Lecture Notes 1 Silicon Photodetectors

• Light Intensity and Photon Flux • Photogeneration in Silicon • Photodiode ◦ Basic operation ◦ Photocurrent derivation ◦ Quantum eﬃciency ◦ Dark current • Direct Integration • Photogate • Appendices ◦ Appendix I: Derivation of Continuity Equation ◦ Appendix II: Depletion Width for PN Junction ◦ Appendix III: MOS Capacitor ◦ Appendix IV: Useful Data

EE 392B: Silicon Photodetectors 1-1 PSfrag replacements

Preliminaries

• Photodetector is the front end of the image sensor. It converts light incident on it into photocurrent that is (hopefully) proportional to its intensity • Conversion is done in two steps: ◦ Incident photons generate e-h pairs in the detector (e.g., silicon) ◦ Some of the generated carriers are converted into photocurrent • Photocurrents are typically very small (10s to 100s of fA) ◦ Direct measurement is difficult ADC ◦ Usually integrated into charge on a capacitor and then converted to Gain voltage before readout DN Photonflux Current density Charge Voltage Quantum Efficiency Integration Conversion space/time Gain

ph/cm2·sec A/cm2 Col V

EE 392B: Silicon Photodetectors 1-2 Visible Light

• We are mainly concerned with visible light image sensors

• Recall that the energy of a photon is given by Eph = hc/λ, where h = 4.135 × 10−15eV.sec is Planck’s constant, c = 3 × 108m/s is the speed of light, and λ is the wavelength • Visible light wavelengths (λ) range from 400 nm to 700 nm

Violet: 400 nm (Eph = 3.1 eV)

Blue: 450 nm (Eph = 2.76 eV)

Cyan: 500 nm (Eph = 2.48 eV)

Green: 550 nm (Eph = 2.27 eV)

Yellow: 600 nm (Eph = 2.08 eV)

Red: 700 nm (Eph = 1.77 eV)

Infrared: > 800 nm (Eph < 1.55 eV)

EE 392B: Silicon Photodetectors 1-3 • The amount of light incident on an image sensor surface depends on ◦ The light source ◦ The surface reflectance of the object being imaged ◦ The imaging optics used • Different visible light sources, e.g., daylight (D65), incandescent, halogen, fluorescent have different power spectra

EE 392B: Silicon Photodetectors 1-4 Radiometry and Photometry

• Two ways to measure the intensity of light incident on a surface: ◦ Radiometry measures it as irradiance E W/m2 2 ◦ Photometry measures it as illuminance Eν in lux or lumens/m , which 1 2 is deﬁned as 683W/m at λ = 555nm • Illuminance takes into account the sensitivity of the human eye to diﬀerent wavelengths; λ = 555nm is the wavelength for which the human eye is most sensitive and the value for which the photopic vision curve is normalized

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EE 392B: Silicon Photodetectors 1-5 • Translating from irradiance to illuminance: Denote the vision photopic curve as Y (λ) and the irradinace density E(λ) W/m2.nm, then the illuminance is given by 700 Eν = 683 Y (λ)E(λ) dλ lux Z400

EE 392B: Silicon Photodetectors 1-6 Photon Flux

2 • Photon ﬂux F0 is the number of photons per cm .sec incident on a surface

• Using the photon energy Eph(λ), we can readily translate irradiance density E(λ) into photon ﬂux 700 10−4E(λ) F = dλ photons/cm2.sec 0 E (λ) Z400 ph • Translating from illuminance to photon ﬂux: −20 ◦ At λ = 555nm, Eph = 35.8 × 10 Joule; thus 1 lux corresponds to 16 11 2 F0 = 10 /683 × 35.8 = 4.09 × 10 photons/cm ·sec, or, 133 photons strike a 1µm × 1µm surface per 1/30 sec ◦ A typical light source (e.g., D65) has a wide range of wavelengths 12 2 and 1 lux roughly corresponds to F0 ≈ 10 photons/cm .sec, or, 333 photons strike a 1µm× 1µm surface per 1/30 sec

EE 392B: Silicon Photodetectors 1-7 • Photon ﬂux values encountered vary over a very wide range:

4 17 clear sky ≈ 10 Lux, or F0 = 10 13 room light ≈ 10 Lux, or F0 = 10 11 full moon ≈ 0.1 Lux, or F0 = 10 −4 8 moonless night ≈ 10 Lux, or F0 = 10

EE 392B: Silicon Photodetectors 1-8 Photocharge Generation in Semiconductors

• Incident photon energy must be > band gap energy (Eg) to generate an electron-hole pair

