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.     

Hg0.56 Cd0.44 Te 1.6- to 2.5-μm avalanchephotodiode and noise study far from resonant impact ionization

An investigation was made on the avalanche multiplication and impact ionization processes in p-n^--n⁺ junctions formed in Hg_0.56Cd_0.44Te solid solutions. Photocurrent multiplication was determined at 300 K in planar p-n^- -n⁺ structures characterized by a breakdown voltage of 30 V. The experimental results were used to calculate the electron, α, and hole, β, ionization coefficients. It was found that α is greater than β because Δ, the spin-orbit splitting energy, is higher than the bandgap energy. These experimental results were in satisfactory agreement with multiplication noise measurements using separate electron and hole injection     

Statistical features of the avalanche multiplication build-up in semiconductors with different coefficients of impact ionization

The statistical fluctuations of the efficiency of the avalanche multiplication are studied for the Geiger- mode avalanche photodiodes. The resulting distribution function of the partial multiplication coefficients is characterized by an anomalously broad variance (on the order of the mean value). The expressions for the partial feedback coefficients are derived in terms of the mean gain, and the corresponding dependences on the over-voltage across the diode are calculated. An algorithm for the Monte-Carlo simulation of the avalanche process is presented, and the results of the statistical tests are compared with the theoretical dependence.     

A new look at impact ionization-Part I: A theory of gain, noise,breakdown probability, and frequency response

Impact ionization in thick multiplication regions is adequately described by models in which the ionization coefficients are functions only of the local electric field. In devices with thin multiplication lengths, nonlocal effects become significant, necessitating new models that account for the path that a carrier travels before gaining sufficient energy to impact ionize. This paper presents a new theory that incorporates history-dependent ionization coefficients, and it is shown that this model can be utilized to calculate the low-frequency properties of avalanche photodiodes (APD's) (gain, noise, and breakdown probability in the Geiger mode) and the frequency response. A conclusion of this work is that an ionization coefficient is not a fundamental material characteristic at a specific electric field and that any experimental determination of ionization coefficients is valid only for the particular structure on which the measurement was performed     

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     

Temperature dependence of the electron impact ionization in InGaP-GaAs-InGaP DHBTs

Temperature dependence of electron impact ionization in InGaP-GaAs-InGaP double heterojunction bipolar transistors (DHBTs) were comprehensively studied in the temperature range of 300 to 450 K. It has been found that, as the temperature increases, the electron multiplication in the InGaP collector is found to be weakly reduced, which results in a relatively small negative temperature dependence of junction breakdown. The temperature dependence of electron impact ionization at elevated temperatures for InGaP material is investigated based on the electron multiplications measured from the InGaP collector region. An empirical expression is obtained to predict the electron ionization coefficients at elevated temperature up to 450 K in the electric field range of 380 to 650 kV/cm. As compared to InP and GaP binaries, the ternary InGaP shows a lower electron ionization coefficient and much weaker temperature dependence. We found that, introducing additional scattering mechanism such as alloy scattering would provide a better interpretation on the low electron impact ionization and its weak temperature dependence observed in InGaP.     

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.     

Dead-space-based theory correctly predicts excess noise factor forthin GaAs and AlGaAs avalanche photodiodes

The conventional McIntyre carrier multiplication theory for avalanche photodiodes (APDs) does not adequately describe the experimental results obtained from APDs with thin multiplication-regions. Using published data for thin GaAs and Al_0.2Ga_0.8As APDs, collected from multiplication-regions of different widths, we show that incorporating dead-space in the model resolves the discrepancy. The ionization coefficients of enabled carriers that have traveled the dead space are determined as functions of the electric field, within the confines of a single exponential model for each device, independent of multiplication-region width. The model parameters are determined directly from experimental data. The use of these physically based ionization coefficients in the dead-space multiplication theory, developed earlier by Hayat et al. provide excess noise factor versus mean gain curves that accord very closely with those measured for each device, regardless of multiplication-region width. It is verified that the ratio of the dead-space to the multiplication-region width increases, for a fixed mean gain, as the width is reduced. This behavior, too, is in accord with the reduction of the excess noise factor predicted by the dead-space multiplication theory     

A new look at impact ionization-Part II: Gain and noise in shortavalanche photodiodes

For Part I see R.J. McIntyre, ibid., vol.46, no.8, pp.1623-31 (1999). In Part I, a new theory for impact ionization that utilizes history-dependent ionization coefficients to account for the nonlocal nature of the ionization process has been described. In this paper, we will review this theory and extend it with the assumptions that are implicitly used in both the local-field theory in which the ionization coefficients are functions only of the local electric field and the new one. A systematic study of the noise characteristics of GaAs homojunction avalanche photodiodes with different multiplication layer thicknesses is also presented. It is demonstrated that there is a definite “size effect” for thin multiplication regions that is not well characterized by the local-field model. The new theory, on the other hand, provides very good fits to the measured gain and noise. The new ionization coefficient model has also been validated by Monte Carlo simulations     

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.     

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     

On the time dependency of the avalanche process in semiconductors

An equation is derived for the time dependent current in the avalanche region of a uniform diode, for the case of one charged carrier of either electron or hole is injected, for unequal ionization rates and drift velocities of electrons and holes. The appearing time-constant differs from earlier results, derived on the basis of the well known intrinsic response time. The noise characteristics of the current are established for a Poissonian injection of an electron or hole. The factor which describes the influence of the statistics of the multiplication for low frequencies, is also found to differ from previous published results. At high current multiplication factors the present study is in agreement with previous work.     

