## Multiplication theory for dynamically biased avalanche photodiodes: new limits for gain bandwidth product |

Optics Express, Vol. 20, Issue 7, pp. 8024-8040 (2012)

http://dx.doi.org/10.1364/OE.20.008024

Acrobat PDF (1440 KB)

### Abstract

Novel theory is developed for the avalanche multiplication process in avalanche photodiodes (APDs) under time-varying reverse-biasing conditions. Integral equations are derived characterizing the statistics of the multiplication factor and the impulse-response function of APDs, as well as their breakdown probability, all under the assumption that the electric field driving the avalanche process is time varying and spatially nonuniform. Numerical calculations generated by the model predict that by using a bit-synchronous sinusoidal biasing scheme to operate the APD in an optical receiver, the pulse-integrated gain-bandwidth product can be improved by a factor of 5 compared to the same APD operating under the conventional static biasing. The bit-synchronized periodic modulation of the electric field in the multiplication region serves to (1) produce large avalanche multiplication factors with suppressed avalanche durations for photons arriving in the early phase of each optical pulse; and (2) generate low avalanche gains and very short avalanche durations for photons arriving in the latter part of each optical pulse. These two factors can work together to reduce intersymbol interference in optical receivers without sacrificing sensitivity.

© 2012 OSA

## 1. Introduction

*pin*photodetector [1, 2]. The gain offered by the APD improves the receiver sensitivity as it amplifies the photocurrent, thereby reducing the relative effect of Johnson noise in the preamplifier stage of an optical receiver and improving the receiver’s sensitivity [3, 4

4. B. L. Kasper and J. C. Campbell, “Multigigabit-per-second avalanche photodiode lightwave receivers,” J. Lightwave Technol. **5**(10), 1351–1364 (1987). [CrossRef]

*pin*photodetector [5

5. T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. **36**(21), 1807–1808 (2000). [CrossRef]

6. P. Sun, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Statistical correlation of gain and buildup time in APDs and its effects on receiver performance,” J. Lightwave Technol. **24**(2), 755–768 (2006). [CrossRef]

6. P. Sun, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Statistical correlation of gain and buildup time in APDs and its effects on receiver performance,” J. Lightwave Technol. **24**(2), 755–768 (2006). [CrossRef]

7. D. S. G. Ong, J. S. Ng, M. M. Hayat, P. Sun, and J. P. R. David, “Optimization of InP APDs for high-speed lightwave systems,” J. Lightwave Technol. **27**(15), 3294–3302 (2009). [CrossRef]

*et al*. [8] demonstrated a sensitivity of −19dBm at 40 Gbps with a bit-error rate of 10

^{−10}, providing approximately a 9dBm improvement over conventional

*pin*diode. This was achieved by including a trans-impedance amplifier with tunable response to boost the gain-bandwidth-product (GBP) from 140 to 270 GHz and operating the APDs with avalanche gain values of 3 to 10. Notably, the biggest breakthrough came as Si/Ge APDs were demonstrated in 2009. Kang

*et al*. [9

9. Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics **3**(1), 59–63 (2009). [CrossRef]

*et al*. [10

10. W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express **17**(15), 12641–12649 (2009). [CrossRef] [PubMed]

*k*, is much less than unity in Si). However, these devices have lower normal-incidence sensitivity compared with InGaAs-InP APDs of the same absorption-layer thickness [11] due to a combination of high dark currents, resulting from the low bandgap and high level of intrinsic carrier concentration of Ge, and reduced quantum efficiency of Ge at longer telecom wavelengths due to its low absorption coefficient at 1.55 μm (3.5x10

^{2}cm

^{−1}for Ge compared to 6.1x10

^{3}cm

^{−1}for InGaAs). To compensate for the reduced photoresponsivity of Ge at longer telecom wavelengths, waveguide structures have been proposed and unity-gain bandwidths of 23 GHz and 29.5 GHz have been achieved for evanescent-coupled and butt-coupled Si/Ge APDs, respectively [11]. While Si/Ge APDs show promise due to their compatibility with SOI CMOS and the almost transit-time-limited avalanche duration associated with Si (since hole ionization is minimal), and despite all numerous other efforts in the past two decades or so that targeted new APD materials and structures [12