◦ Electrons go to the conduction band (EC)

◦ Holes go to the valence band (EV ) • Energy band diagram of silicon:

PSfrag replacements Eg = 1.124eV

• Coincidentally (and luckily) photons in the visible range have enough energy to generate e-h pairs ◦ No photon can generate more than one e-h pair • Energy gap of other semiconductors: Ge (0.66 eV), GaAs (1.42 eV)

EE 392B: Silicon Photodetectors 1-9 Photocharge Generation Rate in Silicon

2 • Assume a monochromatic photon ﬂux F0 photons/cm .sec at wavelength λ incident at the surface (i.e., x = 0) of silicon photon ﬂux 0

PSfrag replacements silicon

e-h pair

• The photon absorption in a material is governed by its absorption coeﬃcient α(λ) cm−1 • Let F (x) be the photon ﬂux at depth x, then the number of photons absorbed per second between x and x + ∆x is given by F (x) − F (x + ∆x) ≈ αF (x)∆x,

EE 392B: Silicon Photodetectors 1-10 We can write this equation in the limit as dF (x) = −αF (x) dx Solving we obtain

−αx 2 F (x) = F0e photons/cm .sec

Thus the rate of e-h pairs generated at x is d G(x) = (F − F (x)) = αF e−αx e-h pair/cm3.sec dx 0 0

EE 392B: Silicon Photodetectors 1-11 Absorption Coeﬃcient of Silicon

7 10

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3 10 rption Abso

2 10 PSfrag replacements

1 10 200 300 400 500 600 700 800 900 1000 Wavelength [nm]

E. Palik, ”Handbook of Optical Constant of Solids,” Academic, New York, 1985

EE 392B: Silicon Photodetectors 1-12 Absorption Length of Visible Light in Silicon

EE 392B: Silicon Photodetectors 1-13 Light Absorption in a Silicon Slab

EE 392B: Silicon Photodetectors 1-14 Comments

• F (x) and G(x) are average values assuming a large ensemble of photons (approaching continuum values) ◦ The photon absorption process is actually discrete and random • Note that: ◦ 99% of blue light is absorbed within 0.6 µm ◦ 99% of red light is absorbed within 16.6 µm • These depths (surprisingly) are quite consistent with the junction and well depths of a CMOS process • But, this is not the whole story . . . ◦ Photocharge needs to be collected and converted into electrical signal

EE 392B: Silicon Photodetectors 1-15 Photodetectors in Silicon

• A photodetector is used to convert the absorbed photon ﬂux into photocurrent • There are three types of photodetectors used, photodiode, which is a reverse biased pn junction, photogate, and pinned diode • In a standard CMOS process there are three types of photodiodes available ◦ nwell/psub ◦ n+/psub ◦ p+/nwell and two types of photogates ◦ nMOS transistor gate to drain ◦ pMOS transistor gate to drain

EE 392B: Silicon Photodetectors 1-16 • In this lecture notes we discuss the photodiode and photogate operation. The pinned diode will be discussed in the following lecture notes

EE 392B: Silicon Photodetectors 1-17 Photodiode Operation

• Assume the depletion approximation of a reverse biased pn junction

PSfrag replacements photon ﬂux quasi-neutral n-type n-region

vD > 0 depletion region iph

quasi-neutral p-region p-type

• The photocurrent, iph, is the sum of three components: ◦ Current due to electrons generated in the depletion (space charge) sc region, iph p ◦ Current due to holes generated in the quasi-neutral n-region, iph n ◦ Current due to electrons generated in the quasi-neutral p-region, iph

EE 392B: Silicon Photodetectors 1-18 • Most electrons generated in the depletion region are converted into current by strong electric field • Carriers generated in the quasi-neutral regions need to diffuse to the depletion region to be collected ◦ Some charge is lost through recombination ◦ The diffusion length determines the fraction of charge that is not recombined

EE 392B: Silicon Photodetectors 1-19 Photocurrent Derivation

• Assumptions ◦ Abrupt pn junction ◦ Depletion approximation ◦ Low level injection, i.e., ﬂux induced carrier densities << majority carrier densities ◦ Short base region approximation, i.e., junction depths << diﬀusion lengths. This is is quite reasonable for advanced CMOS processes • Our results are inaccurate but will help us understand the dependence of

iph on various device parameters

References:

• F. Van de Wiele, “Photodiode Quantum Eﬃciency,” in P. G. Jespers, F. van de Wiele, M. H. White eds. “Solid State Imaging,” p. 47, Noordhoﬀ (1976).

