1. Chapter 6 Photodetectors
6.1 Principles of Photodiodes
The PIN Photo-Detector
Avalanche Photodiodes
6.2 Photo-Detector Noise
Noise Sources
Signal-to-Noise Ratio
6.3 Detector Response Time
6.4 Avalanche Multiplication Noise

2. 6.1.1 The PIN Photo-Detector
The PIN photodiode is structured with p and n regions separated by a lightly n-doped intrinsic (i) region (Fig. 6-1).
Incident photon with energy > band-gap energy of the photodiode will generate free electron-hole pairs, known as photo-carriers (Fig. 6-2).
The high electric field present in the depletion region causes the carriers to separate and be collected across the reverse-biased junction.
This gives rise to a photo-current flow in an external circuit, with one electron flowing for every carrier pair generated.

3. 6.1.1 The PIN Photo-Detector
Figure 6-1. Schematic representation of a PIN photodiode circuit with an applied reverse bias.

4. 6.1.1 The PIN Photo-Detector
Figure 6-2. Simple energy-band diagram for a PIN photodiode. Photons with energy > band-gap energy can generate free electron-hole pairs.

5. 6.1.1 The PIN Photo-Detector
As the charge carriers flow through the material, some electron-hole pairs will recombine and disappear.
On the average, the charge carriers move a diffusion length Ln or Lp for electrons and holes, respectively.
The time it takes for an electron or hole to recombine is known as the carrier lifetime and is represented by tn and tp, respectively.
The lifetimes and the diffusion lengths are related by
Ln = (Dntn)1/ and Lp = (Dptp)1/2
where Dn and Dp are the electron and hole diffusion coefficients, expressed in units of cm2/sec.

6. 6.1.1 The PIN Photo-Detector
Optical radiation is absorbed in the semiconductor material according to the exponential law
P(x) = Po[1 - exp(-as(l)x)] (6-1)
Here, as(l) is the absorption coefficient at wavelength l,
Po is the incident optical power level,
and P(x) is the optical power absorbed in a
distance x.

7. 6.1.1 The PIN Photo-Detector
The optical absorption coefficient versus wavelength is shown in Fig. 6-3 for several photodiode materials.
The cutoff lc is determined by the band-gap energy Eg of the material:
lc(mm) = hc/Eg = 12.4 / Eg(eV) (6-2)
The cutoff wavelength is about 1.06-µm for Si and 1.6-µm for Ge.
For longer wavelengths, the photon energy is not sufficient to excite an electron from the valence to the conduction band.

8. 6.1.1 The PIN Photo-Detector
Figure 6-3. Optical absorption coefficient as a function of wavelength for Si, Ge, and GaAs.

9. 6.1.1 The PIN Photo-Detector
Example 6-1: A photodiode constructed of GaAs has a band-gap energy of 1.43eV at 300oK.
From Eq. (6-2), the long-wavelength cutoff is
lc = hc/Eg
(6.625x10-34J.s)(3x108m/s)
= = 869 nm.
(1.43eV)(1.6x10-19J/eV)
This GaAs photodiode will not operate for photons of l > 869 nm.

10. 6.1.1 The PIN Photo-Detector
At the lower-wavelength end, the photo-response cuts off with very large values of as at the shorter wavelengths.
The photons are absorbed very close to the photo-detector surface, the recombination time of the generated electron-hole pairs is very short.

11. 6.1.1 The PIN Photo-Detector
If the depletion region has a width w, then the total power absorbed in the distance w is
P(w) = Po[1 - exp(-asw)] (6-3)
Take into account the reflectivity Rf at the entrance face of the photodiode, the primary photocurrent Ip resulting from the power absorption of Eq. (6-3) is given by Ip = (q/hn)Po[1-exp(-asw)](1-Rf) (6-4)
where Po is the optical power incident on the photo-detector, q is the electron charge, and hv is the photon energy.

12. 6.1.1 The PIN Photo-Detector
The quantum efficiency h is the number of the photo-carrier pairs generated per incident photon of energy hv and is given by
number of electron-hole generated
h = (6-5)
number of incident photons
= (Ip/q) / (Po/hv).
Here, Ip is the average photocurrent generated by a steady-state average optical power Po incident on the photo-detector.

13. 6.1.1 The PIN Photo-Detector
Example 6-2: In a 100-ns pulse, 6.0x106 photons at 1300-nm fall on an InGaAs photo-detector. On the average, 5.4x106 electron-hole pairs are generated.
The quantum efficiency is found from Eq. (6-5) as
number of e-h pairs generated
h =
number of incident photons
= (5.4x106) / (6x106) = 0.9.
Thus, the quantum efficiency at 1300-nm is 90 %.