12. J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. **10**(4), 777–787 (2004). [CrossRef]

13. D. C. Herbert and E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. **48**(7), 1475–1477 (2001). [CrossRef]

14. R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron. Dev. **13**(1), 164–168 (1966). [CrossRef]

15. M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. **39**(3), 546–552 (1992). [CrossRef]

15. M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. **39**(3), 546–552 (1992). [CrossRef]

16. M. M. Hayat and B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. **10**(10), 1415–1425 (1992). [CrossRef]

## 2. Potential for breaking the traditional gain-bandwidth limits through dynamic biasing

18. M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. **49**(5), 770–781 (2002). [CrossRef]

*early*in the optical pulse experience a period of high electric field in the multiplication region of the APD, where they can generate a strong avalanche current in the early phase of the optical-pulse interval, as depicted schematically by the black curve corresponding to the dynamic-bias response in Fig. 1. Next, as a low electric field period follows the high-field phase within the same optical pulse, carriers in the multiplication region undergo a much weakened impact ionization process, which leads to the termination of the avalanche current with a high probability. Thus, high-gain avalanche pulses are triggered by the “early” arriving photons in each optical pulse, albeit with a much reduced avalanche duration due to the weakening of the impact ionization in the second phase of the bias period. Second, photons that arrive

*late*in the optical pulse period are still detected as the APD remains reverse biased throughout the bias period. However, the resulting avalanche gain is very low and the avalanche durations they trigger are very short, as depicted schematically by the red curve correspondig to the dynamic bias response in Fig. 1. The avalanche pulses in the traditional constant-biasing scheme are also shown schematically in Fig. 1 for comparison.

20. N. Namekata, S. Adachi, and S. Inoue, “1.5 GHz single-photon detection at telecommunication wavelengths using sinusoidally gated InGaAs/InP avalanche photodiode,” Opt. Express **17**(8), 6275–6282 (2009). [CrossRef] [PubMed]

## 3. Impact ionization under dynamic electric fields

*V*(

_{BD}*t*),

*t*≥ 0, is applied to an APD. Consider a charge-depleted multiplication region of the APD extending from

*x*= 0 to

*x*=

*w*, as shown in Fig. 2 , with the convention that the electric field is pointing in the negative

*x*direction. Let E(

*x*,

*t*) denote the dynamic electric field in the multiplication region at position

*x*and at time

*t*. If the field is spatially uniform, then E(

*x*,

*t*) ≡ E(

*t*) =

*V*(

_{BD}*t*)/

*w*. Suppose that a parent hole (electron) is created at an arbitrary location

*x*in the multiplication region of the APD, and assume that the field is sufficiently high so as conduction-band electrons and valence-band holes travel at their material-specific saturation velocities,

*v*and

_{e}*v*, respectively. As the hole travels the multiplication region, it can impact ionize at a stochastic location, say ξ, and at time τ = (ξ–

_{h}*x*)/

*v*. The ability of a carrier to impact ionize is probabilistic and depends upon the time- and position-dependent electric field as well as the carrier’s history of prior impact ionizations (detailed description deferred to Subsection 3.1). Upon impact ionization, the parent hole is replaced with two valence-band offspring holes and a conduction-band offspring electron. Each offspring carrier proceeds, as its parent, to further impact ionize, and so on. As this process continues, it may or may not terminate, depending on the field, device and material properties. The stochastic

_{h}*dynamic multiplication factor*is the total number of electron-hole pairs generated as a result of a parent carrier in the presence of the dynamic field; it can be either finite or infinite. On the other hand, the

*dynamic avalanche duration*is the time measured from the creation of the parent carrier to the time when all carriers have exited the multiplication region. In Subsections 4.1 and 4.2 we will derive equations that enable us to calculate the statistics of the mean multiplication factor and the mean impulse response function (integral equations for the excess-noise factor and the breakdown probability under the dynamic fields are presented in the Appendix). However, before doing so we will need to extend the notions of the ionization coefficient, dead space, and the probability density function of the carrier’s free path (prior to ionization) to a dynamic-electric-field setting.

*x*,

*t*) at the point where and when it impact ionizes. We define α(

*x*,

*t*) and β(

*x*,

*t*) as the electron and hole

*time-varying*non-localized ionization coefficients associated with carriers at location

*x*in the multiplication region and at time

*t*. These are the ionization coefficients for those carriers that have already traveled the dead space, which is the minimum distance a carrier must travel before it acquires sufficient energy to effect an impact ionization. Following the model for non-localized ionization coefficients under a static electric field [22

22. M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. **47**(3), 625–633 (2000). [CrossRef]

*A*,

*E*, and

_{c}*m*, are known for various III-V materials [22

22. M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. **47**(3), 625–633 (2000). [CrossRef]

24. F. Osaka, T. Mikawa, and T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga_{0}_{.}_{33}In_{0.67}As_{0.70}P_{0.30},” Appl. Phys. Lett. **45**(3), 292–293 (1984). [CrossRef]

*x*and of age

*s*relative to the launch instant of the dynamic electric field (at

*t*= 0), and let

*X*and

_{e}*X*be their stochastic free-path distances to their

_{h}*first*impact ionization. As it turns out, the

*age*of a carrier will play a key role in the formulation of the theory for avalanche multiplication under dynamic fields, as described in Section 4. Under a dynamic electric field, the probability density function of the location, ξ, of the first ionization by a parent carrier of age

