Bit-error rates for optical receivers using avalanche photodiodeswith dead space

Bit-error rates are computed for an on-off keying optical communication system using avalanche photodiodes (APDs). We use a model for the APD that includes dead space and the finite response time. Dead space is the minimum distance that a newly generated carrier must travel in order to acquire sufficient energy to become capable of causing an impact ionization in the multiplication region of the APD. The detector's finite impulse response and its randomness are important for high data-rate systems. Using an exact analysis, we show that the presence of dead space enhances the performance at relatively low data rates. Using a Gaussian approximation technique with the exact mean and variance, we demonstrate that dead space degrades the performance at-high data rates since it is responsible for longer tails in the impulse response function of the APD, which in turn increases the effect of intersymbol interference     

Effect of dead space on the excess noise factor and time responseof avalanche photodiodes

The effect of dead space on the statistics of the gain process in continuous-multiplication avalanche photodiodes (APDs) is determined using the theory of age-dependent branching processes. The dead space is the minimum distance that a newly generated carrier must travel in order to acquire sufficient energy to cause an impact ionization. Analytical expressions are derived for the mean gain, the excess noise factor, and the mean and standard deviation of the impulse response function, for the dead-space-modified avalanche photodiode (DAPD), under conditions of single carrier multiplication. The results differ considerably from the well-known formulas derived by R.J. McIntyre and S.D. Personick in the absence of dead space. Relatively simple asymptotic expressions for the mean gain and excess noise factor are obtained for devices with long multiplication regions. In terms of the signal-to-noise ratio (SNR) of an optical receiver in the presence of circuit noise, it is established that there is a salutory effect of using a properly designed DAPD in place of a conventional APD. The relative merits of using DAPD versus a multilayer (superlattice) avalanche photodiode (SAPD) are examined in the context of receiver SNR; the best choice turns out to depend on which device parameters are used for the comparison     

Effect of dead space on gain and noise in Si and GaAs avalanchephotodiodes

The effect of dead space on the mean gain, the excess noise factor, and the avalanche breakdown voltage for Si and GaAs avalanche photodiodes (APDs) with nonuniform carrier ionization coefficients are examined. The dead space, which is a function of the electric field and position within the multiplication region of the APD, is the minimum distance that a newly generated carrier must travel in order to acquire sufficient energy to become capable of causing impact ionization. Recurrence relations in the form of coupled linear integral equations are derived to characterize the underlying avalanche multiplication process. Numerical solutions to the integral equations are obtained and the mean gain and the excess noise factor are computed     

Statistical properties of the impulse response function ofdouble-carrier multiplication avalanche photodiodes including the effectof dead space

The statistical properties of the impulse response function of double-carrier multiplication avalanche photodiodes (APDs) are determined, including the effect of dead space, i.e., the minimum distance that a newly generated carrier must travel in order to acquire sufficient energy to become capable of causing an impact ionization. Recurrence equations are derived for the first and second moments and the probability distribution function of a set of random variables that are related, in a deterministic way, to the random impulse response function of the APD. The equations are solved numerically to produce the mean impulse response, the standard deviation, and the signal-to-noise ratio (SNR), all as functions of time     

Avalanche noise characteristics of thin GaAs structures withdistributed carrier generation [APDs]

It is known that both pure electron and pure hole injection into thin GaAs multiplication regions gives rise to avalanche multiplication with noise lower than predicted by the local noise model. In this paper, it is shown that the noise from multiplication initiated by carriers generated throughout a 0.1 μm avalanche region is also lower than predicted by the local model but higher than that obtained with pure injection of either carrier type. This behavior is due to the effects of nonlocal ionization brought about by the dead space; the minimum distance a carrier has to travel in the electric field to initiate an ionization event     

Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes: theory and experiment [Al/sub 0.6/Ga/sub 0.4/As/GaAs]

The history-dependent recurrence theory for multiplication noise in avalanche photodiodes (APDs), developed by Hayat et al., is generalized to include inter-layer boundary effects in heterostructure APDs with multilayer multiplication regions. These boundary effects include the initial energy of injected carriers as well as bandgap-transition effects within a multilayer multiplication region. It is shown that the excess noise factor can be significantly reduced if the avalanche process is initiated with an energetic carrier, in which case the initial energy serves to reduce the initial dead space associated with the injected carrier. An excess noise factor reduction up to 40% below the traditional thin-APD limit is predicted for GaAs, depending on the operational gain and the multiplication-region's width. The generalized model also thoroughly characterizes the behavior of dead space as a function of position across layers. This simultaneously captures the effect of the nonuniform electric field as well as the anticipatory nature of inter-layer bandgap-boundary effects.     