• J.C. Tandon, D.J. Roulston, S.G. Chamberlain, Solid State Electronics, vol. 15, pp. 669 – 685, (1972).

• R.W. Brown, S.G. Chamberlain, Physica Status Solidi (a), vol. 20, pp. 675 – 685 (1973)

EE 392B: Silicon Photodetectors 1-20 PSfrag replacements

• Consider the depletion approximation for a reverse biased pn junction photon ﬂux v 0 quasi-neutrali n-type n-region x1

depletion region

x2 quasi-neutral p-region p-type x3 x 2 • Assume a monochromatic photon ﬂux F0 photon/cm ·sec incident at the surface (x = 0), the e-h generation rate at depth x is given by −αx 3 G(x) = αF0e ph/cm .sec • Assuming all generated electrons in the space charge region are collected, the current density due to generation in the space charge region is

sc −αx1 −αx2 2 jph = qF0(e − e ) A/cm , where q = 1.6 × 10−19Col is the electron charge

EE 392B: Silicon Photodetectors 1-21 • The current density due to generation in n-type quasi-neutral region, which is diﬀusion current (since there is no ﬁeld in this region), is given by 0 p ∂pn(x) jph = −qDp ∂x x=x1

0 where pn is the photogenerated minority carrier (hole) density, and Dp is the diffusion constant of holes (in cm2/sec) • To find the current density due to generation in the n-type quasi-neutral 0 region, we first need to find pn(x). This can be done by solving the continuity equation (see derivation in Appendix I) with current density p substituted for by the jph expression above 0 0 ∂p ∂2p n = D n + G(x) − R(x), ∂t p ∂x2 where G(x) is the hole photogeneration rate and R(x) is their recombination rate By the short base assumption, the recombination rate is negligible and we set R(x) = 0

EE 392B: Silicon Photodetectors 1-22 Now, assuming steady state, the continuity equation simpliﬁes to 0 d2p 0 = D n + G(x), p dx2 which has solution of the form

0 F0 −αx pn(x) = a + bx − e αDp To ﬁnd a and b, we assume that: 0 ◦ at x = 0, we have an ohmic contact, which gives pn(0) = 0 0 ◦ at x = x1, i.e., at the edge of the depletion region, pn(x1) = 0 Substituting, we obtain

0 F x 0 −αx1 −αx pn(x) = (1 − (1 − e ) − e ) αDp x1

EE 392B: Silicon Photodetectors 1-23 0 0 pn(x) quasi-neutral n-region

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x1 depletion region

We can now ﬁnd the diﬀusion current density 0 p ∂pn(x) jph = −qDp ∂x x=x1 qF 0 − −αx1 = (1 (αx1 + 1)e ) αx1

EE 392B: Silicon Photodetectors 1-24 • The current density due to generation in the p-type can be similarly found, and we obtain qF n 0 −αx2 −αx3 jph = ((α(x3 − x2) − 1)e + e ) α(x3 − x2) Here we assumed that an ohmic contact at x = x3, which is quite arbitrary (you will derive it with more reasonable assumptions in HW1) • The total photogenerated current density is thus given by qF (1 − e−αx1) (e−αx2 − e−αx3) j = 0 − A/cm2 ph α x (x − x ) 1 3 2 • To ﬁnd x1 and x2, we can use the simplifying assumptions to derive the depletion region width (see Appendix II), and we obtain 2 1 1 x − x = s (v + φ + φ ) + , 2 1 q D n p N N s a d and use the fact that xn/xp = Na/Nd, where −13 s = 10.45 × 10 F/cm is the permittivity of Si −3 Nd and Na are the donor and acceptor densities in cm φn and φp are the potentials in the n and p regions

EE 392B: Silicon Photodetectors 1-25 Example

• Consider the nwell/psub diode in the generic 0.5µm CMOS process 12 described in Handout 4 with vD = 2V and F0 = 4.09 × 10 photons/cm2·sec at λ = 555nm (room light), ﬁnd the photocurrent density components

• Using the depletion equation, we ﬁnd that x1 = 2 − 0.176 = 1.824µm and x2 = 3.76µm The photocurrent density components are

sc 2 jph = 120 nA/cm p 2 jph = 192 nA/cm n 2 jph = 28 nA/cm 2 Thus the total photocurrent density jph = 340 nA/cm 2 So, for a photodiode of area 30µ , iph = 102fA