*s*and at location

*x*is zero before the dead space is travelled. After the dead space, however, it is exponential with a nonuniform rate that not only depends upon the path of the carrier from its birth location

*x*to ξ but also on the history of the time- and space-varying electric field from the birth instant of the carrier to the instant of its first impact ionization. Along these lines, we can extend the shifted exponential model for the carrier’s free path [25

25. M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. **28**(5), 1360–1365 (1992). [CrossRef]

*age-dependent*probability density function of

*X*and

_{e}*X*as

_{h}*h*(ξ;

_{e}*x*,

*s*) and

*h*(ξ;

_{h}*x*,

*s*), respectively,andwhere

*d*(

_{e}*x*,

*s*) and

*d*(

_{h}*x*,

*s*) represent the

*age-dependent*dead spaces for an electron and hole, respectively, that were created at position

*x*and of age

*s*. Here,

*h*(ξ;

_{e}*x*,

*s*)Δξ approximates the probability that an electron born at location

*x*and with age

*s*(relative to the launch instant of the electric field) impact ionizes for the first time anywhere in the location ξ to ξ + Δξ. The electron’s age-dependent dead space is computed by equating the ionization threshold energy to the energy gained from the ballistic transport of the carrier in the time- and space-varying electric field. Thus, the age-dependent dead space is the minimum

*d*value that satisfies the equationwhere for any 0 ≤

*x*≤

*w*, E

*(*

_{th,e}*x*) is the ionization threshold energy for electrons for the material at position

*x*in the multiplication region. Similarly, the hole’s age-dependent dead space is computed as the minimum

*d*value that satisfies the equationwhere E

*(*

_{th,h}*x*) is the ionization threshold energy for holes for the material at position

*x*in the multiplication region. (Note that Eqs. (3a) and (3b) above respectively collapse to Eqs. (4) and (5) in [26

26. M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. **49**(12), 2114–2123 (2002). [CrossRef]

*x*and

*s*.

## 4. Avalanche multiplication theory under dynamic electric fields

*et al*. in [15

15. M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. **39**(3), 546–552 (1992). [CrossRef]

16. M. M. Hayat and B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. **10**(10), 1415–1425 (1992). [CrossRef]

*et al*. [27

27. C. H. Tan, P. J. Hambleton, J. P. R. David, R. C. Tozer, and G. J. Rees, “Calculation of APD impulse response using a space- and time-dependent ionization probability distribution function,” J. Lightwave Technol. **21**(1), 155–159 (2003). [CrossRef]

*x*,

*t*), launched at time

*t*= 0, then an electron born at location

*x*with age 0 will experience this time-varying field in its lifetime. In contrast, if an electron is born at location

*x*with age

*s*(relative to the launch time of the field at

*t*= 0) then it will experience in its lifetime a

*clipped*version of the dynamic field starting at

*t*=

*s*, namely E(

*x*,

*t*)

*u*(

*t-s*), where

*u*(.) is the unit-step function. To take the carrier’s “age” into account while modeling the avalanche multiplication process, we must formulate a model in which the ionization probability is parameterized by the age of the carrier at the time when it triggers the avalanche process. We term such formulation an

*age-dependent*analysis. In what follows (including the Appendix), we will derive sets of age-dependent recurrence equations that enable us to calculate the mean gain, the excess noise factor, the probability mass function of the gain, the mean of the impulse-response function, as well as the breakdown probability, all under a dynamic electric field.

### 4.1. Mean gain

*Z*(

*x*,

*s*) (

*Y*(

*x*,

*s*)) as the totality of all electrons and holes, including the parent carrier, initiated by an electron (hole), injected at location

*x*with age

*s*. Note that by convention

*Z*(

*w*,

*s*) =

*Y*(0,

*s*) = 1 since an electron (hole) placed at the right (left) edge of the multiplication region will exit the multiplication region without ionizing. Now consider a parent electron-hole pair at location

*x*and of age

*s*. The age-dependent stochastic multiplication factor,

*M*(

*x*,

*s*), defined as the total number of electron-hole pairs generated as a result of an electron-hole pair whose initial location in the multiplication region is

*x*and whose ages are

*s*, is simply

*x*=

*w*and with age

*s*≥ 0 (recall that the age is always measured with respect to the launch instant of the dynamic electric field at time

*t*= 0), then the stochastic age-dependent gain is given by

*Z*(

*x*,

*s*) and

*Y*(

*x*,

*s*). Once we find the first and second moments of

*Z*and

*Y*we can relate them to the age-dependent mean gain via Eqs. (4) and (5), and to the age-dependent excess noise factor as shown in the Appendix. Suppose that the first ionization for a parent electron of age

