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Ionisation Avalanche Noise

Background Theory

Chapter Introduction

This chapter summarises the background theory of the impact ionisation process including avalanche multiplication and the associated avalanche excess noise. Several authors have attempted to explain the energy gain and loss mechanisms and the carrier ionisation behaviour in semiconductors since the 1950s. However the scattering mechanisms are not well known in even well characterised semiconductors and due to the lack of accurate knowledge of the band structure beyond the conduction and valence minima, several assumptions were made in these approximate theories, each with its own merits and degree of accuracy. Wolff [1] assumed carriers gain energy through many collisions before reaching the ionisation threshold while in contrast Shockley [2] and [3] assumed carriers that ionised are “lucky” in avoiding energy losing collisions. A good exposition of these theories can be found in two excellent reviews by Stillman and Wolff [4] and by Capasso [5]. The emphasis of this chapter is to describe the local and non-local models that retain the essential physics of the ionisation process without going into the microscopic details of the transport properties of semiconductor physics. These would enable the experimental results to be accurately replicated using the developed models and the material dependent parameters.

Impact Ionisation Theory

Impact ionisation is the key mechanism responsible for avalanche multiplication of carriers in semiconductor devices subjected to high reverse fields. Impact ionisation is a process when an energetic carrier (electron or hole) in a high reverse junction field, in collision with atoms in the crystal lattice, loses energy to create an electron-hole pair.

for a typical direct band gap semiconductor structure. An electron in the conduction band gains energy from the applied field across the semiconductor, and moves to higher states in the conduction band. However in the process of acceleration, the free carrier also experience non-ionisation collision processes such as phonon scattering events, which randomises the carrier momentum. Phonon scattering may involve carriers gaining energy (absorptive), losing energy (emissive) or exchanging momentum (elastic). Carriers on average will lose energy by phonon scattering since the emissive phonon scattering rate is proportional to n + 1 whereas the phonon absorption rate is proportional to n, where n is the phonon occupation number that depends on the phonon energy,

(2.1)

where is Boltzmann's constant and T is the absolute temperature.

When the electron gains sufficient energy above the threshold energy, which must at least be the band gap energy, the electron may either transfer all or some of its kinetic energy to a valence electron, promoting it to the conduction band and leaving a hole behind in the valence band. Subjected to the electric field, the primary carrier together with the newly created electron-hole pair can subsequently gain energy to cause further ionisation events to produce additional electron-hole pairs as they move through the lattice.

Since both energy and momentum needs to be conserved, carriers can only initiate impact ionisation when they gain the threshold energy, E_th required for generation of electron-hole pairs, from the electric field. By applying these and assuming simple parabolic bands with equal electron and hole effective mass, Wolff [1] showed that a direct band gap structure require a minimum energy of 1.5 E_gto cause impact ionisation. In reality, the electron and hole mass are different, and energy bands are non-parabolic. Furthermore, at high electric fields, most of the ionising carriers have kinetic energy much greater than the threshold energy [6]. The energy at which ionisation becomes important is not directly related to that calculated via minimum energy and momentum requirements, but to the high energy band structure [7]. Values of E_th are frequently adjusted to obtain agreement with experimental data when used in different theoretical models.

Each carrier type has a characteristic ionisation coefficient dependent on the material, temperature and field. These coefficients describe the rate of creation of electron-hole pairs, and are defined as the reciprocal of the average distance travelled by electrons and holes in the direction of the electric field to initiate an ionisation event. In general, the ionisation coefficient of electrons and holes are represented by α and β respectively and are position and field dependent.

Carriers that enter the high field region with energy less than the ionisation threshold energy must traverse a dead space distance before they acquire sufficient energy to impact ionise. Models which disregard the dead space, such as Stillman and Wolfe's model [4] are called local models. Models that account for the dead space are called non-local models. Both models will be discussed in the following sections.

