Chapter 4 - Calculation of V_BR for Diffused Diodes

4.1 - Introduction

The purpose of this thesis is to develop step recovery diodes that can operate at voltages much higher than those currently available. Hence, they must also have higher breakdown voltages, and a rapid method for estimating V_BR is required. Despite the widespread use of diffused psn rectifiers (where the "s" represents a lightly doped region) few results have been published on calculating their breakdown voltage. Numerous authors [Koko66], [Warn72], [Wils73], [Bali87] have considered the breakdown voltage of single diffused junctions of the form:

(4.1)

Bulucea [Bulu91] has extended this by calculating the breakdown voltages for low-voltage ps_nn rectifiers with both a single diffused p region and an epitaxial n region. However high voltage psn diodes often have two diffused regions, both of which influence the breakdown voltage.

This chapter considers the breakdown voltage of rectifiers with the structure shown in Figure 4.1, corresponding to:

(4.2)

with N(x) > 0 taken to mean net acceptor doping, and N(x) < 0 net donor doping. N_S1 and N_S2 are the surface concentrations, and N_B is the wafer background doping. This structure can be made in practice by implanting and diffusing p and n type dopants into opposite sides of a wafer of thickness L.

Figure 4.1. Doping profile of the devices considered in this chapter.

Although breakdown voltages can be calculated by numerically solving the ionization integrals using simulators such as Medici [Medi93], it is still of interest to have a simple, fast method of estimating the breakdown voltage for use in device optimization, particularly when several parameters must be optimized. A new method of estimating V_BR is presented in this chapter.

4.2 - Basic Method

Determining the breakdown voltage due to impact ionization to a high degree of precision is a task best suited to numerical simulation, given the nonlinear nature of the problem. However, it is fairly straightforward to obtain an estimate by assuming that breakdown occurs at a given critical electric field E_C, rather than by considering the field dependent ionization coefficients.

The breakdown voltage will be determined using the following assumptions:

1. Breakdown occurs when a given critical electric field E_C is exceeded at the metallurgical junction. The method of determining E_C is discussed below.

2. Avalanche breakdown is the only breakdown mechanism considered. Tunneling and thermal effects are not included. (The doping levels and gradient are much too low in the devices considered in this thesis for tunneling to be of any significance. The justification for ignoring thermal effects in this thesis is provided in Appendix E.)

3. The mobile charge in the space charge regions is negligible. Hence the space charge is determined entirely by the doping profile. This is a reasonable assumption for slowly-varying conditions or near the end of a reverse recovery transient. This assumption also implies that there is only one space charge region, with the electrostatic junction at the metallurgical junction x_mj.

4. The space charge region is assumed to have sharp boundaries at x₁ and x₂, with x₁ < x_mj < x₂. This is the usual depletion region approximation.

If the metallurgical junction location is represented by x_mj, which can be determined by numerically solving (4.2) for N(x_mj) = 0 and given N_S1, N_S2, N_B, λ₁, λ₂, and L, then the boundary x₁ (the left boundary) can be determined by using Poisson's equation and integrating the space charge between x₁ and x_mj, varying x₁ until the maximum electric field (which is at x_mj) is equal to the critical field. Analytically,

(4.3)

Substituting (4.2) gives

(4.4)

Equation (4.4) can be rewritten as:

(4.5)

The boundary x₁ can then be determined, knowing Ec and the doping profile factors (N_S1, N_S2, N_B, λ₁, λ₂, and L).

The boundary x₂ can be obtained by balancing the positive and negative space charge such that:

(4.6)

Substituting (4.2) into (4.6) and evaluating the integrals yields:

(4.7)

Thus given x₁ and the doping profile factors, x₂ can be determined.

The breakdown voltage can then be calculated using

(4.8)

If (4.2) is substituted into (4.8), one obtains

(4.9)

Thus, given the doping profile of the diode and the critical electric field, the breakdown voltage can be determined.