Avalanche multiplication properties of GaAs calculated from spatially transient ionisation coefficients

In this paper the high field phenomenon of avalanche multiplication in a GaAs p-i-n infrared detector is studied using a Monte-Carlo simulation. The Lucky-Drift model of impact ionization is used to give the characteristic lengths for transport through the device. The transport is then modelled by generating motion consistent with the probability functions derived from the mean free paths. This produces a spatially transient ionization coefficient for each carrier and allows the realistic statistical simulation of avalanche multiplication. Properties such as mean gain, multiplication noise and the transient response to a photonic pulse have been calculated and explained for a length of i-GaAs, with an emphasis on short active region phenomena. The effect on the ionization coefficients of a periodic field change has been investigated. It has been found that the effective carrier deadspace is approx. 1.35 times the absolute deadspace. The transient current calculations indicate the narrow bandwidth of this type of device. The presence of a periodic field change, caused by periodic δ-doping, was found to increase both electron and hole ionization coefficients by different proportions.     

Mean gain of avalanche photodiodes in a dead space model

Based on a first order expansion of the recursive equations, we derive approximate analytical expressions for the mean gain of avalanche photodiodes accounting for dead space effects. The analytical solutions are similar to the popular formula first obtained in local approximation, provided that the ionization coefficients, α and β, are replaced with suitable effective ionization coefficients depending on dead space. The approximate solutions are in good agreement with the exact numerical solutions of the recursive equations for p-i-n devices as well as for photodiodes with nonconstant electric field profile. We also show that dead space causes non negligible differences between the values of the effective ionization coefficients entering in carrier continuity equations, the carrier ionization probability per unit length and the ionization coefficients derived by experimenters from multiplication measurements     

A simplified approach to time-domain modeling of avalanchephotodiodes

A simplified algorithm for calculating time response of avalanche photodiodes (APDs) is presented. The algorithm considers the time course of avalanche processes for the general case of position-dependent double-carrier multiplications including the dead space effect. The algorithm is based on a discrete time setting ideally suited for computer modeling and can be applied to any APD structure. It gives a fast and accurate estimation of the time and frequency response of APDs. As an example, the present method is applied to InP-InGaAs separate absorption, grading, charge, and multiplication (SAGCM) APDs. The variation of multiplication pain with bias voltage and 3-dB electrical bandwidth at different multiplication gain obtained using the new algorithm show good agreement with experimental results. The algorithm can be used to study temperature dependence of APD characteristics and can be easily extended to calculate the excess noise factor     

Effect of stochastic dead space on noise in avalanche photodiodes

A stochastic dead-space model for impact ionization is developed and used to study the effect of the soft nature of the ionization capability of carriers on the excess noise factor of avalanche photodiodes. The proposed model is based on the rationale that the gradual, or soft, transition in the probability density function (PDF) for the distance from birth to impact ionization can be viewed as that resulting from uncertainty in the dead space itself. The resulting soft PDF, which is parameterized by a tunable softness parameter, is used to establish the limitations of the existing hard-threshold ionization models in ultrathin multiplication layers. Calculations show that for a fixed operational gain and fixed average dead space, the excess noise factor tends to increase as a result of the softness in the PDF in very thin multiplication layers (viz, <70 nm), or equivalently, under high applied electric fields (viz., >800 kV/cm). A method is proposed for extracting the softness parameter from noise versus multiplication measurements.     