Local Impact Ionisation Model

Avalanche Multiplication Theory

A finite number of successive impact ionisation events result in an increase in the number of carriers or the multiplication of primary carriers to produce current gain. Figure 2.2 illustrates the chain of impact ionisation events resulting due to an electron being injected into a high field region of length w, in which a uniform electric field is applied across the region. The electron injected at x = 0, travelling from left to right can generate a new electron hole pair via impact ionisation after travelling a random ionisation path length. The newly generated electron and the original electron behave in a statistically identical but independent manner. The newly generated hole can also create a new electron hole pair after travelling a random ionisation path length. The new electrons and holes may create more electron hole pairs as they traverse the avalanche region until they exit. This multiplication process ends when all the electrons and holes have exited the avalanche region at x = w and x = 0, respectively.

Assuming that the carrier's ionisation probability depends solely on the local electric field, Stillman & Wolfe's analytical expression [4] for the position dependent average multiplication for a primary carrier pair generated within w, at x is given by

.(2.2)

Due to an arbitrary injection at x within the avalanche region, a distributed generation of electron hole pairs in the avalanche region at a rate G(x), is obtained. The mean multiplication factor due to this mixed carrier injection, M_mix, is expressed as a weighted average of the gain:

. (2.3)

The multiplication factor for pure electron injection where an electron is injected at x = 0, reduces equation (2.2) to

. (2.4)

The multiplication factor for pure hole injection where a hole is injected at x = w, is similarlyexpressed as

. (2.5)

If the electric field is constant across the high field region, for the case of ideal p⁺-i-n⁺structures, a and b can be treated as constants, hence equation (2.5) and (2.6) can be simplified further to give

(2.6)

and

(2.7)

respectively.

If a = b , electrons and holes have equal probability to impact ionise. Equation (2.6) is thus simplified to:

(2.8)

Avalanche breakdown occurs when aw = 1, when each injected carrier undergoes on average one impact ionisation event as it traverses the multiplication region.

For the case when b = 0, which implies there is no carrier feedback, equation (2.8) reduces to

. (2.9)

In this case, there is no avalanche breakdown owing to absence of feedback from the holes and M rises exponentially with a.

Rearranging equations (2.6) and (2.7), a and b can be determined by measuring M_e and M_h in ideal p⁺-i-n⁺structures with uniform electric fields based on the simplified assumptions outlined above, using the equations below:

(2.10)

and

.(2.11)

Excess Noise Theory

The stochastic nature of impact ionisation process leads to fluctuation of the gain of the avalanching device about the mean value, . This fluctuation introduces noise associated with impact ionisation, in addition to shot noise since the ionisation of any individual carrier has a certain probability of occurrence due to the randomness in the position of ionisation carriers and in the number of secondary carriers produced by the initiating carrier. This gives rise to a distribution of gain around the mean gain.

The excess noise is quantified by a figure-of-merit known as the excess noise factor, F. Considering a standard deviation for one multiplication event, with mean gain, F is given by:

(2.12)

Since , the expression of F can be rearranged to be as follows:

(2.13)

such that the mean square noise current is given by

(2.14)

where I_pr is the primary current and B is the measurement bandwidth.

The excess noise due to carriers generated arbitrarily within the ionisation region is given by:

(2.15)

Assuming uniform field and local theory [8], the excess noise factor can be mathematically related to the gain and ratio of the electron and hole ionisation coefficients, . The excess noise factor for pure electron injection is given by:

(2.16)

and excess noise for pure hole injection is given by:

(2.17)

For a material to exhibit low avalanche noise, k must be much smaller or much greater than unity, with the carrier type with the higher ionisation coefficient initiating multiplication. A material with dissimilar ionisation coefficients will have lower carrier feed back since only one carrier type has a high probability to ionise. This reduces the statistical fluctuation in the number of multiplication events as the ionisation process becomes more deterministic, and hence reduces the level of excess noise. However it is a prerequisite that the dominant carrier initiates the ionisation process or else the secondary carriers may generate long multiplication chains and result in a large fluctuation in current, seen as large excess noise.

If k equals unity, where a = b , for the case of electron-initiated multiplication, the ionisation events due to holes serve as feed back mechanism to the avalanche process. The physical process of carrier feedback has a large effect on noise. The statistical fluctuation in the multiplication in this case is large and therefore corresponds to a large amount of excess noise.

