An analytical approach to study the effect of carrier velocities onthe gain and breakdown voltage of avalanche photodiodes

A simplified model for calculating gain and breakdown voltage of avalanche photodiodes (APDs) having constant ionization coefficients in their multiplication layer is presented. Good agreement is seen between the calculated results and the experimental data for published InP-InGaAs separate absorption, grading, charge, and multiplication (SAGCM) APDs. The model denotes that the gain and the breakdown voltage have a dependence on the carrier velocity ratio that is not predicted by conventional models. Hence, by comparing the calculated and measured static characteristics of the APD, one can estimate the velocity of minority carriers in the multiplication region of the device     

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.     

Effect of dead space on avalanche speed [APDs]

The effects of dead space (the minimum distance travelled by a carrier before acquiring enough energy to impact ionize) on the current impulse response and bandwidth of an avalanche multiplication process are obtained from a numerical model that maintains a constant carrier velocity but allows for a random distribution of impact ionization path lengths. The results show that the main mechanism responsible for the increase in response time with dead space is the increase in the number of carrier groups, which qualitatively describes the length of multiplication chains. When the dead space is negligible, the bandwidth follows the behavior predicted by Emmons but decreases as dead space increases     

Excess noise analysis of separate absorption multiplication regionsuperlattice avalanche photodiodes

The operation of a separate absorption multiplication region avalanche photodiode (SAM-APD) introduces noise as results of randomness in the number and in the position at which dark carrier pairs are generated, randomness in the photon arrival number, randomness in the carrier multiplication, and the number and the position of the photogenerated carriers in the bulk of the diode. The dark current results in a smaller mean multiplication gain in excess noise factor versus mean multiplication plot due to the partial multiplication process of these generated carriers compared to the usual values associated with carriers injected at one edge of the diode. Previous analyses of mean multiplication and excess noise factor for an arbitrary superposition of injected carriers are extended to allow the presence of dark carriers in the multiplication region under the model, which admits variation (with position) of the band-gap, dark generated rate, and ionization coefficients with each stage for the superlattice APD, and the presence of impact ionization in the absorption region. The calculations reveal the presence of impact ionization carriers in the absorption region which results in a larger excess noise factor than the usual values associated with carriers injected at one edge of the device, and fits well with experimental results     

Calculation of the diffusion curvature related avalanche breakdown in high-voltage planar p-n junctions

Avalanche multiplication calculations are performed in high-voltage planar p-n junctions to determine breakdown voltage limitations imposed by curvature effects. The issue of choice of ionization coefficient for avalanche multiplication is discussed. From the calculations, a series of design curves and equations are generated which relate the breakdown voltage and peak electric field to those of an ideal junction of the same doping profile, the critical parameters being the substrate doping concentration, the diffusion profile, and the ratio of the radius of curvature to the substrate depletion width for the ideal one-dimensional case. With appropriate distance normalization, these curves and equations can be reduced to a single curve and a single equation. The agreement between theory and experiment is consistently good provided the correct ionization coefficients are used in the theory.     

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     

On the small-signal equivalent circuit of p-n junctions in the condition of finite carriers multiplication

A new version of the equivalent circuit of the space-charge region of the reverse biased p-n junction is evaluated and the expressions for the circuit parameters are given. The basic idea is to separate the equivalent circuit in to the “conductive” and “displacement” branches and in this way the equivalent circuit parameters have physical meaning. The results are applicable to conditions of the finite multiplication factors and unequal ionization coefficients in a wide range of frequency and in the presence of the generation of carriers (thermal or outside induced) in the space-charge region. The numerical results for two complementary abrupt silicon p-n junctions and for low-frequency are given. The equivalent circuit of the multiplication noise source of the p-n junction is discussed.     

Effects of Ionization Velocity and Dead Space on Avalanche Photodiode Bit Error Rate

A model is presented for the bit error rate (BER) contributed by the receiver in an optical telecommunications system that includes the effects of ionizing carrier velocity and dead space in the avalanche photodiode (APD) and of additive circuit noise. The probability distribution functions of bit charge used to calculate BER are not, as is commonly assumed, Gaussian, confirming the need to directly compute the receiver statistics. Integrating the current over the central section of the bit period can minimize intersymbol interference. The assumption that carriers travel to ionization with infinite velocity underestimates BER in InP APDs with short avalanche region widths, and overestimates BER when . Models assuming constant carrier velocity or allowing for velocity enhancement predict distinctly different BER over a wide range of avalanche width and multiplication because of the manner in which the current evolves during the bit period.     

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     

Switching response of graded-base PN junction diodes

This paper extends the earlier analysis by Kingston of the switching response of a uniform-base diode to a graded-base diode. It concerns the time required to switch a diode from a forward-biased to a reverse-biased condition. The current transient can be separated into two phases: 1) the constant current phase during which the carrier density at the junction changes gradually from a forward-biased to a reverse-biased condition, and 2) the nonconstant current phase during which the injected carriers stored in the base region gradually disappear. In the present analysis, it is found that in a graded-base diode where the impurity concentration decreases from the emitter junction towards the base contact, the time for the constant current phase is greatly shortened because of favorable initial carrier distribution. The effect is already significant if the impurity concentration changes by a factor from 3 to 1 from the emitter junction to the base contact. To shorten the nonconstant current phase, however, a much larger change of impurity concentration, say of the order from 500 to 1, from the emitter junction to the base contact is needed.     