EE 392B: Silicon Photodetectors 1-26 Factors Aﬀecting Photocurrent

• iph is linear in F0, i.e., proportional to illumination

• iph is nonlinear in α and λ

• iph increases as x1 decreases and x2 increases, i.e., as the depletion width (x2 − x1) increases, which can be achieved by a combination of: ◦ shallow pn junction, ◦ low doping, and/or ◦ by increasing reverse bias voltage • Depletion region width, however, increases slowly with reverse bias voltage (high reverse bias voltage also increases dark current as we shall soon see)

EE 392B: Silicon Photodetectors 1-27 Quantum Eﬃciency

• Quantum efficiency QE(λ) is the fraction of photon flux that contributes to photocurrent as a function of the wavelength λ • Using our derived photocurrent equation, we obtain 1 (1 − e−αx1) (e−αx2 − e−αx3) QE(λ) = − electrons/photons α x (x − x ) 1 3 2 • This, in addition to being inaccurate due to the approximations we made, ignores: ◦ reflection at the surface of the chip ◦ reflections and absorptions in layers above the photodetector

◦ variation of jph over the photodetector area (edge eﬀects)

EE 392B: Silicon Photodetectors 1-28 Example

Consider the nwell/psub diode with vD = 2V

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EE 392B: Silicon Photodetectors 1-29 Dark Current

• There are sources other than photon ﬂux that lead to current in the photodetector – the sum of these currents is called ”dark current” • It is called ”dark current” because it is the the photodetector current with no illumination present (in the dark) • Dark current is bad. It limits the image sensor performance: ◦ Introduces unavoidable shot noise ◦ Can vary substantially over the image sensor array causing Dark Signal Non-uniformity (DSNU) ◦ Reduces signal swing

EE 392B: Silicon Photodetectors 1-30 Sources of Dark Current

EE 392B: Silicon Photodetectors 1-31 Dark Current Contributions

• Dark current due to thermal generation (dominated by traps with energy in the middle of the bandgap) – can be calculated ◦ Generation current is exponentially dependent on temperature • Dark current due to interface defects, material (crystal) defect, and metal contamination, such as: ◦ Edge of the STI or LOCOS

◦ Si/SiO2 interface ◦ Edge of junctions (end-of-range implant damage) These sources are diﬃcult to model and can only be experimentally measured ◦ Highly fabrication process dependent ◦ Mitigated by careful pixel layout and dopant proﬁling (more on this later)

EE 392B: Silicon Photodetectors 1-32 Calculation of Generation Dark Current

• Thermally generated dark current density due to bulk defects consists of three components: p ◦ Current due to carrier diffusion from the quasi-neutral regions, jdc n and jdc (vD > 0) sc ◦ Current due to generation in the space charge region, jdc • We first analyze the first two components in the same way we analyzed p n jph and jph, assuming abrupt pn junction and short base approximation let

pn(x) be the thermally generated minority carriers in the n-type quasi-neutral region Ignoring recombination and assuming steady state, the continuity equation reduces to d2p (x) 0 = D n , p dx2 with the general solution

pn(x) = ax + b

EE 392B: Silicon Photodetectors 1-33 PSfrag replacements photon Assuming ohmic contact at x = 0 we get pn(0) = pn0, the minority carrier concentration at thermal equilibrium, and assuming no free carriers at the

edge of the depletionv region, we have pn(x1) = 0 Thus i x p (x) = p 1 − n n0 x 1

p 0 0 n p n-type n quasi-neutral n-region x1 x xn depletion region xp

x2 quasi-neutral p-region p-type x3 np np0

EE 392B: Silicon Photodetectors 1-34 The (diﬀusion) current density is given by

p dpn(x) jdc = −qDp dx x=x1 2 pn0 ni = qDp = qD p x1 Ndx1 Similarly 2 n ni jdc = qDn Na(x3 − x2) • Derivation of the current due to generation in the space charge region, sc jdc, is more complicated (see below). It yields

qn x x jsc ≈ i n + p , dc 2 τ n τ p 0 0 n p where τ0 and τ0 are the excess carrier lifetimes in the n and p type material, respectively

EE 392B: Silicon Photodetectors 1-35 In practice the depletion region is much wider on one side. In this case we can express the current density as

sc qnixd jdc ≈ , 2τo where xd = xn + xp is the depletion width and τo = τn = τp is the excess carrier lifetime of the wider side