This local noise model provides useful design rules for APDs employing thick avalanche widths but no longer holds true as the avalanche width decreases. As the device size shrinks leading to higher electric fields, the dead space may become a significant fraction of the total distance travelled by the carriers before ionising. McIntyre's local noise model will not be able to account for the reduced multiplication at low fields and will overestimate the excess noise for thin devices.

Non-local Impact Ionisation Model

Okuto and Crowell [9], pointed out that in thin devices where the electric field is high, a and b are not only functions of the electric field but also depend on the position within the device.

Several groups have proposed modifying the local analysis to account for the dead space and determine a and b. Okuto and Crowell [10] used a first order approximation that assumes the ionisation coefficient for any carrier is equal to zero until it travels the dead space distance and thereafter depends on the local electric field. Bulman [11] simplified [9]'s equations by ignoring hole dead space effects which did not contribute to multiplication at low electric fields. Flitcroft et al. [12] has shown that the multiplication and breakdown voltages from submicron devices could be accurately predicted using simple correction to the local analysis as described by Wood et al. [13]

Monte Carlo (MC) models, in general, provide the most rigorous approach to analysing high field carrier transport but are considered to be time consuming and uneconomical for device design and analysis. Complex MC models which incorporate the full band structure [14] or the simplified versions [15], [16] have correctly described experimental results but necessitate many adjustable parameters.

Hayat et al. [17] developed a technique that uses the PDF of the ionisation path length to formulate a closed form expression of integral equations for predicting multiplication and excess noise in APDs. McIntyre [18] proposed an extension of Hayat's model [17], corroborated by measurements on GaAs APDs [19]. A model that is equivalent to Hayat's recurrence method [17] is the random path length model (RPL) formulated by Ong et al. [15], which is a stochastic technique that relies on randomly chosen ionisation path lengths to calculate the probability of carrier ionising and thus determine the multiplication and excess noise of APDs.

Within these two models, the local effective ionisation coefficients are replaced by enabled ionisation coefficients, a* and b*. These enabled coefficients describe the carriers that have already travelled the dead space and are generally different from those deduced using local analysis.

In both the recurrence model and RPL model, the ionisation path length PDF of a carrier in an uniform electric field is typically represented by a displaced exponentially decaying function at position x, h_e(x) for electrons and h_h(x) for holes as:

(2.18)

(2.19)

where the dead space, d_eand d_h for electrons and holes respectively,can be estimated in the most simplistic way as the distance travelled ballistically by the electrons and holes:

(2.20)

where q is the electronic charge, ξ is the electric field, E_the(h) is the threshold energy for electrons (holes).

In the approach used by Hayat et al. [17], the total number of carriers produced by an electron and hole injected including the primary carrier at 0 < x < w, are defined as the random variables Z(x) and Y(x) respectively. The recurrence equation describing the electron ensemble average is written as:

. (2.21)

The first term in equation (2.21) describes the case when the primary electron does not ionise at all. If the primary electron ionises at x < x' < w, the primary electron and the newly generated electron and hole, can further impact ionise to give random sum of electrons and holes. These values are statistically dependent and the total number of carriers produced by the original electron is the sum of the three values. The second term of equation (2.21) integrates the product of the mean sum of the carriers produced by an ionisation at location x' and the probability that it occurs. The same deduction for holes gives

. (2.22)

A similar set of expressions of the second moment of the random variables can also be formulated by taking the ensemble average of the mean square values:

(2.23)

(2.24)

where in the second term of equation (2.21) and (2.22) is replaced by the mean square sum, taking into account that each carrier is independent.

The set of coupled linear integral equations (equations 2.21-2.24) can be solved numerically by successive iteration method and using boundary conditions z(x) = y(0) = z₂(w) = y₂(0) = 1. The mean gain at position x can be computed from

(2.25)

The excess noise can be calculated by:

(2.26)

The derivations of the excess noise expression in equation (2.26) can be found in Appendix A.

References

[1] P. A. Wolff, "Theory of electron multiplication in silicon and germanium", Phys. Rev., vol. 95, no. 6, pp. 1415-1420, 1954.