This method relies heavily on the proper choice of E_C. Unfortunately, E_C is neither a physical parameter (i.e. the underlying solid-state physics is not described by E_C, rather it is governed by ionization coefficients), nor a constant. Empirical expressions relating E_C to doping parameters are available for the textbook cases of n+ p abrupt diodes and linearly graded diodes [Bali87], but no expressions are available for the case of the diffused doping profile considered in equation (4.2). This chapter presents a new empirical method of estimating E_C, and hence breakdown voltages, for diffused profiles. This method provides good accuracy and very fast speed compared to device simulators.

4.3 - Calculating E_C

Figure 4.2 shows the constant V_BR contours in λ-L space that result when a constant critical breakdown field of E_C = 225 kV/cm is assumed, and V_BR is calculated using the procedure outlined in the previous section. (For Fig. 4.2, N_S1 = N_S2 = 10¹⁷ cm^-3, N_B = 1.85x10¹⁴ cm^-3, and it has been assumed that λ₁ = λ₂ = λ, which is a reasonable assumption if boron and phosphorus are the p and n dopants, respectively.) Figure 4.3 shows a similar plot generated from Medici simulations, and Figure 4.4 shows the difference of the values in Figure 4.2 relative to these Medici simulations. The Medici simulations used the Van Overstraeten and De Man data [Van70] for the ionization rates in silicon. (The breakdown voltages were determined using Medici by noting the diode voltage at which the reverse current exceeded 100 μA/mm². Between 0 V and 200 V the voltage was incremented in 2V steps, and above 200 V it was incremented in 10V steps.)

The Medici simulations used in this thesis were almost all defined as one-dimensional structures with 300 equally-spaced nodes. This proved to be inadequate for only a very few structures which had very steep doping gradients. In these cases, the gridding was modified to decrease the spacing in the vicinity of the steep gradients. The validity of the simulation results in these special cases was confirmed by running several simulations with different gridding for each structure, to ensure that the gridding was sufficiently fine that a small change in grid spacing would not affect the simulation results.

λ, μm

L, μm

Figure 4.2 The breakdown voltage contours (labeled in Volts) calculated using E_C= 225 kV/cm.

λ, μm

L, μm

Figure 4.3. The breakdown voltage contours (labeled in Volts) calculated using the Medici device simulator.

Figure 4.4. The relative difference between the breakdown voltages presented in Figure 4.2 and the Medici simulations in Figure 4.3.

The results of the two sets of data agree within ±15% for voltages above 170V, and both show the same general trends. However, the quantitative agreement for lower voltage devices is rather poor. The agreement is poorest when there is a steep doping gradient at the metallurgical junction. This is not surprising since the constant critical electric field approximation works best when comparing devices with similar doping levels [Bulu91],[Sze66]. Thus, if the approximation works well for the high voltage devices, where the depletion region extends mostly through regions where N(x)>> N_B, it is unlikely to hold for the low voltages devices which have steep doping gradients in the depletion region.

The results are fairly sensitive to the value of E_C. Choosing a different value for E_C, E_C = 210 kV/cm, provides a lower difference for the higher voltage devices, but the voltage above which the agreement is ±15% or better is raised to 270V. The limitations of assuming a constant E_C for an extended range of doping profiles are clearly demonstrated by these two examples.

To determine a method of choosing a geometry-dependent critical field E_C(λ,L), values for x₁, x₂, and E_C were determined by working backwards from the V_BR(λ,L) results obtained in the Medici simulations. The results for E_C(λ,L) are shown in Figure 4.5. The contours of the data shown in Figure 4.5, Figure 4.3 and Figure 4.2 show a striking resemblance, leading to the unexpected result that there is an approximate one-to-one correspondence between breakdown voltage and critical field for a wide range of structures. This suggests that better estimates for E_C can be obtained by first calculating a V_BR0(λ,L) using a constant critical field E_C0, and then choosing a new E_C based on the V_BR0(λ,L) calculation. In practice, it has been found that the empirical function

(4.10)

proposed here for the first time, produces excellent results. The desired V_BR(λ,L) are then calculated as before, but using E_C(λ,L) rather than E_C0. (The crookedness of the contours in Figure 4.5 is due to the discretization used in the simulator, and not the physical phenomenon itself.)