• Example: again consider the nwell/psub diode with vD = 2V, at room temperature

sc 2 jdc = 3.977 nA/cm p n 2 jdc + jdc = 1.9611 nA/cm 2 jdc = 5.938 nA/cm

2 So, for a photodiode of area 30µm , idc ≈ 1.78fA

EE 392B: Silicon Photodetectors 1-36 Factors Aﬀecting idc

• idc increases dramatically with temperature T , since it increases with the E 1.5 − g intrinsic carrier concentration ni, which is proportional to T e 2kT sc • idc is the dominant component of idc ◦ 1 it increases with doping concentration, since τ is proportional to N ◦ it decreases with decrease in depletion width, thus reducing reverse

bias voltage reduces idc (but also reduces iph!) sc ◦ the calculated idc is only valid for low electric ﬁeld, at higher electric ﬁelds (which occurs in the shallower and more highly doped junctions sc of advanced processes), idc increases much faster with reverse bias voltage

EE 392B: Silicon Photodetectors 1-37 Generation-Recombination in Depletion Region

• Here, we derive the generation-recombination current in the depletion region of a reverse biased pn-junction • The analysis is referred to as the Shockley, Read, Hall (SRH) model • Salient features of the SRH model: ◦ Generation and recombination of carriers occur through localized states (recombination centers) with energy within the bandgap ◦ Overall population of the recombination center is fairly constant ◦ Recombination centers quickly capture the majority carriers, but have to wait for the arrival of a minority carrier

EE 392B: Silicon Photodetectors 1-38 • The generation-recombination rate is given by N v σ σ (pn − n2) U = t th n p i Ei−Et Et−Ei σp p + ni exp kT + σn n + ni exp kT h (pni − n2)h i = i , Ei−Et Et−Ei τn p + ni exp kT + τp n + ni exp kT

where Nt is the generation-recombinationh i centerh density, σn,p iis the capture cross section, and the minority carrier lifetimes are given as

τn,p = 1/Ntvthσn,p

We assume that τn = τp ≡ τo (see Handout 4) Recombination: U > 0 Generation: U < 0 • In the depletion region E − E (x) n = n exp fn i n i kT i E (x) − E p = n exp i fp n i kT i EE 392B: Silicon Photodetectors 1-39 • Then −n2 U ∼= i Et−Ei Ei−Et τpni exp kT + τnni exp kT −ni = Et−Ei Ei−Et τo exp kT + exp kT • Deﬁne h i E − E U = exp i t T kT Then, the maximum generation rate is obtained when ∂U = 0 ⇒ UT = 1 ⇒ Et = Ei, ∂UT and is given by ni Gmax = , 2τo • The generation current is x2 qn x jsc = q G dx ≈ i d dc max 2τo xZ1

EE 392B: Silicon Photodetectors 1-40 Generation Current at Si/SiO2 Surface

• The semiconductor surface has plenty of localized states having energies within the bandgap • The kinetics of generation-recombination at the surface is similar to trap

states in the bulk except that the trap density Nst is an areal density (# / cm2) • Again, using the SRH model, the surface generation-recombination rate is given by 2 Nstvthσnσp(psns − ni ) Us = Ei−Est Est−Ei σp ps + ni exp kT + σn ns + ni exp kT 2 ∼ h (psns i− ni ) h i = Nstvthσ , Ei−Est ps + ns + 2ni cosh kT

where again we assumedh that σn = σp ≡ σ i

Recombination: Us > 0

Generation: Us < 0

EE 392B: Silicon Photodetectors 1-41 • Surface generation depends on carrier density ◦ When the surface has plenty of carriers, either due to inversion or accumulation, 2 ∼ (psns − ni ) Us = Nstvthσ Ei−Est ps + ns + 2ni cosh kT is small h i

◦ When the surface is depleted, ps and ns are small, and N v sn U ∼= − st thσn = − i, s 2 i 2

Here we assume that Est = Ei and s = Nstvthσ has the unit of velocity (cm·sec−1) Surface generation current density is thus qsn js = i dc 2

EE 392B: Silicon Photodetectors 1-42 Activation Energy of Dark Current Components

• Notice that the generation currents due to bulk traps and surface traps

are proportional to ni 2 • And the diﬀusion current in the quasi-neutral region is proportional to ni • Therefore, in a plot of dark current vs log(1/T )

◦ the activation energy of bulk trap and surface trap dark current is Eg ◦ the dark current that is due to diﬀusion in the quasi-neutral region

has activation energy of Eg/2

EE 392B: Silicon Photodetectors 1-43 Surface Recombination Velocity

• It is diﬃcult to ”derive” the surface recombination velocity • It is obtained experimentally

