n o i se i s in f er r e d f r o m t h e d y nami c r an g e)$
Noise of a pixel	$N_{t} = N_{f}^{2} + N_{s}^{2} + N_{d}^{2} + N_{r}^{2}$	$N_{s} = S$ $N_{d} = I_{d} t$	$N_{t} : t o t a l n o i se$ $N_{f} : f i x e d - p a tt er n n o i se (z er o f or a s in g l e p i x e l)$ $N_{s} : s h o t n o i se o f t h e s i g na l S in e l ec t r o n s$ $N_{d} : s h o t n o i se o f t h e d a r k c ha r g e$ $N_{r} : r e a d n o i se$ $I_{d} : d a r k c u r r e n t in e l ec t r o n s p er seco n d$ $t : e x p os u r e$ $(10000 e^{-} g i v e N_{s} = 100 e^{-} r m s; w i t h N_{r} = 6 e^{-} t h e t o t a l i s 100.2 e^{-} an d t h e r e a d n o i se i s n e g l i g ib l e, co m p u t e d; t h e r e a d n o i se o f a C C D i s a f e w t o ab o u t 10 e^{-} r m s)$
Dark current and temperature	$I_{d} (T) = I_{d} (T_{0}) 2^{(T - T_{0}) /Δ T}$	$N_{d a r k} = I_{d} t$ $t_{e q} = \frac{σ _{r}^{2}}{I _{d}}$	$I_{d} : d a r k c u r r e n t in e l ec t r o n s p er p i x e l an d seco n d$ $T_{0} : r e f er e n ce t e m p er a t u r e$ $Δ T : t e m p er a t u r e s t e p t ha t ha l v es t h e c u r r e n t (5 t o 7^{\circ} C in t h e n o t e, 7 K in t h e s l i d es)$ $N_{d a r k} : d a r k e l ec t r o n s$ $t_{e q} : t im e a t w hi c h t h e d a r k s h o t n o i se e q u a l s t h e r e a d n o i se σ_{r}$ $(20 e^{-} / pixel/s a t 0^{\circ} C b eco m e 2.8 (7^{\circ} C s t e p) t o 1.25 (5^{\circ} C s t e p) a t - 20^{\circ} C, an d 640 t o 238 a t 25^{\circ} C; t_{e q} = 1.8 s a t 0^{\circ} C f or 6 e^{-}, a l l co m p u t e d)$
Quantum efficiency of a thinned device	$η = (1 - R) (1 - e^{- α d})$	$N_{e} = η Φ t_{in t}$ $η_{e x t} \leq 1 - R$	$η : q u an t u m e f f i c i e n cy$ $R : r e f l ec t an ce o f t h e s u r f a ce$ $α : ab sor pt i o n coe f f i c i e n t$ $d : t hi c k n ess o f t h e ab sor bin g s i l i co n$ $N_{e} : e l ec t r o n s co l l ec t e d$ $Φ : p h o t o n f l ux o n t h e p i x e l$ $t_{in t} : in t e g r a t i o n t im e, t h e f i r s t f or m i s t h e s t an d a r d o n e an d w a s n o t co p i e d f r o m a so u r ce$ $(R = 0.70 a t 250 nm w i t h o u t co a t in g l imi t s η t o 0.30, co m p u t e d; f r o n t - i l l u mina t e d d e v i ces r e a c h ab o u t 40% a t 600 t o 700 nm, ba c k - t hinn e d o n es 50% t o 90%; t h e t hinn e d s i l i co n i s ab o u t 10 μ m (s l i d es) or 20 μ m (n o t e))$
Photon counting of X-rays	$N_{e} = \frac{E}{w}, w = 3.65 eV$	$σ_{N} = F N_{e}$ $Δ E_{F W H M} = 2.355 w F N_{e}$	$E : e n er g y o f t h e p h o t o n$ $w : m e an e n er g y p er e l ec t r o n - h o l e p ai r in s i l i co n$ $F : F an o f a c t or$ $σ_{N} : s t a t i s t i c a l s p r e a d o f t h e n u mb er o f e l ec t r o n s$ $Δ E_{F W H M} : w i d t h o f t h e l in e a t ha l f ma x im u m$ $(5.9 k e V g i v e 1616 e^{-}, t h e n o t e p r in t s 1620; F = 0.1 (a ss u m e d) g i v es 109 e V, t h e p h y s i c a l l imi t o f t h e n o t e; t h e 140 e V a c hi e v e d l e a v e 37 e V r m s or 10 e^{-} r m s o f o t h er n o i se, co m p u t e d)$