碲镉汞雪崩光电二极管的发展

雪崩光电二极管(APD)工作在足够的反偏下,光生载流子到达耗尽区并诱发放大过程.碲镉汞(MCT)电子和重空穴有效质量之间的非对称性会产生不相等的电子电离系数和空穴电离系数,提供一个由单独一类载流子诱发的碰撞电离过程,从而形成具有包括接近无噪声增益在内的“理想”APD特征的电子雪崩光电二极管(EAPD).对于包括低光子数...     

A model for dark current and multiplication in HgCdTe avalanche photodiodes

In this paper we report the calculated results of the dark current and multiplication factor in MBE grown HgCdTe avalanche photodiodes with separate absorption and multiplication (SAM-APD). The device architecture used for this analysis comprises the following layers: p⁺ contact, p junction, n⁻ multiplication, n charge sheet, n⁻ absorber, and n⁺ contact. Various leakage current mechanisms are considered and the generation-recombination term is found to be the dominant one for this device structure. However, experimental reverse bias I-V characteristics reported earlier by T. de Lyon et al. shows a large deviation from ideality, which can not be explained in terms of bulk leakage current mechanism. To explain the large difference between experimental and theoretical data we consider that the dominant generation-recombination current is multiplied through impact ionization process. To validate this assumption, multiplication is calculated as a function of reverse bias. Electric field profile is obtained and the multiplication is computed using the ionization coefficients and avalanche gain equations. Breakdown voltage is found to be 85 V for room temperature operation in agreement with available data in the literature. The theoretical I-V curves considering multiplication are compared with the experimental ones and a close agreement is found which validate this model.     

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     

Normalized theory of impact ionization and velocity saturation in nonpolar semiconductors via a Markov chain approach

This paper presents an investigation of the velocity, energy, and impact ionization distributions in nonpolar semiconductors at very high fields. The treatment uses a finite Markov chain formulation. When optical phonon collisions and impact ionization are the major scattering mechanisms in the semiconductor, a transition matrix which characterizes the transition probabilities between virtual states defined by small discrete energy intervals can be easily computed. The resulting matrix provides the means not only to study the impact ionization phenomenon but also the steady state transport velocity and energy distribution of the charge carriers at high electrical fields and a given lattice temperature. In addition, the effects on the transport properties due to either an abrupt infinite (AI) or a finite energy dependent (FED) ionization cross-section above the ionization threshold energy are examined. The calculated avalanche transport velocity shows excellent agreement with the experimental data in Si obtained by Duh and Moll. The resulting calculations when extrapolated to a lower field also agree favorably with existing saturation drift velocity data in n and p type Si and p type Ge. The energy distribution is shown to be strongly affected by the choice of the model for the energy dependence of the ionization cross-section. One of the main applications of the results is to assist investigation of the non-localized nature of electron and hole avalanche ionization coefficients previously noted by Okuto and Crowell (O-C). The present results for this spatial distribution can replace O-C's intuitively chosen exponential approximation. The spatial ionization distribution generated by the present calculation is essentially exponential with a threshold energy dark space. This result provides a useful kernel for a more precise formulation in studies that relate impact ionization coefficients to charge multiplication data. The normalized ionization coefficients obtained from the AI model are very similar to Baraff's calculation as are the FED model results after appropriate normalization. Simple analytical expressions with meaningful asymptotic results for the average ionization energy and the ionization coefficient are also derived from the present data. These results are applicable for a range of different energy dependence of the ionization cross section provided that the average energy for pair production is used as the effective threshold parameter.     

Temperature dependence of impact ionization in GaAs

The temperature dependence of electron and hole impact ionization in gallium arsenide (GaAs) has been determined from photomultiplication measurements at temperatures between 20 K and 500 K. It is found that impact ionization is suppressed by increasing temperature because of the increase in phonon scattering. Temperature variations in avalanche multiplication are shown to decrease with decreasing avalanching region width, and the effect is interpreted in terms of the reduced phonon scattering in the correspondingly reduced ionization path length. Effective electron and hole ionization coefficients are derived and are shown to predict accurately multiplication characteristics and breakdown voltage as a function of temperature in p/sup +/in/sup +/ diodes with i-regions as thin as 0.5 /spl mu/m.     

Wavelength dependence of multiplication noise in silicon avalanche photodiodes

Theoretical and experimental results on wavelength dependence of multiplication noise in silicon avalanche photodiodes are described. When the photodiode has a p-n⁺-junction and is illuminated from the n⁺-side, multiplication noise increases by decreasing optical wavelength. Effective ionization coefficient ratio keffis equal tokexp (2Kw_{a}) for a uniform junction electric field, wherekis the ratio of ionization coefficients of electrons α and holes β. The multiplication noise depends on the product of optical absorption coefficientKand the avalanche-region width wa. Calculations show that there exists an optimum wafor minimizing multiplication noise at a given wavelength. Theoretical results are shown to agree with results of experiments on diodes with a low-high-low impurity profile. Measured ionization coefficient ratiokvalues are 0.04 and 0.08 at 0.811- and 0.633-µm wavelength, respectively.     

The integral for multiplication factor M with single or equal ionization coefficients

General expressions for the multiplication factorMare derived for both step and linear graded junctions by integrating the common form for the ionization coefficientalpha(E)= alpha_{infin}exp(-b/E). Equal ionization coefficients are assumed. The results are compared with a commonly used approximation.