14. 6.1.1 The PIN Photo-Detector
To achieve a high quantum efficiency, the depletion layer must be thicker.
However, the thicker the depletion layer, the longer it takes for the photo-generated carriers to drift across the reverse-biased junction.
Compromise has to be made between response speed and quantum efficiency.

15. 6.1.1 The PIN Photo-Detector
The performance of a photodiode is often characterized by the responsivity R.
This is related to the quantum efficiency h by R = Ip/Po = hq/hn (6-6)
Typical PIN responsivities are shown in Fig. 6-4.
Representative values are 0.65-A/W for Si at 900- nm and 0.45-A/W for Ge at 1.3-µm.
For InGaAs, typical values are 0.9-A/W at 1.3-µm and 1.0-A/W at 1.55-µm.

16. 6.1.1 The PIN Photo-Detector
Example 6.3: Photons of energy 1.53x10-19J are incident on a photodiode which has a responsivity of 0.65A/W.
If the optical power level is 10mW, then from Eq. (6-6) the photo-current is Ip = RPo = (0.65A/W)(10mW) = 6.5mA
The responsivity is a linear function of the optical power. The photocurrent Ip is directly proportional to the optical power Po incident upon the photo-detector, so that the responsivity R is constant at a given wavelength.

17. 6.1.1 The PIN Photo-Detector
For a given material, as the wavelength of the incident photon becomes longer, the photon energy becomes less than that required to excite an electron from the valence band to the conduction band.
The responsivity thus falls off rapidly beyond the cutoff wavelength, as can be seen in Fig. 6-4.

18. 6.1.1 The PIN Photo-Detector
Figure 6-4. Comparison of the responsivity and quantum efficiency as a function of wavelength for different PIN photodiodes.

19. 6.1.1 The PIN Photo-Detector
Example 6-4 : As shown in Fig. 6-4, for the wavelength range nm < l < 1600-nm, the quantum efficiency for InGaAs is around 90 %. Thus, in this wavelength range the responsivity is
R = hq/hn = hql/hc
= (0.90)(1.6x10-19C)l/(6.625x10-34J.s)(3x108m/s)
= 7.25x105 l
For example, at 1300-nm we have R = [7.25x105(A/W)/m](1.30x10-6m)
= 0.92 A/W

20. 6.1.1 The PIN Photo-Detector
At wavelengths higher than 1600-nm, the photon energy is not sufficient to excite an electron from the valence band to the conduction band.
For example, In0.53Ga0.47As has an energy gap Eg = 0.73 eV, so that from Eq. (6-2) the cutoff wavelength is
lc = 1.24/Eg = 1.24/0.73 = 1.7 mm
At wavelengths < 1100-nm, the photons are absorbed very close to the photo-detector surface, where the recombination rate of the generated electron-hole pairs is very short. The responsivity thus decreases rapidly for smaller wavelengths.

21. 6.1.2 Avalanche Photodiodes
APDs internally multiply the primary signal photocurrent in a mechanism known as impact ionization.
The created carriers are accelerated by the high electric field, gaining enough energy to cause further impact ionization. This phenomenon is the avalanche effect.
A commonly used structure for achieving carrier multiplication with very little excess noise is the reach-through construction shown in Fig. 6-5.

22. 6.1.2 Avalanche Photodiodes
Figure 6-5. Reach-through avalanche photodiode structure and the electric fields in the depletion and multiplication regions.

23. 6.1.2 Avalanche Photodiodes
In normal usage, the RAPD is operated in the fully depleted mode. Light enters the device through the p+ region and is absorbed in the p material, which acts as the collection region for the photo-generated carriers.
The photo-generated electrons drift through the p region in the pn+ junction, where a high electric field exists.
It is in this high-field region that carrier multiplication takes place.

24. 6.1.2 Avalanche Photodiodes
The average number of electron-hole pairs created by a carrier per unit distance traveled is called the ionization rate.
Most materials exhibit different electron ionization rates a and hole ionization rates b.
Values of a and b for five different semiconductor materials are shown in Fig  
The ratio k = b/a of the electron and hole ionization rates is a measure of the photo-detector performance.
APDs constructed of materials in which one type of carrier largely dominates impact ionization exhibit low noise and large gain-bandwidth products.

25. 6.1.2 Avalanche Photodiodes
Figure 6-6. Carrier ionization rates obtained experimentally for Si, Ge, Ga, As, GaAsSb, and InGaAs.

26. 6.1.2 Avalanche Photodiodes
The multiplication M for all carriers generated in the photodiode is defined by M = IM / Ip (6-7) where IM is the average value of the total multiplied output current and Ip is the primary unmultiplied photocurrent defined in Eq. (6-4).
The performance of an APD is characterized by the responsivity given by
RAPD = (hq/hn)M = RoM (6-8)
where Ro is the unity gain responsivity.