Using this method, and the empirically determined parameters E_C0 = 208 kV/cm, M = 0.33, and V₁ = 1000 V, the difference plot of Figure 4.6 was obtained, with an overall average difference magnitude of 3.3%. This plot is considerably better than that of Figure 4.4. These parameter values produced the best results for the particular combination of N_S1, N_S2 and N_B considered above, but the parameters can vary over a fairly large range and still produce good results. The parameters E_C0 = 190 kV/cm, M = 0.47, and V₁ = 1000 V produced the best results on average for a wider range of power structures. For instance, using these parameters on the structure discussed above yielded 6.8% average difference magnitude, and a second structure with N_S1 = N_S2 = 10¹⁹ cm^-3 and N_B = 10¹³ cm^-3, yielded an average difference magnitude of 8.2%.

Figure 4.5. The critical electric field, E_C(λ,L), contours (labeled in kV/cm) as determined from the Medici simulations.

Figure 4.6. The relative difference between the breakdown voltages calculated using the E_C(λ,L) given in equation (4.10) and Medici simulations.

A fifty by fifty array of λ-L points was used to generate the data for Figure 4.6. It required 3.5 minutes of processor time on a 90 MHz Pentium-class personal computer to calculate all 2500 voltages. In contrast, using Medici on a Sun Sparc 10 workstation to calculate breakdown voltage for a single λ-L combination typically takes several minutes of computer time (depending on grid spacing and other simulation parameters), so a very large time saving is realized by use of the approximations presented here.

The equations presented above assume that only λ and L are being varied, however, since equation (4.3) does not depend explicitly on any doping or geometry parameters, any such parameter can be varied during optimization.

4.4 - Conclusions Regarding the Method of Calculating V_BR

A new two-step method of rapidly estimating the breakdown voltage of diffused rectifiers has been presented, which should prove useful for device optimization. By using the critical field approximation rather than the more rigorous ionization integral approach, significant time savings are realized. The first step is used to determine a value for E_C(λ,L), and the second step uses this value to determine V_BR(λ,L). This method takes advantage of the fact that there is an approximate one-to-one relationship between the device breakdown voltage and the critical field for a wide range of device structures. These results have been compared to Medici simulations, and have been shown to agree very well for a wide range of power diodes.

The interesting fact that E_C and V_BR are linked through an approximate one-to-one relationship is explored from a theoretical standpoint in Appendix C.

4.5 - Qualitative Observations on the Nature of V_BR(λ,L)

The contours of Figure 4.2 (and Figure 4.3) have several interesting features. For instance, consider the breakdown voltage variation for λ = 25 μm as L is varied. For the limiting case of L → 0 the breakdown voltage will be large, since the two gaussian profiles will largely cancel each other, leading to a region of low doping which extends across most of the structure. As L increases, V_BR falls as this compensation decreases. The breakdown voltage eventually reaches a minimum value, at L>> 35 μm. The cause of this minimum can be seen by referring to Figure 4.7. The minimum occurs when the doping gradient at the junction is largest. Figure 4.8 shows the doping profile and electric field profile at breakdown for this case. The high gradient produces a very narrow depletion region.

Figure 4.7. The doping gradient at the junction.

Figure 4.8. Doping and field profiles for λ = 25 μm, L = 35 μm.

For a given λ, the L that produces the lowest breakdown voltages can be obtained by setting

(4.11)

Using (4.2), this can be written as

(4.12)

The locus of points given by (4.12) is shown in Figure 4.2, labeled "A". This curve clearly follows the V_BR minima.