*s*positioned at location

*x*occurs at location

*X*= ξ, where

_{e}*x*≤ξ≤

*w*. Note that the instant of this ionization (measured from the launch instant of the dynamic electric field at time

*t*= 0) is

*s*+ τ, where τ = (ξ-

*x*)/

*v*is the transit time of this electron in the multiplication region. Note that the two offspring electrons at ξ, who are born with common age

_{e}*s +*τ, will independently generate

*Z*

_{1}(ξ,

*s +*τ) and

*Z*

_{2}(ξ,

*s +*τ) carriers, respectively. On the other hand, the offspring hole, whose age at birth is also

*s +*τ, will generate

*Y*(ξ,

*s +*τ) carriers independently of

*Z*

_{1}(ξ,

*s +*τ) and

*Z*

_{2}(ξ,

*s +*τ). Thus, conditional on the event that the first impact ionization for the parent electron occurs at location ξ, the sum

*Z*

_{1}(ξ,

*s +*τ) +

*Z*

_{2}(ξ,

*s +*τ) +

*Y*(ξ,

*s +*τ) will simply amount to

*Z*(

*x*,

*s*). Since we can always express the mean of

*Z*(

*x*,

*s*) as the expectation of the conditional mean (given that the first ionization of the parent electron occurs at location

*X*), we can write E[

_{e}*Z*(

*x*,

*s*)] as an iterated expectation E[E[

*Z*(

*x*,

*s*) |

*X*]] by conditioning first on the location of the first ionization. Hence,where the symbol “E” denotes expectation. Now if we define the notation for the mean of the quantities as

_{e}*z*(

*x*,

*s*) = E[

*Z*(

*x*,

*s*)] and

*y*(

*x*,

*s*) = E[

*Y*(

*x*,

*s*)], then the expression on the right hand side of Eq. (6) can be cast as

*X*in (7) while taking into account the scenario when the parent electron may not impact ionize at all, we obtain the integral equation

_{e}*s*instead of a parent electron. In this case, we realize that the stochastic location of the first ionization of the parent hole,

*X*, can be in the interval [0,

_{h}*x*] (instead of [

*x*,

*w*] as in the case of the parent electron). The resulting integral equation for

*y*(

*x*,

*s*) is

*m*(

*x*,

*s*) = E[

*M*(

*x*,

*s*)], can be calculated using the identity

*m*(

*x*,

*s*) = 0.5[

*z*(

*x*,

*s*) +

*y*(

*x*,

*s*)], and the age-dependent mean gain,

*g*(

_{a}*s*) = E[

*G*(

*s*)], is simply

*m*(

*w*,

*s*) in the case of a hole-injection APD. As a special case, the usual mean gain,

*g*, of an APD under a static bias is simply

*g*=

*g*(0).

_{a}### 4.2. Mean impulse response function

*I*(

_{e}*t*,

*x*,

*s*), the age-dependent

*stochastic*impulse-response function at time

*t*initiated by an

*electron*injected at location

*x*and with age

*s*. Similarly,

*I*(

_{h}*t*,

*x*,

*s*) is the stochastic age-dependent impulse-response function at time

*t*, initiated by a

*hole*injected at location

*x*with age

*s*. Mathematically, if we define

*i*(

_{e}*t*,

*x*,

*s*) and

*i*(

_{h}*t*,

*x*,

*s*) as the mean quantities of

*I*(

_{e}*t*,

*x*,

*s*) and

*I*(

_{h}*t*,

*x*,

*s*), respectively, then we can write that

*conditional*mean of

*I*(

_{e}*t*,

*x*,

*s*) given that the first ionization of the parent electron occurs at location

*X*aswhere

_{e}*x*≤

*X*≤

_{e}*w*. On the other hand, when no ionization occurs (when

*X*>

_{e}*w*), thenwhich is simply a rectangular pulse of duration equal to the transit time of the parent electron at

*x*as it drifts across the remainder of the multiplication region. When we average Eq. (9) over all possible values of

*X*, we obtain the following integral equation

_{e}*i*(

_{h}*t*,

*x*,

*s*):

*i*(

*t*,

*s*), (in the case of a hole injection to the multiplication region at

*x*= 0) for a photon absorbed at time

*s*is then obtained using

*i*(

*t*,

*s*) =

*i*(

_{h}*t*,

*w*,

*s*).