27. 6.2 PHOTODETECTOR NOISE
The power SNR at the output of an optical receiver is
(6-9)
To achieve a high SNR,
1. PD must have a high h to generate a large signal power.
2. PD and amplifier noises should be kept as low as
possible.
The sensitivity of a photodiode is describable in terms of the minimum detectable optical power. This is the optical power necessary to produce a photo-current of the same magnitude as the total rms noise current, or equivalently, a SNR of 1.

28. Noise Sources
In the receiver model of Fig. 6-8, the photodiode has a small series resistance Rs, a total capacitance Cd consisting of junction and packaging capacitances, and a bias (or load) resistor RL.
The amplifier following the photodiode has an input capacitance Ca and a resistance Ra.

29. Noise Sources
Figure 6-8. (a) Simple model of a photo-detector receiver, and (b) its equivalent circuit.

30. Noise Sources
If a modulated signal of optical power P(t) falls on the detector, the primary photo-current iph(t) generated is
iph(t) = (hq/hn)P(t) (6-10)
The primary current consists of a dc value Ip -- the average photo-current due to the signal power, and a signal component ip(t).
For PINs, the mean-square signal current <is2> for a sinusoidally varying input signal of modulation index m is
<is2> = s2s,PIN = <ip2(t)> (6-11a) = sp2 = m2Ip2/ (6-12)
where s2 is the variance.

31. Noise Sources
For APDs, the mean-square signal current <is2> is <is2> = s2s,APD = <ip2(t)>M (6-11b) where M is the average avalanche gain.
The quantum or shot noise follow a Poisson process. The quantum noise current has a mean-square value
<iQ2> = sQ2 = 2qIpBM2F(M) (6-13)
where F(M) = Mx, 0 < x < 1.0, is a noise figure associated with the random nature of the avalanche process. For PIN photodiodes, M and F(M) are unity.

32. Noise Sources
The mean-square value of the bulk dark current iDB arisen from thermally generated electrons and/or holes is given by
<iDB2> = sDB2 = 2qIDBM2F(M) (6-14)
where ID is the primary (unmultiplied) detector bulk dark current.
The surface dark current is simply referred to as leakage current. The mean-square value of this current is given by
<iDS2> = sDS2 = 2qILB (6-15)
where IL is the surface leakage current. The surface dark current is not affected by the avalanche gain.

33. Noise Sources
The dark currents and the signal current are uncorrelated, the mean-square PD noise current <iN2> can be written as
<iN2> = sN2 = <iQ2> + <iDB2> + <iDS2>
= sQ2 + sDB2 + sDS (6-16)
= 2q(Ip + ID)M2F(M)B + 2qILB

34. Noise Sources
The PD load resistor contributes a mean-square thermal (Johnson) noise current
<iT2> = sT2 = 4kBTB/RL, (6-17)
where kB is Boltzmann's constant and T is the absolute temperature.
This Johnson noise can be reduced by using a load resistor which is large but still consistent with the receiver bandwidth requirements.

35. 6.2.2 Signal-to-Noise Ratio
Substituting Eqs. (6-11), (6-16), and (6-17) into Eq. (6-9) for the SNR at the input of the amplifier, we have
(6-18)
For PINs, the dominating noise currents are those of the detector load resistor (the thermal current iT) and the active elements of the amplifier circuitry (iamp).
For APDs, the thermal noise is of lesser importance and the photo-detector noises usually dominate.
From Eq. (6-18), it is seen that the signal power is multiplied by M2 and the quantum noise plus bulk dark current is multiplied by M2F(M).

36. 6.2.2 Signal-to-Noise Ratio
The optimum gain at the maximum SNR can be found by differentiating Eq. (6-18) with respect to M, setting the result equal to zero, and solving for M.
For a sinusoidally modulated signal, with m = 1 and F(M) approximated by Mx, will yield
(6-19)

37. 6.3 Detector Response Time
The response time of a photodiode together with its output circuit depends on the factors:
The transit time of the photo-carriers in the depletion region.
2. The diffusion time of the photo-carriers generated outside the depletion region.
3. The RC time constant of the photodiode and its associated circuit.

38. 6.3 Detector Response Time
The transit time td the photo-carriers take to travel across the depletion region depends on the carrier drift velocity vd and the depletion layer width w, and is given by
td = w / vd (6-27)
The electric field in the depletion region is large enough so that the carriers have reached their scattering-limited velocity.

