Appendix C - The Relationship Between V_BR and E_C

C.1 - Introduction

The results of Chapter 4 rely on the empirical observation that there is an approximate one-to-one relationship between the breakdown voltage, V_BR, and the peak electric field at breakdown, E_C. This appendix briefly considers this relationship from a theoretical viewpoint. It is shown that this one-to-one relationship can be predicted mathematically for the case of the two-sided abrupt junction, with uniform doping levels N_A and N_D. It is also shown that the mathematics rapidly become intractable for cases more complex than this, such as the diffused rectifier discussed in Chapter 4.

C.2 - Theory

The avalanche breakdown process is most accurately described by ionization equations of the form [Van70]:

(C.1)

where α_n and α_p are the numbers of electron-hole pairs produced by ionization per unit length (in the direction of the electric field) per electron and hole, respectively. For silicon, the relevant ionization coefficients are [Van70]:

(C.2)

The total number of electron hole pairs generated in the depletion region (of width W) of a p-n junction by a single pair initially generated at x is then given by [Ghan77]:

(C.3)

Breakdown occurs when M(x) becomes infinite.

Using equations (C.1) in (C.3) for the case of M(x) → ∞ does not produce simple mathematical results. However, a simpler "effective" ionization rate equation is also available, which combines the effects of hole and electron ionization [Van70]:

(C.4)

where

(C.5)

These values are valid in the range 1.75 x 10⁵ V/cm < |E| < 6.40 x 10⁵ V/cm.

With this simplified ionization equation, a simple condition for M(x) → ∞ can be obtained [Koko66]:

(C.6)

For the purposes of examining the relationship between V_BR and E_C, consider a diode junction of arbitrary profile, with the proviso that:

(C.7)

In this case, E > 0 in the depletion region, so our equations can be simplified slightly by removing the absolute value operators from equation (C.4). Also, the depletion region will be defined as:

(C.8)

with the junction at x = 0.

Then equation (C.6) can be rewritten, taking into consideration (C.4), (C.7) and (C.8), as:

(C.9)

Consideration of Poisson's equation for this structure:

(C.10)

linked with the conditions in (C.7) and (C.8) shows that the electric field is monotonic with distance in the range -x₁ < x < 0, and is also monotonic (in the opposite direction) in the separate range 0 < x < x₂. This allows us to establish a one-to-one relationship between E(x) and x in each of these ranges (separately), and thus rearranging (C.10) to find an expression for dx in terms of dE allows (C.9) to be rewritten in the form:

(C.11)

where the variable of integration has been changed. The fact that E = E_C at x = 0, and that E = 0 at x = x₁ and x = x₂ has also been used in deriving (C.11). Since, as noted above, E(x) and x are related uniquely in each region, N(x) can be written as N(E), eliminating the distance coordinate entirely from (C.11). this yields, with some rearranging:

(C.12)

where N₁(E) is the doping as a function of electric field in the range -x₁ < x < 0, and where N₂(E) is the doping as a function of electric field in the range 0 < x < x₂.

In a pn junction with uniform doping on the opposite sides of the junction, N₁(E) and N₂(E) are constants, equal to -N_D and +N_A, respectively. This greatly simplifies the mathematics, since (C.12) can be reduced to:

(C.13)

It is now simple to show that E_C and V_BR are related by noting that for an abrupt junction as described above, the voltage applied across the junction can be written from simple diode theory as:

(C.14)

(The approximation comes from the fact that the diode built-in voltage has been ignored, since it is typically orders of magnitude smaller than the breakdown voltages considered in this thesis.) Thus, combining (C.13) and (C.14) yields:

(C.15)

Evaluating the integral and using the tabulated "exponential integral" function E₁(z), which is given by [Abra65]:

(C.16)

allows (C.15) to be rewritten as:

(C.17)

This expression shows that V_BR is directly related to E_C, regardless of the choice of N_A or N_D. Furthermore, it is simple to show that V_BR decreases monotonically with increasing E_C, so V_BR and E_C have a one-to-one relationship.

(The expression in (C.17) is similar to one in [Koko66] which was derived for the case of a one-sided junction. However, since there is only one geometrical parameter to vary in a one-sided junction, the light-side doping N, both V_BR and E_C can be written as a function of N. Thus N can be eliminated from these two expressions and V_BR can be written in terms of E_C. To show convincingly that the relationship between E_C and V_BR is independent of doping parameters, more than one doping parameter must be variable, as is the case for the two-sided junction considered above.)

The fact that V_BR decreases with increasing E_C is worth some reflection, because at first glance it appears to be counter-intuitive, since (C.14) predicts that V_BR ∝ E_C². This can be explained by examining the nature of equation (C.4). The ionization rate α_EFF increases very rapidly with increasing electric field due to the exponential nature of (C.4). Thus, the depletion region width, W_DR, required to satisfy (C.6) falls very rapidly with increasing E_C. Since

(C.18)

an increase in E_C will be more than offset by the corresponding decrease in W_DR. Comparing (C.17) and (C.18) shows that W can be written as a function of E_C:

(C.19)

A plot of W_DR versus E_C is for the critical electric field range of interest in this thesis is shown in Figure C.1. It is also compared to a simple 1/E_C function, showing that W_DR decreases proportionately faster than E_C increases.

Figure C.1 - Depletion region width as a function of E_C. The 4000/E_C curve shows that W_DRdecreases proportionately faster than E_Cincreases. E_C is in V/cm, W is in cm.

In the asymptotic limit of E_C → ∞, W_DR → 1/a. However, this is of limited interest, since the electric fields required to approach this limit are much higher than any field that can be practically generated in a simple pn junction.

A quick survey of the mathematics in this chapter shows that obtaining similar analytical results for a diffused structure, or a generalized structure, is not possible. The mathematics rapidly become intractable. Furthermore, we know from the simulations of Chapter 4 that there is only an approximate one-to-one relationship between E_C and V_BR for the diffused devices, which is more difficult to prove mathematically than the exact one-to-one relationship that exists for the abrupt structures discussed above. However, it is not too great a leap of logic to say the that one-to-one relationship that has been demonstrated theoretically for the abrupt junctions is likely to hold (approximately) for diffused junctions as well, since one can look at the abrupt junction as a limiting case of a diffused structure.