*s*will absent and all the integral equations in this section collapse to their static-field counterparts. For example, Eqs. (10a) and (10b) collapse respectively to Eqs. (5) and (6) in [27

27. C. H. Tan, P. J. Hambleton, J. P. R. David, R. C. Tozer, and G. J. Rees, “Calculation of APD impulse response using a space- and time-dependent ionization probability distribution function,” J. Lightwave Technol. **21**(1), 155–159 (2003). [CrossRef]

**39**(3), 546–552 (1992). [CrossRef]

### 4.3. Pulse response, pulse-response bandwidth and pulse-integrated mean gain

*i*(

*t*,

*s*), is dependent on the birth time

*s*of the photogenerated parent carrier triggering the avalanche (or equivalently on the arrival time of the absorbed photon if we ignored the time it takes the carrier to be transported from the absorber to the edge of the multiplication region). Now in on-off keying optical communication, photons arrive randomly within each optical pulse according to a temporal distribution that is governed by the pulse’s power profile within the bit. Hence, the appropriate quantity to look at when assessing ISI would be the

*pulse*-

*response*function,

*i*(

_{p}*t*), rather than an impulse response. If we assume that the time origin (namely the launch time of the electric field) is the beginning of the bit, we can express

*i*(

_{p}*t*) aswhere

*p*(

_{ph}*s*) is the probability density function of arriving photons within an optical pulse of duration

*T*. Note that if the electric field is static, then

*i*(

*t*,

*s*) is simply

*i*(

*t*-

*s*), and

*i*(

_{p}*t*) would become simply the convolution of

*i*(

*t*) and

*p*(

_{ph}*t*). An alternative way to view

*i*(

_{p}*t*) is to regard it as a photon-arrival time averaged impulse response. Since early and late photons have long and short impulse responses, respectively, it would make sense to look at the average of the impulse response functions over all possible photon arrival times within each received optical pulse. In any event, by calculating the 3dB bandwidth of the Fourier transform of

*i*(

_{p}*t*), we can obtain the pulse-response bandwidth,

*B*, which combines the APD’s avalanche-duration limited bandwidth with the bandwidth of the optical pulse in each bit of duration

_{p}*T*. By defining the

*pulse-averaged gain*,which is simply the average of the age-dependent mean gain,

*g*(

_{a}*s*), we can introduce the

*pulse-integrated gain-bandwidth product*,

*GBP*, as

_{p}*GBP*collapses to the usual gain bandwidth product whenever the biasing is static.

_{p}*n*of detected photons in the optical pulse in each bit. In other words, under bit-synchronous periodic biasing, the output (charge) of an integrate-and-dump receiver

*remains proportional*to the energy in the optical pulse in each bit. Thus, while the dynamically biased APD may not be directly applicable to simple analog detection since the APD’s avalanche gain is time variant, it is a perfect fit to digital communications as the integrate-and-dump receiver maintains its linearity.

## 5. Results

*f*is set to be equal to the bit transmission rate,

_{c}*f*=

_{c}*R*. The quantities

*B*,

*C*and ϕ are free parameters that can be selected to control the overall multiplication factor and also to maximize the benefit of dynamic biasing in minimizing the tail of the pulse response of the APD. As described earlier, it is implicitly assumed that the bias signal is synchronous with the optical pulse stream. In practice, a clock recovery circuit and a phase lock loop must be employed to maintain synchronization. In addition, for simplicity we shall assume a spatially uniform electronic field, E(

*t*) =

*V*(

_{BD}*t*)/

*w*, in the multiplication region of the APD.

### 5.1. Mean impulse-response function under sinusoidal biasing

*w*= 200 nm. The ionization parameters for InP were extracted from [28

28. L. J. J. Tan, J. S. Ng, C. H. Tan, and J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. **44**(4), 378–382 (2008). [CrossRef]

^{6}cm/s. For reference, we first calculated the mean impulse-response function, triggered by a hole injected at position

*x*=

*w*= 200 nm, under a static bias of

*V*= 14.30 V. The 3dB bandwidth was then extracted from the Fourier transform of the impulse response. The calculations of the mean impulse-response function were performed according to the integral Eqs. (6), (10), (11) and (12) in [16

_{B}16. M. M. Hayat and B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. **10**(10), 1415–1425 (1992). [CrossRef]

**39**(3), 546–552 (1992). [CrossRef]

*g*= 28 for this device under the same constant bias of

*V*= 14.30 V. The usual gain-bandwidth product for this device is found to be 238 GHz, which is the same as the pulse-integrated gain-bandwidth product,

_{B}*GBP*, since the field is static in this case.