Low multiplication noise thin Al0.6Ga0.4Asavalanche photodiodes

Avalanche multiplication and excess noise were measured on a series of Al_0.6Ga_0.4As p⁺in⁺ and n⁺ip⁺ diodes, with avalanche region thickness, w ranging from 0.026 μm to 0.85 μm. The results show that the ionization coefficient for electrons is slightly higher than for holes in thick, bulk material. At fixed multiplication values the excess noise factor was found to decrease with decreasing w, irrespective of injected carrier type. Owing to the wide Al_0.6Ga_0.4As bandgap extremely thin devices can sustain very high electric fields, giving rise to very low excess noise factors, of around F~3.3 at a multiplication factor of M~15.5 in the structure with w=0.026 μm. This is the lowest reported excess noise at this value of multiplication for devices grown on GaAs substrates. Recursion equation modeling, using both a hard threshold dead space model and one which incorporates the detailed history of the ionizing carriers, is used to model the nonlocal nature of impact ionization giving rise to the reduction in excess noise with decreasing w. Although the hard threshold dead space model could reproduce qualitatively the experimental results, better agreement was obtained from the history-dependent model     

Gain-bandwidth characteristics of thin avalanche photodiodes

The frequency-response characteristics of avalanche photodiodes (APDs) with thin multiplication layers are investigated by means of a recurrence technique that incorporates the history dependence of ionization coefficients. In addition, to characterize the autocorrelation function of the impulse response, new recurrence equations are derived and solved using a parallel computer. The mean frequency response and the gain-bandwidth product are computed and a simple model for the dependence of the gain-bandwidth product on the multiplication-layer width is set forth for GaAs, InP, Al_0.2Ga_0.8As, and In_0.52Al_0.48 As APDs. It is shown that the dead-space effect leads to a reduction (up to 30%) in the bandwidth from that predicted by the conventional multiplication theory. Notably, calculation of the power-spectral density of the photocurrent reveals that the presence of dead space also results in a reduction in the fluctuations in the frequency response. This result is the spectral generalization of the reduction in the excess noise factor in thin APDs and reveals an added advantage of using thin APDs in ultrafast receivers     

Impact-ionization and noise characteristics of thin III-V avalanchephotodiodes

It is, by now, well known that McIntyre's localized carrier-multiplication theory cannot explain the suppression of excess noise factor observed in avalanche photodiodes (APDs) that make use of thin multiplication regions. We demonstrate that a carrier multiplication model that incorporates the effects of dead space, as developed earlier by Hayat et al. provides excellent agreement with the impact-ionization and noise characteristics of thin InP, In_0.52 Al_0.48As, GaAs, and Al_0.2Ga_0.8As APDs, with multiplication regions of different widths. We outline a general technique that facilitates the calculation of ionization coefficients for carriers that have traveled a distance exceeding the dead space (enabled carriers), directly from experimental excess-noise-factor data. These coefficients depend on the electric field in exponential fashion and are independent of multiplication width, as expected on physical grounds. The procedure for obtaining the ionization coefficients is used in conjunction with the dead-space-multiplication theory (DSMT) to predict excess noise factor versus mean-gain curves that are in excellent accord with experimental data for thin III-V APDs, for all multiplication-region widths     

A new approach for computing the bandwidth statistics of avalanchephotodiodes

A new approach for characterizing the avalanche-buildup-time-limited bandwidth of avalanche photodiodes (APD's) is introduced which relies on the direct knowledge of the statistics of the random response time. The random response time is the actual duration of the APD's finite buildup limited random impulse response function. A theory is developed characterizing the probability distribution function (PDF) of the random response time. Recurrence equations are derived and numerically solved to yield the PDF of the random response time. The PDF is then used to compute the mean and the standard deviation of the bandwidth. The dependence of the mean and the standard deviation of the bandwidth on the APD mean gain and the ionization coefficient ratio is investigated. Exact asymptotics of the tail of the PDF of the response time are also developed to aid the computation efficiency. The technique can be readily applied to multiplication models which incorporate dead space and can be extended to cases for which the carrier ionization coefficient is position dependent     

The frequency response of avalanching photodiodes

The Townsend equations for avalanche breakdown in back biased p-n junctions may be derived from the transport equations for semiconductors. Integral solutions of the time independent equations are well known. An integral solution of the time dependent equations is given for multiplication by one carrier only. An exact solution is given for multiplication by two carriers with equal ionization coefficients in a constant junction field. The Townsend equations are nonlinear because of space charge effects. It is shown, however, that the nonlinearity, which imposes an upper limit on the current multiplication possible, is not important until the total multiplied current approaches the space charge limited current for the junction. Assuming multiplication is due to one carrier, frequency response curves are calculated for constant and linear junction fields and for a generation rate, due to photon absorption, which is either uniform or given by a delta function at the junction boundary. The curves indicate a relatively slight dependence of the frequency response on multiplication. Frequency response curves are also given for multiplication by both carriers with equal ionization coefficients when the junction field is constant. In this case the frequency response decreases continuously as the multiplication is increased. For multiplication by two carriers with unequal ionization coefficients, the frequency response is independent of multiplication until the product of the multiplication and the ratio of the ionization coefficients approaches one. Thereafter the frequency response decreases with multiplication.     