EE 392B: Silicon Photodetectors 1-44 • Experimental procedure: ◦ Reverse bias the gated diode ◦ Sweep gate bias from inversion to accumulation

◦ Measure DC current from substrate in inversion (I3), depletion (I2), and accumulation (I1)

EE 392B: Silicon Photodetectors 1-45 • The currents are given by:

2 qniW AB qnisASB qni Dp I1 = + + AB 2τo(diode) 2 NB s τp qniWGAG qnisASG I2 = I1 + + 2τo(gate) 2 qn sA I = I − i SG 3 2 2 qn sA I − I = i SG 2 3 2 qniWGAG I3 − I1 = 2τo(gate)

EE 392B: Silicon Photodetectors 1-46 Direct Integration

• As discussed earlier, photocurrent is typically too small to measure directly • The most commonly used mode of photodiode operation in an image sensor is direct integration, where the photocurrent (and dark current) are directly integrated over the diode capacitance vD Q PSfrag replacements PSfrag replacements reset high light vD reset Qmax CD iphvo+ idc vo Q low light Qmaxiph + idc C high light D low light tint t tint t

◦ The photodetector is reset to the reverse bias voltage vD

◦ The diode current discharges CD for tint seconds, which is called integration time or exposure time

◦ At the end of the integration time the accumulated charge Q(tint) (in electrons) or voltage vo(tint) is read out

EE 392B: Silicon Photodetectors 1-47 • Assuming that the photo and dark currents do not change with reverse bias voltage, we obtain 1 Q(t ) = (i + i )t electrons int q ph dc int

Assuming that CD does not vary with reverse bias voltage, we get

(iph + idc)tint vo(tint) = vD − V CD • The maximum nonsaturating photocurrent is thus given by

max qQmax iph = − idc tint

Qmax is called the well capacity • To avoid blooming, i.e., overﬂowing of charge to neighboring photodetectors in the image sensor, we ensure that the diode is reverse

biased , i.e., vo(tint) > 0V Thus qQmax ≤ vD × CD (very often the voltage swing is lower than vD resulting in well capacity lower than vD × CD)

EE 392B: Silicon Photodetectors 1-48 Example

2 • Consider the nwell/psub diode with vD = 2.2V, and area AD = 30µm the photodiode capacitance s CD = AD = 1.55fF xn + xp Note: this is unrealistically small since it does not include edge capacitance and the capacitances of interconnect and other devices connected to the photodetetcor • Thus (ignoring these other capacitances) the well capacity −15 −19 Qmax = 3.41 × 10 /1.6 × 10 = 21312 electrons

• Assuming tint = 20ms and dark current idc = 2fA we get that the maximum nonsaturating photocurrent max iph = 167.55fA, which corresponds to 16.5 lux at λ = 555nm • To put this in perspective, a typical DRAM cell holds ∼ 160,000 to 200,000 electrons

EE 392B: Silicon Photodetectors 1-49 Finding Q(tint) and vo(tint) Numerically

• Since the depletion region width changes with the reverse bias voltage,

CD, jph, and jdc are not constant during integration • The output charge and voltage can be found numerically

◦ Set λ and F0 to desired values

◦ Set vo(0) = vD and Q(0) = 0 and calculate vo(k∆t) and Q(k∆t) tint iteratively beginning with k = 1 and ending with k = ∆t ◦ To calculate vo((k + 1)∆t) and Q((k + 1)∆t): k 1. Calculate the depletion region width and CD (using vo(k∆t)) 2. Calculate the current densities jph(k∆t) and jdc(k∆t) and the k charge accumulated ∆Q = (jph(k∆t) + jdc(k∆t))∆t ∆Qk 3. Set vo((k + 1)∆t) = vo(k∆t) − k and CD Q((k + 1)∆t) = Q(k∆t) + ∆Qk

• The following graphs provide computed Q(tint) and vo(tint) as a function of F0 for vD = 2.2V

EE 392B: Silicon Photodetectors 1-50 −9 x 10 nwell/psub Diode, Wavelength=600nm 7

) Direct Integration 2

3 Photo Charge (C/cm

0 0 0.5 1 1.5 2 2.5 3 3.5 2 12 Photon Flux (photons/cm s) x 10

EE 392B: Silicon Photodetectors 1-51 −9 x 10 nwell/psub Diode, Wavelength=600nm 6