39. 6.3 Detector Response Time
The A typical response time of a partially depleted photodiode is shown in Fig
The fast carriers allow the device output to rise to 50 % of its maximum value in approximately 1 ns, but the slow carriers cause a relatively long delay before the output reaches its maximum value.

40. 6.3 Detector Response Time
Figure Typical response time of a photodiode that is not fully depleted.

41. 6.3 Detector Response Time
To achieve a high quantum efficiency, the depletion layer width must be much larger than 1/a, so that most of the light will be absorbed.
The response to a rectangular input pulse of a low-capacitance photodiode having w >> 1/a is shown in Fig. 6-13b.
If the photodiode capacitance is larger, the response time becomes limited by the RC time constant of the load resistor RL and the photodiode capacitance.
The photodetector response then begins to appear as that shown in Fig. 6-13c.

42. 6.3 Detector Response Time
Devices with very thin depletion regions tend to show distinct slow- and fast-response components, as shown in Fig. 6-13d.
The fast component in the rise time is due to carriers generated in the depletion region, whereas the slow component arises from the diffusion of carriers that are created with a distance L from the edge of the depletion region.

43. 6.3 Detector Response Time
Figure Photodiode pulse responses under various detector parameters.

44. 6.3 Detector Response Time
If RT is the combination of the load and amplifier input resistances and CT is the sum of the photodiode and amplifier capacitances,
as shown in Fig. 6-8,
the detector behaves like a simple RC low-pass filter with a passband given by
B = 1/(2pRTCT) (6-29)

45. 6.3 Detector Response Time
Example 6-7 : If the photodiode capacitance is 3 pF, the amplifier capacitance is 4 pF, the load resistor is 1 kW, and the amplifier input resistance is 1 MW,
then CT = 7 pF and RT = 1 kW, so that the circuit bandwidth is B = 1/(2pRTCT) = 23 MHz (6-30)
If we reduce the photo-detector load resistance to 50W, then the circuit bandwidth becomes B = 455 MHz.

46. 6.4 AVALANCHE MULTIPLICATION NOISE
If m denotes the statistically varying APD gain, then 〈m2〉>〈m〉2 = M (6-31) where the symbols〈.〉denote an ensemble average and 〈m〉= M is the average carrier gain defined in Eq. (6-7).
Since the noise created by the avalanche process depends on the mean-square gain〈m2〉, the noise in an APD can be relatively high.
It has been found that,〈m2〉can be approximated by 〈m2〉= M2+x (6-32) where the exponent x varies between 0 and 1.0 depending on the photodiode material and structure.

47. 6.4 AVALANCHE MULTIPLICATION NOISE
The ratio of the actual noise generated in an APD to the noise that would exist if all carrier pairs were multiplied by M is called the excess noise factor F and is defined by
F =〈m2〉/〈m〉2 =〈m2〉/ M (6-33)
This excess noise factor is a measure of the increase in detector noise resulting from the randomness of the multiplication process.

48. 6.4 AVALANCHE MULTIPLICATION NOISE
McIntyre has shown that, for injected electrons and holes, the excess noise factors are
(6-34)
(6-35)
where the subscripts e and h refer to electrons and holes, respectively.

49. 6.4 AVALANCHE MULTIPLICATION NOISE
The weighted ionization rate ratios k1 and k2 take into account the nonuniformity of the gain and the carrier ionization rates in the avalanche region.
To a first approximation k1 and k2 can be considered as constant and equal. Eqs. (6-34) and (6-35) can be simplified as
(6-38)
for electron injection,

50. 6.4 AVALANCHE MULTIPLICATION NOISE
and (6-39)
for hole injection,
where the effective ionization rate ratios are
and (6-40)

51. 6.4 AVALANCHE MULTIPLICATION NOISE
Figure 6-14 shows Fe as a function of the average electron gain Me for various values of the effective ionization rate ratio keff.
If the ionization rates are equal, the excess noise is at its maximum so that Fe is at its upper limit of Me.
As the ionization rates ratio b/a decreases from unity, the electron ionization rate a starts to be the dominant contributor to impact ionization, and the excess noise factor becomes smaller.
If only electrons cause ionization, b = 0 and Fe reaches its lower limit of 2.

52. 6.4 AVALANCHE MULTIPLICATION NOISE
Figure Variation of the electron excess noise factor Fe as a function of the electron gain for various values of the effective ionization rate ratio keff.

53. 6.4 AVALANCHE MULTIPLICATION NOISE
From the empirical relationship for the mean-square gain given by Eq. (6-32), the excess noise factor can be approximated by F = Mx (6-41)
The parameter x takes on values of 0.3 for Si, 0.7 for InGaAs, and 1.0 for Ge avalanche photodiodes.