Effect of impact ionization in the InGaAs absorber on excess noise of avalanche photodiodes

The effects of impact ionization in the InGaAs absorption layer on the multiplication, excess noise and breakdown voltage are modeled for avalanche photodiodes (APDs), both with InP and with InAlAs multiplication regions. The calculations allow for dead space effects and for the low field electron ionization observed in InGaAs. The results confirm that impact ionization in the InGaAs absorption layer increases the excess noise in InP APDs and that the effect imposes tight constraints on the doping of the charge control layer if avalanche noise is to be minimized. However, the excess noise of InAlAs APDs is predicted to be reduced by impact ionization in the InGaAs layer. Furthermore the breakdown voltage of InAlAs APDs is less sensitive to ionization in the InGaAs layer and these results increase tolerance to doping variations in the field control layer.     

Monte Carlo simulations of the bandwidth of InAlAs avalanche photodiodes

We present a Monte Carlo simulation of the bandwidth of an InAlAs avalanche photodiode with an undepleted absorber. The carrier velocities are simulated in the charge layer and the multiplication region. It is shown that the velocity overshoot effect is not as significant as simpler models have suggested. At high electric field intensity, the electron effective saturation velocity is only slightly higher when impact ionization is significant, compared with when impact ionization is absent. The simulated 3 dB bandwidth is consistent with experiments for gains up to 50.     

Dead space effect on the wavelength dependence of gain and noise inavalanche photodiodes

We extend the dead space model proposed by Hayat et al. in order to determine the wavelength-dependent multiplication mean gain 〈G(λ)〉 and excess noise factor F(λ) in the case of mixed electron and hole injection, as it is the case when photons are absorbed within the multiplication region. We compare the predictions of the model with measurements performed on a silicon ultraviolet-selective avalanche photodiode with submicron thick multiplication region. We show that the multiplication gain is constant in the visible and near-infrared part of the spectrum, and increases in the UV range by a factor of 1.8. Furthermore, the excess noise factor is minimal for UV radiation and increases rapidly for longer wavelengths. It appears that the extended dead space model is very adequate at predicting the gain and noise measurement results. In order to unambiguously determine the effect of the dead space, we compare the predictions of our model with those of McIntyre's local noise model. The latter qualitatively describes the wavelength dependence of the gain, but greatly overestimates the excess noise factor     

Avalanche Noise Characteristics in Submicron InP Diodes

We report excess noise factors measured on a series of InP diodes with varying avalanche region thickness, covering a wide electric field range from 180 to 850 kV/cm. The increased significance of dead space in diodes with thin avalanche region thickness decreases the excess noise. An excess noise factor of F = 3.5 at multiplication factor M = 10 was measured, the lowest value reported so far for InP. The electric field dependence of impact ionization coefficients and threshold energies in InP have been determined using a non-local model to take into account the dead space effects. This work suggests that further optimization of InP separate absorption multiplication avalanche photodiodes (SAM APDs) could result in a noise performance comparable to InAlAs SAM APDs.     

Avalanche multiplication in GaInP/GaAs single heterojunctionbipolar transistors

The electron multiplication factors in GaInP/GaAs single heterojunction bipolar transistors (HBT's) have been measured as a function of base-collector bias for a range of GaAs collector doping densities. In the lowest doped (5×10¹⁴ cm^-3) thick collector the multiplication is determined by the local electric field. As the collector doping increases, the measured multiplication is found to be significantly reduced at low values of multiplication from that predicted by the electric field profile. However, good agreement is always found at high multiplication, close to breakdown. This reduction in multiplication at low electric fields is attributed to the dead space, the minimum distance over which carriers must travel before gaining the ionization threshold energy. A simple correction for the dead space is proposed, allowing the multiplication to be accurately predicted even in heavily doped structures