Direct Integration

4 ) 2

3 Total Charge (C/cm 2

0 0 0.5 1 1.5 2 2.5 3 3.5 2 12 Photon Flux (photons/cm s) x 10

EE 392B: Silicon Photodetectors 1-52 nwell/psub Diode, Wavelength=600nm 2.4

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EE 392B: Silicon Photodetectors 1-53 nwell/psub Diode, Wavelength=600nm 2

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EE 392B: Silicon Photodetectors 1-54 Photogate

• Photogate is used in CMOS sensors (CMOS APS), frame transfer CCD (FT-CCD) and time-delay-and-integration CDD (TDI-CCD)

EE 392B: Silicon Photodetectors 1-55 Photogate Operation

• Gate voltage vG is set high enough to bias the MOS capacitor into the deep depletion regime (this requires vG >> vT ) (see Appendix) • Electrons generated in the depletion region are collected in the potential well • Electrons generated in the quasi-neutral region will ◦ Recombine with holes ◦ Diffuse to depletion region and get collected in the potential well if it is within the diffusion length of the minority carriers • Holes will be collected in the substrate • How many of the photo-generated carriers are collected depends on: ◦ Diffusion length of minority carriers ◦ Location and length of the depletion region

EE 392B: Silicon Photodetectors 1-56 EE 392B: Silicon Photodetectors 1-57 Quantum Eﬃciency of Photogate

• Photocurrent has two components sc ◦ Current due to generation in the depletion region, iph, again almost all carriers contribute to the current n ◦ Diffusion current due to generation in the quasi-neutral p-region, iph • To calculate the current we make the depletion approximation and use the basic MOS capacitor equations to find the depletion region width (see Appendix III) (you will derive it in HW2) • A disadvantage of the photogate is lower quantum efficiency, especially for shorter wavelengths (blue), due to absorption in the polysilicon gate (which has the same α as crystalline silicon) • Photogate is also used in direct integration mode; charge accumulated on gate is transferred to another capacitor (as we shall see later)

EE 392B: Silicon Photodetectors 1-58 Quantum Eﬃciency of Photogate

• It can be shown that QE for photogate, not including absorption in the polysilicon gate, is given by 2 −αL −αx αL e − e d cosh ((L − xd)/Ln) QE(λ) = 1−e−αxd + n αe−αxd + α2L2 − 1 L sinh ((L − x )/L ) n n d n • Limiting Cases:

◦ Very long diﬀusion length (Ln → ∞): 1 − e−α(L−xd) QE = 1 − e−αxd + e−αxd 1 − α(L − x ) d ◦ Very thick substrate (L → ∞): −αx αLne d QE = 1 − e−αxd + (αLn + 1)

EE 392B: Silicon Photodetectors 1-59 EE 392B: Silicon Photodetectors 1-60 Appendix I – Derivation of 1-D Continuity Equation

• Consider minority carrier (electron) current ﬂow in p-type silicon • In a slab x to x + ∆x

Gn(x) PSfrag replacements

jn(x) jn(x + ∆x)

x Rn(x) x + ∆x

jn(x): electron current density at x 3 Gn(x): generation rate (electrons/cm ·s) 3 Rn(x): recombination rate (electrons/cm ·s) n(x): electron density at x (electrons/cm3)

EE 392B: Silicon Photodetectors 1-61 • The rate of electron density increase in the slab ∂n(x) 1 ∆x ≈ − (j (x) − j (x + ∆x)) + (G (x) − R (x))∆x, ∂t q n n n n which in the limit, gives ∂n(x) 1∂j (x) = n + (G (x) − R (x)) ∂t q ∂x n n assuming no electric field, the current is only due to diffusion and is given by ∂n(x) j (x) = qD , n n ∂x 2 where Dn is the diffusion constant for electrons in cm /s substituting, we get the continuity equation ∂n(x) ∂2n(x) = D + (G (x) − R (x)) dt n ∂x2 n n • Similarly for holes, ∂p(x) j (x) = −qD p p ∂x and the continuity equation is ∂p(x) ∂2p(x) = D + (G (x) − R (x)) ∂t p ∂x2 p p

EE 392B: Silicon Photodetectors 1-62 • Assuming low level injection, i.e., that excess carrier concentration << majority carrier concentration, we get that

np − np0 Rn = τn

where np0 is the intrinsic minority carrier concentration, and τn is the carrier lifetime

EE 392B: Silicon Photodetectors 1-63 Appendix II – Depletion Width for PN Junction