_{p}*s*. These curves were obtained by solving (10a) and (10b) numerically using the method of iterations. The dynamic electric field profile is also shown in Fig. 4 (bottom). The transit time is simply

*v*/

*w*= 2.985 ps. Note that the period of dynamic bias is equal to 5.5 transit times. (Here, the width of the optical pulse is 8.3 ps, as in a 60-Gbps NRZ bit stream.) The sinusoidal-biasing parameters were selected as follows:

*B*= 13 V,

*C*= 6 V, and ϕ = 0. These parameters were chosen, in part, so that the pulse-integrated gain,

*s*= 4 transit times (green curve) corresponds to a mean impulse response due to a parent hole of age 4 transit times after the launch of the bias at time

*t*= 0. On the other hand, the curve with

*s*= 0 transit times (red curve) corresponds to a parent hole of age 0. It is important to note the change in the shape of the impulse response function as a result of dynamic biasing. Unlike the static-bias case, the impulse response corresponding to

*s*= 0 no longer peaks at the parent hole’s transit time (i.e., one transit time after the trigger instant) but instead it peaks at a later time in response to the increase in the instantaneous electric field in the multiplication region. Another interesting point is that the weakest impulse response is obtained when the age of the hole is approximately

*s*= 2.6 transit times (magenta curve). This observation suggests that the sinusoidal dynamic bias should be delayed appropriately in order for the last photon in the pulse to have the weakest impulse response. The point to be made here is that the time delay in the dynamic bias must be optimized to produce the best pulse-integrated gain-bandwidth product. We have found empirically that a phase angle of ϕ = π/3 gives good results.

*s*values is quite high as seen below. This is due to the rise in the field initially, where a high gain is built up, followed by a drop in the field causing the shortening of the impulse response as the probability of the avalanche terminating is high. For example, when

*s*= 0 the age-dependent bandwidth is 62 GHz and the age-dependent mean gain is 82, while in the static-bias case the bandwidth is 8.5 GHz and the gain is 28. Meanwhile, if we look at larger age values (corresponding to photons arriving late in the pulse), we will see that the gain is generally small and so is the bandwidth. For example, at approximately

*s*= 5 the age-dependent bandwidth is 23 GHz while the age-dependent mean gain is 3. This is because carriers have a reduced probability of impact ionizing due to the low field in the second half of the pulse.

### 5.2. Gain-bandwidth-product improvement

*i*(

_{p}*t*), defined in (11). Figure 6 shows

*i*(

_{p}*t*) once with the sinusoidal dynamic-field profile used in Fig. 5, and once with the static reverse bias. (For simplicity, in this example

*p*(s) in (11) is assumed to be constant at 1/

_{ph}*T*, namely, we assume a uniformly distributed random stream of photons). Note that this is simply the average of the age-dependent impulse responses. We observe that in comparison to

*i*(

_{p}*t*) for the static bias, the tail of

*i*(

_{p}*t*) in the sinusoidal-bias case is much reduced. The pulse-integrated bandwidth in the sinusoidal-bias case is 43.3 GHz, compared to 8.5 GHz in the static-bias case. The pulse-integrated gain in all three cases is about 28. Hence, the calculations predict an enhancement in the pulse-integrated gain-bandwidth product from 238 GHz in the static-bias case to a pulse gain-bandwidth product of 1169 GHz in the dynamic-bias case. This shows that a sinusoidally biased APD with the bias parameters described earlier can increase the pulse gain-bandwidth product of an APD by a factor of 5 compared to the same APD operated under the conventional static biasing scheme. We expect the results do extend to thicker multiplication regions provided that the ratio of the bit duration to the transit time is kept constant.

13. D. C. Herbert and E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. **48**(7), 1475–1477 (2001). [CrossRef]

## 6. Conclusions

*any*APD that has a poor avalanche-duration performance beyond traditional limits inherited from the conventional static biasing. Another projected benefit of dynamic biasing is that it may allow the relaxation of the often stringent requirements on the minimum width of the multiplication region, as normally done to enhance the APD speed. This, in turn, would lead to a reduction of the electric field in the multiplication region, which reduces tunneling current. Future efforts will focus on implementation of the proposed dynamic scheme as well as the rigorous analysis of the receiver sensitivity under bit-synchronous dynamic biasing.

## Appendix

## A.1. Age-dependent excess noise factor

*z*

_{2}(

*x*,

*s*) and

*y*

_{2}(

*x*,

*s*). Since we can always express the second moment of

*Z*(

*x*,

*s*) as the expectation of the conditional second moment of

*Z*(

*x*,

*s*) given that the first ionization of the parent electron (triggering

*Z*(

*x*,

*s*)) occuring at location

*X*, we can write

_{e}*X*explicitly in terms of the pdf of

_{e}*X*, we obtain

_{e}*y*

_{2}(

*x*,

*s*):

*z*(

*x*,

*s*) and

*y*(

*x*,

*s*) have already been computed by first solving the pair of equations in (8).