Gain, Binning and Uniformity

Name	Formula	Related Formulae	Variables/Constants
Electron-multiplying gain	$G = (1 + g)^{N}$	$σ_{e f f} = \frac{σ _{r}}{G}$ $S_{x} = \frac{σ _{r}^{2}}{F _{E M}^{2} - 1}$	$G : m u l t i pl i c a t i o n g ain$ $g : p r o babi l i t y o f a seco n d a r y e l ec t r o n p er s t a g e$ $N : n u mb er o f s t a g es o f t h e g ain r e g i s t er$ $σ_{e f f} : r e a d n o i se r e f er r e d t o t h e in p u t$ $F_{E M} : e x cess n o i se f a c t or (t h e s h o t n o i se i s m u l t i pl i e d b y 1.41 in t h e so u r ce)$ $S_{x} : s i g na l in e l ec t r o n s a t w hi c h t h e m u l t i pl i e d an d t h e co n v e n t i o na l o u tp u t ha v e e q u a l n o i se, o b t ain e d b y se tt in g F_{E M}^{2} S = S + σ_{r}^{2}$ $(g = 0.01 t o 0.016 an d N = 590 g i v e G = 354 t o 11700, co m p u t e d, a g ain s t m or e t han 1000 in t h e so u r ce; a r e a d n o i se o f 60 e^{-} b eco m es 6, 0.6 an d 0.06 e^{-} f or G = 10, 100 an d 1000; F_{E M}^{2} = 2 g i v es S_{x} = σ_{r}^{2})$
Binning	$S_{bin} = \sum_{i = 1}^{n} S_{i}$	$S N R_{bin} \leq n \cdot S N R_{1} (shot limit)$ $S N R_{bin} \approx n \cdot S N R_{1} (read limit)$	$S_{bin} : s i g na l o f t h e s u mm e d c ha r g e p a c k e t$ $S_{i} : c ha r g e o f o n e p i x e l$ $n : n u mb er o f p i x e l s t ha t a r e s u mm e d o n t h e c hi p$ $S N R_{1} : s i g na l - t o - n o i se r a t i o o f o n e p i x e l, t h e g ain h o l d s i f t h e c ha r g es a r e a dd e d b e f or e t h e r e a d - o u t an d t h e r e a d n o i se i s co u n t e d o n ce$ $(a binnin g o f 2 \times 2 t u r n s 1024 \times 1024 in t o 512 \times 512 an d s u m s n = 4 p i x e l s, i . e . a g ain o f 4 in t h e r e a d - n o i se l imi t an d 2 in t h e s h o t - n o i se l imi t, co m p u t e d; t h e s u mmin g w e l l m u s t h o l d t h e s u m)$
Uniformity and linearity	$P R N U (%) = \frac{N}{S} \times 100$	$L R (%) = (1 - \frac{S / T}{S _{m} / T _{m}}) \times 100$ $P R N U_{pp} \approx 3 \cdot P R N U_{r m s}$	$P R N U : p h o t o - r es p o n se n o n - u ni f or mi t y$ $N : r m s v a r ia t i o n o f t h e s i g na l am o n g p i x e l s$ $S : s i g na l n e a r ha l f o f t h e f u l l w e l l$ $L R : r es i d u a l o f t h e l in e a r i t y (t h e l a y o u t o f t h e so u r ce i s u n c l e a r, t h e f or m i s r eco n s t r u c t e d)$ $T : e x p os u r e t im e, t h e in d e x m ma r k s t h e r e f er e n ce p o in t$ $(t h e t y p i c a l v a l u e i s ab o u t 1% r m s or \pm 3% p e ak t o p e ak; t h e r a t i o 3 f or p e ak t o p e ak i s m y r e a d in g)$

Important Questions

Ques: Why does self-induced drift alone not empty a well, and what carries the last charge?

With only the field of the charge itself, the remaining fraction is $1/ (1 + t / τ_{s i})$ and falls as a power of time. The table gives $τ_{s i} = 0.26$ ns for a surface channel, but a residual of $1 0^{- 5}$ needs $26 μ s$ (computed), far more than a transfer. Thermal diffusion decays exponentially, $τ_{t h} = 4 L^{2} / (π^{2} D_{n})$ , and 11.5 time constants, 2.9 μs for 25 μm (computed), give $1 0^{- 5}$ . The fringing fields of the neighbouring gates work in the whole time and, in some designs, dominate; the thesis says so for the final stage. The expression of $τ_{s i}$ was reconstructed from a scan that was unclear, and which mechanism dominates depends on the electrode length and the charge. Reference: Barbe, Charge-Coupled Devices: Transfer Speed vs. Charge-Handling Capacity, International Conference on the Application of Charge-Coupled Devices (1974); Daimon-Hagihara, doctoral thesis on charge-coupled devices, Caltech (1975); Keller, Astronomical Instrumentation 2010, Lecture 12: Detectors, Leiden University.

Ques: Are the transfer time constants of the sources consistent?