PSfrag replacements • Energy band diagrams at thermal equilibrium

Ec Ec

Efn qφn Ei Ei qφp Efp

Ev Ev

p-type n-type

kT Nd kT Na Here φn = ln and φp = ln , where q ni q ni k = 8.62 × 10−5eV K−1 is the Boltzman constant T is the temperature in Kelvin q = 1.6 × 10−19Col is the electron charge 10 −3 ni is the intrinsic carrier concentration ≈ 1.45 × 10 cm at room temperature −3 Nd and Na are the donor and acceptor densities in cm

EE 392B: Silicon Photodetectors 1-64 PN Junction Energy Band Diagram

• The energy band diagram for reverse biased pn junction p-type n-type PSfrag replacements

qφp Efp Ec qvD

Efn qφn Ei

Ev 3 ρ col/cm

− qNd xp x xn −qNa E V/cm x

Emax

φ V

EE 392B: Silicon Photodetectors 1-65 E and φ are found by solving the Poisson equation d2φ dE ρ(x) − − 2 = = , dx dx s −13 where s = 10.45 × 10 F/cm is the permittivity of Si So in the p-type region, we obtain

qNa E(x) = − (x + xp) s and qNa Emax = − xp s Similarly, in the n-type region we have

qNd E(x) = (x − xn) s and qNd Emax = − xn s Thus x N n = a xp Nd

EE 392B: Silicon Photodetectors 1-66 Now xn φ(xn) = − Edx Z−xp 2 2 qN x qNax = d n + p 2s 2s = vD + φn + φp Combining the last two equations, we obtain that the depletion width 2 1 1 x = x + x = s(v + φ + φ )( + ) d n p q D n p N N r a d • Example (nwell/psub diode): assuming vD = 2V, φn = 0.3486V, and φp = 0.289V, we get xn = 0.176µm and xp = 1.76µm • The (small signal) diode capacitance per unit area is deﬁned as dQ C = , dvD where the charge Q = qNdxn = qNaxp. Thus, C = s F/cm2 xn + xp For the previous example C = 5.4 × 10−9F/cm2

EE 392B: Silicon Photodetectors 1-67 Appendix III – MOS Capacitor

• First consider the energy band diagrams under thermal equilibrium for PSfragpolysilicon,replacementsoxide, and silicon

E0 0.95eV Ec 4.05eV

Ef ≈ Ec Ec

Ei qφp Ef polysilicon Ev

polsilicon oxide p-type

E0 is the free electron energy E0 − Ec = 4.05eV is the semiconductor electron aﬃnity oxide E0 − Ec = 0.95eV is the oxide electron aﬃnity

EE 392B: Silicon Photodetectors 1-68 • The PSfragenergyreplacementsband diagram for the MOS system under thermal equilibrium

assuming vG = 0

qv0

3.1eV

3.1eV Ec

Ei qφp Ef Ev qψs0 tox poly oxide p-type ρ

0 xd x −Q −qNa

EE 392B: Silicon Photodetectors 1-69 We can ﬁnd v0, ψ0, and xd by writing the ﬂat-band voltage in two ways and solving the Poisson equation E v = g + φ = v + ψ , F B 2q p 0 s0

qNaxd v0 = , and Cox 2 qNaxd ψs0 = 2s ox 2 −14 Cox = F/cm , and ox = 34.5 × 10 F/cm tox PSfrag replacements • Energy band diagram in the deep depletion regime

qv0

Ec Ei Ef Ev qvG qψs

Ef tox poly oxide p-type

EE 392B: Silicon Photodetectors 1-70 The MOS system is in deep depletion when ψs > 2φp (this is the same condition as for strong inversion except that here we are interested in the transient response before the onset of strong inversion)

This gives that 1 vG > 2φp − vF B + 4qNasφp = vT , Cox where vT is the threshold voltage (assumingp no threshold adjust implant is used)

• To ﬁnd the depletion region depth xd, note that

v0 + ψs = vG + vF B, where 2 qNaxd ψs = , and 2s qNaxd v0 = Cox solving for ψs we get 2 ψs = v1 + v2 − v2 + 2v1v2, q EE 392B: Silicon Photodetectors 1-71 where

v1 = vG + vF B, and qNas v2 = 2 Cox the depletion width can then be determined

Note: for the MOS capacitor to stay in the deep depletion we set vD = ψs

EE 392B: Silicon Photodetectors 1-72 Appendix IV Bulk Mobility

EE 392B: Silicon Photodetectors 1-73 Minority Carrier Diﬀusion Length

EE 392B: Silicon Photodetectors 1-74 Minority Carrier Lifetime

EE 392B: Silicon Photodetectors 1-75