## A.2. Age-dependent breakdown probability

**39**(3), 546–552 (1992). [CrossRef]

*x*in the multiplication region and of age

*s*(relative to the launch instant of the dynamic electric field) impact ionizes for the first time at location ξ, then the probability that the parent electron generates a

*finite*number of offspring carriers is precisely the product of the probabilities that each of the two offspring electrons and offspring hole created at ξ with age

*s*+ (ξ-

*x*)/

*v*generates a

_{e}*finite*number of offspring carriers. (Implicit in this statement is that each carrier acts independently of the other carries, which is a correct assumption since in this paper we do not include any feedback effect from the created charges on the electric field.) Note that in the special case when the parent electron exits the multiplication region without ionizing, the conditional probability that produces a finite number of carriers is trivially equal to one.

*P*(

_{Z}*x*,

*s*) = P{

*Z*(

*x*,

*s*) < ∞} and

*P*(

_{Y}*x*,

*s*) = P{

*Y*(

*x*,

*s*) < ∞}, and note that P{

*Z*(

*x*,

*s*) < ∞ |

*X*= ξ} = P{

_{e}*Z*(ξ,

*s*+ (ξ-

*x*)/

*v*)) < ∞}

_{e}^{2}P{

*Y*(ξ,

*s*+ (ξ-

*x*)/

*v*.)) < ∞}, while P{

_{e}*Z*(

*x*,

*s*) < ∞ |

*X*>

_{e}*w*} = 1. By averaging the above equation over all possible

*X*we obtain

_{e}*P*(

_{Z}*x*,

*s*) and

*P*(

_{Z}*x*,

*s*) are calculated by solving the nonlinear coupled integral equations in (A5) numerically (using iterations, for example), the breakdown probability is calculated. For example, for a hole-injection APD, the age-dependent breakdown probability for a photon absorbed at time

*s*, is simply

## A.3. Probability mass function of the age-dependent multiplication factor

*Z*(

*x*,

*s*) and

*Y*(

*x*,

*s*). Here we state the equations without proof. Let

*f*(

_{Z}*x*,

*s*,

*m*) = P{

*Z*(

*x*,

*s*) =

*m*} and

*f*(

_{Y}*x*,

*s*,

*m*) = P{

*Y*(

*x*,

*s*) =

*m*},

*m*= 1, 2, 3, …. Then, they must satisfy the following coupled integral equations

*= if*

_{k}*k*=0 and it is zero otherwise, and “*” denotes discrete convolution with respect to the variable

*m*. An alternative form of (A7) can be obtained by taking the discrete (time) Fourier transform with respect to the discrete variable

*m*, which yields the characteristic function. More precisely, define

## Acknowledgments

## References and links

1. | B. E. A. Saleh and M. C. Teich, |

2. | P. Bhattacharya, |

3. | R. G. Smith and S. D. Personick, “Receiver Design for Optical Fiber Communication Systems,” |

4. | B. L. Kasper and J. C. Campbell, “Multigigabit-per-second avalanche photodiode lightwave receivers,” J. Lightwave Technol. |

5. | T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. |

6. | P. Sun, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Statistical correlation of gain and buildup time in APDs and its effects on receiver performance,” J. Lightwave Technol. |

7. | D. S. G. Ong, J. S. Ng, M. M. Hayat, P. Sun, and J. P. R. David, “Optimization of InP APDs for high-speed lightwave systems,” J. Lightwave Technol. |

8. | K. Makita, T. Nakata, K. Shiba, and T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. |

9. | Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics |

10. | W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express |

11. | Y. Kang, Z. Huang, Y. Saado, J. Campbell, A. Pauchard, J. Bowers, and M. J. Paniccia, “High performance Ge/Si avalanche photodiodes development in Intel,” Opt. Fiber Comm. Conf. & Expo. (OFC/NFOEC), 1–3 (2011). |

12. | J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. |

13. | D. C. Herbert and E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. |

14. | R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron. Dev. |

15. | M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. |

16. | M. M. Hayat and B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. |

17. | G. Agrawal, |

18. | M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. |

19. | N. Yasuoka, H. Kuwatsuka, M. Makiuchi, T. Uchida, and A. Yasaki, “Large multiplication-bandwidth products in APDs with a thin InP multiplication layer,” Proc. IEEE Laser & Electro Opt. Soc. Ann. Meeting LEOS' 2003, 999–1000 (2003). |

20. | N. Namekata, S. Adachi, and S. Inoue, “1.5 GHz single-photon detection at telecommunication wavelengths using sinusoidally gated InGaAs/InP avalanche photodiode,” Opt. Express |

21. | J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE |

22. | M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. |

23. | J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In |

24. | F. Osaka, T. Mikawa, and T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga |

25. | M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. |

26. | M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. |

27. | C. H. Tan, P. J. Hambleton, J. P. R. David, R. C. Tozer, and G. J. Rees, “Calculation of APD impulse response using a space- and time-dependent ionization probability distribution function,” J. Lightwave Technol. |

28. | L. J. J. Tan, J. S. Ng, C. H. Tan, and J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. |

**OCIS Codes**

(040.0040) Detectors : Detectors

(040.5160) Detectors : Photodetectors

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.4510) Fiber optics and optical communications : Optical communications

(040.1345) Detectors : Avalanche photodiodes (APDs)

**ToC Category:**

Detectors

**History**

Original Manuscript: December 20, 2011

Revised Manuscript: March 2, 2012

Manuscript Accepted: March 9, 2012

Published: March 22, 2012

**Citation**

Majeed M. Hayat and David A. Ramirez, "Multiplication theory for dynamically biased avalanche photodiodes: new limits for gain bandwidth product," Opt. Express **20**, 8024-8040 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-8024

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### References

- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).
- P. Bhattacharya, Semiconductor Optoelectronic Devices (Prentice Hall, 1996).
- R. G. Smith and S. D. Personick, “Receiver Design for Optical Fiber Communication Systems,” Semiconductor Devices for Optical Communication, H. Kressel ed. (Springer-Verlag, 1980).
- B. L. Kasper, J. C. Campbell, “Multigigabit-per-second avalanche photodiode lightwave receivers,” J. Lightwave Technol. 5(10), 1351–1364 (1987). [CrossRef]
- T. Nakata, I. Watanabe, K. Makita, T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. 36(21), 1807–1808 (2000). [CrossRef]
- P. Sun, M. M. Hayat, B. E. A. Saleh, M. C. Teich, “Statistical correlation of gain and buildup time in APDs and its effects on receiver performance,” J. Lightwave Technol. 24(2), 755–768 (2006). [CrossRef]
- D. S. G. Ong, J. S. Ng, M. M. Hayat, P. Sun, J. P. R. David, “Optimization of InP APDs for high-speed lightwave systems,” J. Lightwave Technol. 27(15), 3294–3302 (2009). [CrossRef]
- K. Makita, T. Nakata, K. Shiba, T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. 2, 234–240 (2005).
- Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009). [CrossRef]
- W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express 17(15), 12641–12649 (2009). [CrossRef] [PubMed]
- Y. Kang, Z. Huang, Y. Saado, J. Campbell, A. Pauchard, J. Bowers, and M. J. Paniccia, “High performance Ge/Si avalanche photodiodes development in Intel,” Opt. Fiber Comm. Conf. & Expo. (OFC/NFOEC), 1–3 (2011).
- J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004). [CrossRef]
- D. C. Herbert, E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. 48(7), 1475–1477 (2001). [CrossRef]
- R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron. Dev. 13(1), 164–168 (1966). [CrossRef]
- M. M. Hayat, B. E. A. Saleh, M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. 39(3), 546–552 (1992). [CrossRef]
- M. M. Hayat, B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. 10(10), 1415–1425 (1992). [CrossRef]
- G. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
- M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002). [CrossRef]
- N. Yasuoka, H. Kuwatsuka, M. Makiuchi, T. Uchida, and A. Yasaki, “Large multiplication-bandwidth products in APDs with a thin InP multiplication layer,” Proc. IEEE Laser & Electro Opt. Soc. Ann. Meeting LEOS' 2003, 999–1000 (2003).
- N. Namekata, S. Adachi, S. Inoue, “1.5 GHz single-photon detection at telecommunication wavelengths using sinusoidally gated InGaAs/InP avalanche photodiode,” Opt. Express 17(8), 6275–6282 (2009). [CrossRef] [PubMed]
- J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).
- M. A. Saleh, M. M. Hayat, B. E. A. Saleh, M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. 47(3), 625–633 (2000). [CrossRef]
- J. S. Ng, C. H. Tan, J. P. David, G. Hill, G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003). [CrossRef]
- F. Osaka, T. Mikawa, T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga0.33In0.67As0.70P0.30,” Appl. Phys. Lett. 45(3), 292–293 (1984). [CrossRef]
- M. M. Hayat, W. L. Sargeant, B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. 28(5), 1360–1365 (1992). [CrossRef]
- M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002). [CrossRef]
- C. H. Tan, P. J. Hambleton, J. P. R. David, R. C. Tozer, G. J. Rees, “Calculation of APD impulse response using a space- and time-dependent ionization probability distribution function,” J. Lightwave Technol. 21(1), 155–159 (2003). [CrossRef]
- L. J. J. Tan, J. S. Ng, C. H. Tan, J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. 44(4), 378–382 (2008). [CrossRef]

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