Not directly. The slides print $τ_{S i} = 0.14 μ s$ for self-induced drift and 0.25 μs for thermal diffusion with $D_{n} \approx 10 c m^{2} /s$ ; the table of the 1974 paper gives 0.26 ns, 0.13 ns and 0.09 ns for self-induced drift. Thermal diffusion agrees: $4 L^{2} / (π^{2} D_{n}) = 0.25 μ s$ for $L = 25 μ m$ , the spacing of the slides (computed). The nanosecond values follow from the formula for $l = 0.4$ mil, one third of the 1.2 mil cell (assumed), but give ten times too much for $l = 1.2$ mil (computed). The slides’ 0.14 μs may be a different device, charge or unit; the scan does not say, so the two are not reconciled. Reference: Keller, Astronomical Instrumentation 2010, Lecture 12: Detectors, Leiden University; Barbe, Charge-Coupled Devices: Transfer Speed vs. Charge-Handling Capacity, International Conference on the Application of Charge-Coupled Devices (1974).

Ques: What do the sources say about surface-channel against buried-channel devices?

The table of Barbe gives for the surface, shallow buried and deep buried channels the capacities 6, 5 and $1.4 \times 1 0^{11} c m^{- 2}$ , the mobilities 500, 1000 and $1500 c m^{2} /Vs$ and the fringing-field times 3.8, 1.4 and 0.26 ns, so that the buried channel has less charge and is faster (the exponents were uncertain in the scan). The NASA report found buried-channel devices free of interface-state trapping up to $5 \times 1 0^{5}$ rad(Si), while surface-channel devices degraded below $1 0^{5}$ rad(Si). The thesis states that diffusion limits surface-channel devices to clock rates below 10 MHz, and that interface states of $1 0^{11} states/c m^{2} eV$ can require regeneration after as few as about 5 to 100 transfers. Reference: Barbe, Charge-Coupled Devices: Transfer Speed vs. Charge-Handling Capacity, International Conference on the Application of Charge-Coupled Devices (1974); NASA Technical Reports Server 19750020771, Radiation effects on charge-coupled devices; Daimon-Hagihara, doctoral thesis on charge-coupled devices, Caltech (1975).

Ques: Is the charge under a gate the full well of a real pixel?

No. For $10 \times 20 μ m^{2}$ , 0.1 μm of oxide and 10 V the slides give $3.6 \times 1 0^{6}$ electrons; with $ε_{r} = 3.9$ it is $4.3 \times 1 0^{6}$ (computed), which implies $ε_{r} = 3.39$ in the source. This is 21550 electrons per $μ m^{2}$ (computed). The capacity of 1974 is 6000 per $μ m^{2}$ for 5 V (computed), and the vertical full well of the Hamamatsu devices, 300000 electrons for a pixel of 24 μm if the unit is micrometres, as printed without it, is 520 per $μ m^{2}$ (computed). The real well is set by the voltage difference of the phases, the channel stops and the horizontal register, not by the total oxide charge. Reference: Keller, Astronomical Instrumentation 2010, Lecture 12: Detectors, Leiden University; Barbe, Charge-Coupled Devices: Transfer Speed vs. Charge-Handling Capacity, International Conference on the Application of Charge-Coupled Devices (1974); Hamamatsu, Technical Information SD-25: Characteristics and Use of FFT-CCD Area Image Sensor (KMPD9002E06).

Ques: What does a charge transfer efficiency of 0.99999 mean for a large device?

The total efficiency is $C T E^{p}$ : for $p = 2048$ transfers, 0.99999 gives 0.980 and 0.9999 gives 0.815 (computed), so a tenfold worse efficiency per transfer loses 19% instead of 2%. The first-order loss $pC T I = 2.05%$ (computed) is close to the exact 2.03%. Hamamatsu’s table of line-binning lag gives 0.00064 and 0.00128 for 0.99999, which is 64 and 128 transfers of $1 0^{- 5}$ (inferred). The note quotes 0.99999 as typical and does not say whether the efficiency is per gate or per pixel, which differ by the number of phases, so the totals above need that definition. Slides from Leiden also call 0.99999 typical of a good device. Reference: Hamamatsu, Technical Information SD-25: Characteristics and Use of FFT-CCD Area Image Sensor (KMPD9002E06); Keller, Astronomical Instrumentation 2010, Lecture 12: Detectors, Leiden University.

Ques: Are the sensitivity, the full well and the dynamic range of the Hamamatsu note consistent?

The note gives $S_{v} = 2.0 μ V / e^{-}$ and $C_{f d} = 44$ fF for one series, and $S_{v} = q A_{v} / C_{f d}$ then requires $A_{v} = 0.55$ (computed), a plausible gain of a source follower that the note does not print. Table 2-2 prints the conversion efficiency in ‘V/e-’ (1.7 and 2.2), evidently μV, which gives a saturation of 0.51 V and 0.66 V for 300000 electrons (computed). The dynamic ranges 50000 and 37500 match read noises of 6 and 8 electrons (94.0 and $91.5 dB$ , computed); the noise row was garbled, so these values are inferred. The pixel size also lacks its unit. Reference: Hamamatsu, Technical Information SD-25: Characteristics and Use of FFT-CCD Area Image Sensor (KMPD9002E06).