The Effect of Temperature on the Ionization Coefficient in Silicon†

The avalanche breakdown of p-n junction diodes across their space charge regions is known to be analogous to the Townsend mechanism for gases. Electric charge carriers, which gain sufficient energy from the field, are able to produce secondary electron-hole pairs. Both the holes and the electrons can themselves have ionizing collisions and thus the process leads to an avalanche. An important factor, controlling the breakdown, is the ionization coefficient a, defined as the number of electron-hole pairs produced by a carrier moving unit distance in the direction of the field.

This paper presents the results of nn investigation into the effect of lattice temperature on the ionization coefficient. This has been achieved by observing the breakdown voltage of a range of silicon diodes with either step or linear graded junctions, and applying simple and well-known relationships between ionization coefficient and breakdown voltage.

Measurements have been made over the temperature range 77 to 400°K, and for field strengths from 4 to 9 × 10⁵ volts/cm. Results show the ionization to become more efficient with decrease in temperature over this range of field strength. Temperature is found to have a greater effect at the lower field strength. This is shown to be consistent with modern theory.     

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     

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     

Time and frequency response of avalanche photodiodes with arbitrarystructure

A method is developed for solving the coupled transport equations that describe the electron and hole currents in a double-carrier multiplication (DCM) avalanche photodiode (APD) of arbitrary structure. This solution makes it possible to determine the time and frequency response of the device. The injection can be localized to one or both ends of the multiplication region, or distributed throughout an extended region where multiplication can occur concurrently. The results are applied to conventional APDs with position-dependent carrier ionization rates (e.g., a separate-absorption-grading-multiplication APD) as well as to superlattice multiquantum-well (MQW) structures where the ionizations are localized to bandgap transition regions. The analysis may also be used to determine the dark current and include the carrier trapping at the heterojunction interfaces. The results indicate that previous time-dependent theories only account for the tail of the time response under high-gain conditions and are inaccurate for high-speed devices     

Multiplication noise in uniform avalanche diodes

A general expression is derived from which the spectral density of the noise generated in a uniformly multiplying p-n junction can be calculated for any distribution of injected carriers. The analysis is limited to the white noise part of the noise spectrum only, and to diodes having large potential drops across the multiplying region of the depletion layer. It is shown for the special case in whichbeta = kalpha, wherekis a constant and α and β are the ionization coefficients of electrons and holes, respectively, that the noise spectral density is given by2eI_{in}M^{3}[1 + (frac{1 - k}{k})(frac{M - 1}{M})^{2}]where M is the current multiplication factor and Iinthe injected current, if the only carriers injected into the depletion layer are holes, and by2eI_{in}M^{3}[1 - (1 - k)(frac{M - 1}{M})^{2}]if the only injected carriers are electrons. An expression is also derived for the noise power which will be delivered to an external load for the limitM rightarrow infin.     

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.     

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.     

Theory of carrier multiplication and noise in avalanche devices—Part I: One-carrier processes

A new general theory is given for the carrier multiplication factor M and for the noise in devices in which avalanching occurs due to impact ionization by one type of carriers, such as in the channel of JFET's at sufficiently high channel fields. The theory involves the consideration of the discrete statistical process thatNionizations can occur per carrier transit in an avalanche region of finite lengthw. ForN rarr infin, the known results of the various continuous type theories, due to Tager, van der Ziel, and Chenette (also McIntyre, and Personick if the ionization coefficient of one type of carriers is set equal to zero) are recovered; these formulations are thus shown to be asymptotic theories. For finiteN, the results show a continuous transition from the onset of avalanche (N = 1), as recently measured by Rucker and van der Ziel, to the asymptotic case. Curves covering the entire region are presented.     

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     

Computer studies of the effect of electron and hole current multiplication factors on the d.c. and microwave properties of symmetrical Si DDR IMPATT devices

Computer studies are presented on the effect of carrier current multiplication on the d.c. field and current profiles and the small-signal admittance of a symmetrical Si double-drift region (DDR) IMPATT diode, taking into account the realistic field dependence of ionization rate and drift velocity of charge carriers and also the effect of mobile space-charge. The d.c. field and current profiles indicate that the lowering of the electron current multiplication (M_n) is more effective than the lowering of hole current Multiplication factor (M_p) in modifying the d.c. properties of Si DDR devices. The computer-aided small-signal analysis carried out for the same structure shows that, a lowering of M_n leads to a sharp decrease of the peak value of the small-signal negative conductance at a fixed d.c. current density which is accompanied by a shift of the frequency range of oscillation towards the higher frequency side.