[2] W. Shockley, "Problems related to P-N junctions in Silicon and Germanium", Solid State Elect., vol. 2, pp. 35-67, 1961.

[3] B. K. Ridley, "Lucky drift mechanism for impact ionisation in semiconductor", J. Phys. C: Solid State Phys., vol. 16, 1983

[4] G. E. Stillman and C. M. Wolfe (editors R. K.Willardson and A. C Beer), "Avalanche photodiodes", Semiconductors and semimetals, Academic Press, Inc., vol. 12, pp. 291-393, 1977.

[5] F. Capasso, "Physics of avalanche photodiodes", Semiconductors and Semimetals, Lightwave Communications Technology, Part D: Photodetectors, W. T. Tsang, Ed., Orlando: Academic Press. Inc., vol. 22, pp. 3-172, 1985.

[6] J. Bude and K. Hess, "Thresholds of impact ionization in semiconductors", J. Appl. Phys., vol. 72, no. 8, pp. 3554-3561, Oct. 1992.

[7] J. Allam, "Universal dependence of avalanche breakdown on bandstructure: choosing materials for high-power devices", Jpn. J. Appl. Phys., vol. 36, no. 1(3B), pp. 1529-1542, Mar. 1997.

[8] R. J. McIntyre, "Multiplication noise in uniform avalanche diodes", IEEE Trans. Electron Dev., vol. ED-13, no. 1, pp. 164-168, Jan. 1966.

[9] Y. Okuto and C. R. Crowell, "Ionization coefficients in semiconductors: A nonlocalized property", Phys. Rev. B, vol. 10, no. 10, pp. 4284-4296, 1974.

[10] Y. Okuto and C. R. Crowell, "Threshold energy effect on avalanche breakdown voltage in semiconductor junctions", Solid-State Elect., vol. 18, no. 2-D, pp. 161-168, 1975.

[11] G. E Bulman, V. M. Robbins, and G. E. Stillman, "The determination of impact ionisation coefficients in (100) Gallium Arsenide using avalanche noise and photocurrent multiplication measurements", IEEE Trans. Electron Dev., vol. ED-32, no. 11, pp. 2454-2466, 1985.

[12] R. M. Flitcroft, J. P. R. David, P. A. Houston, and C. C. Button, "Avalanche multiplication in GaInP/GaAs single heterojunction bipolar transistors", IEEE Trans. Electron Dev., vol. 45, no. 6, pp. 1207-1212, Jun. 1998.

[13] M. H. Woods, W. C. Johnson, and M. A. Lampert, "Use of a schottky barrier to measure impact ionization coefficients in semiconductors", Solid State Electron., vol. 16, no. 3, pp. 381-394, Mar. 1973.

[14] V. Chandramouli, C. M. Maziar, and J. C. Campbell, "Design considerations for high performance avalanche photodiode multiplication layers", IEEE Trans. Electron Dev., vol. 41, no. 5, pp. 648-654, May 1994.

[15] D. S. Ong, K. F. Li, G. J. Rees, J. P. R. David, and P. N. Robson, "A simple model to determine multiplication and noise in avalanche photodiodes", J. Appl. Phys., vol. 83, no. 6, pp. 3426-3428, Mar. 1998.

[16] S. A. Plimmer, J. P. R. David, D. S. Ong, and K. F. Li, "A simple model including the effects of dead space", IEEE Trans. Electron Dev., vol. 46, no. 4, pp. 769-775, Apr. 1999.

[17] 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., vol. 39, no. 3, pp. 546-552, Mar. 1992.

[18] R. J. McIntyre, "A new look at impact ionization - part I: a theory of gain, noise, breakdown probability, and frequency response", IEEE Trans. Electron Dev., vol. 46, no. 8, pp. 1623-1631, Aug. 1999.

[19] P. Yuan, K. A. Anselm, C. Hu, H. Nie, C. Lenox, A. L. Holmes, B. G. Streetman, J. C. Campbell, and R. J. McIntyre, "A new look at impact ionization - part II: gain and noise in short avalanche photodiodes", IEEE Trans. Electron Dev., vol. 46, no. 8, pp. 1632-1639, Aug. 1999.

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