Ques: How strongly does cooling reduce the dark current, and what does it cost?

The note says the dark current halves for every 5 to 7 °C of cooling, the slides say 7 K. For 20 electrons per pixel and second at 0 °C, cooling to -20 °C leaves 2.8 (7 °C step) or 1.25 (5 °C step) (computed), and the dark shot noise equals a read noise of 6 electrons after 1.8 s at 0 °C (computed). The values of the data sheet are measured in MPP mode, and the low-dark-current mode holds about half of the full well. The second device has 200 electrons per pixel and second at 0 °C, ten times more, and the note does not give the reason. Cooling also lowers the response above 800 nm, 0.75% per °C at 1000 nm. Reference: Hamamatsu, Technical Information SD-25: Characteristics and Use of FFT-CCD Area Image Sensor (KMPD9002E06); Keller, Astronomical Instrumentation 2010, Lecture 12: Detectors, Leiden University.

Ques: What are the quantum efficiencies of front and back illumination?

Front illumination passes the light through the gates (polysilicon of 0.5 μm in the slides), and gives a maximum of about 40% at 600 nm to 700 nm and almost nothing below 400 nm. The slides state 50% rising to about 90% for back illumination with a coating, and 70% reflection loss at 250 nm without it, so $η \leq 0.30$ (computed). The note cites a paper with more than 90% but gives no figure of its own. Thinning is to about 10 μm in the slides and about 20 μm in the note; both are as printed. Deep-depletion X-ray devices use 30 μm to 50 μm. The absorption of the gate material is not given in the sources and is a background explanation. Reference: Hamamatsu, Technical Information SD-25: Characteristics and Use of FFT-CCD Area Image Sensor (KMPD9002E06); Keller, Astronomical Instrumentation 2010, Lecture 12: Detectors, Leiden University.

Ques: How good is the energy resolution for X-rays, and what limits it?

A 5.9 keV photon gives $5900/3.65 = 1616$ electrons (computed; the note prints 1620). The statistical limit $2.355 w F N$ gives 109 eV for $F = 0.1$ , which reproduces the 109 eV of the note; the Fano factor is not printed and the value is my assumption. The measured 140 eV at -70 °C means $14 0^{2} - 10 9^{2} /2.355 = 37$ eV rms of other noise, or 10 electrons (computed), larger than the 6 electrons of read noise that the dynamic range implies. The note does not give the composition of the rest, which could include incomplete charge collection or losses in transfer; that is not stated. Reference: Hamamatsu, Technical Information SD-25: Characteristics and Use of FFT-CCD Area Image Sensor (KMPD9002E06).

Ques: When does the electron-multiplying gain help, and are the numbers consistent?

The gain is $(1 + g)^{N}$ with $g = 0.01$ to 0.016; for 590 stages that is 354 to 11700 (computed), consistent with ‘more than 1000’. The read noise is divided by the gain (60 electrons become 0.06 at 1000), but the shot noise is multiplied by 1.41, so that the multiplied output is better only for $S < σ_{r}^{2} / (F^{2} - 1)$ , i.e. $S < σ_{r}^{2}$ (derived here). The crossover of 42 photons per pixel at 1 MHz means $σ_{r} = 6.5$ electrons, and 2900 photons at 10 MHz means 54 electrons (both computed, assuming unit quantum efficiency). The source of the 1.41 is a vendor article; the statistics paper has no excess noise factor. Reference: Teledyne Vision Solutions, EMCCDs: The Basics; Andor (Oxford Instruments), Understanding Noise Sources in EMCCDs; Charge-coupled-device multiplication-gain statistics (arXiv:2405.17622).

Ques: Does binning improve the signal-to-noise ratio, and what limits it?

Summing $n$ pixels in the register before the read-out multiplies the signal by $n$ while the read noise is counted once, so the ratio rises by up to $n$ when the read noise dominates and by $n$ when the shot noise does (computed); for 2×2 binning, 4 and 2. The sum has to fit the wells: Hamamatsu designs the horizontal well larger than the vertical one and the summing well larger again, but gives no numbers, so a sum of four packets of 300000 electrons cannot be assumed to fit. The dark charge and the lag also add. Hamamatsu’s example turns 1024×1024 pixels into 512×512. Reference: Hamamatsu, Technical Information SD-25: Characteristics and Use of FFT-CCD Area Image Sensor (KMPD9002E06).

Quartz 5

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CCD Pixel (MOS Photogate)