As L increases along the λ = 25 μm line the two gaussian profiles move apart and the doping gradient at the junction decreases, so the breakdown voltage steadily increases. Figure 4.9 shows the doping and electric field for L = 125 μm. The depletion region is much wider than in the previous case. Eventually, however, the breakdown voltages saturate, as seen by the horizontal contours lines in Figure 4.2. This effect is shown in Figures 4.10 and 4.11 for L = 200 μm and L = 250 μm respectively. At L = 200 μm the electric field to the left of the junction is built up over an area where N(x)>> N_B, and breakdown occurs when

(4.13)

For the values used in this simulation, equation (4.13) yields x_mj - x₁ = 79 μm. Figure 4.10 is in excellent agreement with this estimate. When L is increased to 250 μm, the doping level between x₁ and x_mj is essentially unaffected, so the electric field shape is essentially unchanged (other than a uniform movement to the right), and the breakdown voltage does not change.

Figure 4.9. Doping and field profiles for λ = 25 μm, L = 125 μm.

Figure 4.10. Doping and field profiles for λ = 25 μm, L = 200 μm.

Figure 4.11. Doping and field profiles for λ= 25 μm, L = 250 μm.

The locus of points where V_BR saturates can then be calculated by requiring that N(x)>> N_B between x₁ and x_mj. If the gaussian profile is considered separately from the background doping, and the point x_a is defined (referring to Figure 4.1) as the point where

(4.14)

or, equivalently,

(4.15)

then N(x)>> N_B for x₁ < x < x_mj if x₁ ≥ x_a. The locus of points given by x₁ = x_a is shown in Figure 4.2 by the curve labeled "B". The points above this curve are punchthrough structures, and the points below are non-punchthrough structures.

Now consider the breakdown variation for a fixed L, as λ is varied. As λ is increased from zero for L = 200 μm, the breakdown voltage slowly increases. This is due to the decreased doping gradient to the right of the junction, and the accompanying widening of electric field on the right side of the junction. Since these points lie below curve "B", the electric field shape on the left side of the junction remains essentially unchanged, for the reasons discussed above. This is shown in Figures 4.12 and 4.13 for L = 200 μm, λ = 10 μm and L = 200 μm, λ = 20 μm respectively. In Figure 4.12, x₂ - x_mj>> 10 μm and in Figure 12, x₂ - x_mj>> 15 μm. Although x_mj - x₁ must also change, to satisfy the charge balance, the relative change is only a few percent, compared to the 50% increase in x₂ - x_mj. Since most of the voltage is developed to the left of the junction, the breakdown voltage varies relatively slowly.

Figure 4.12. Doping and field profiles for L = μm, λ = 10 μm.

Figure 4.13. Doping and field profiles for L = 200 μm, λ = 20 μm.

As λ is increased further the breakdown voltage responds as it would to a decreasing L, as discussed above. A maximum breakdown voltage is reached, close to the curve "B" (this corresponds to the structure in Figure 4.9). For larger values of λ, the diode become a punchthrough structure and the voltage begins to fall, until the curve "A" is reached, where the doping gradient at the junction is largest. Then the breakdown voltage will tend to rise again.

It can be seen from Figure 4.2 that the voltage variation below curve "B" is a slowly-varying function of λ, and that V_BR appears to approach a non-zero limiting value as λ → 0. This value can be calculated by noting that this situation corresponds to a one-sided junction with a uniform doping of N_B. The width of the depletion region in this limiting case can be found using (4.13), which gives

(4.16)

The voltage developed across a one-sided junction with uniform doping can be found by integrating Poisson's equation, yielding

(4.17)

Combining (4.16) and (4.17) gives the limiting V_BR:

(4.18)

For the case of Figure 4.2, with E_C = 225 kV/cm and N_B = 1.85x10¹⁴ cm^-3, the limiting V_BR is 892 Volts.

Figures 4.2 and 4.3 show that when designing high-voltage rectifiers for a given breakdown voltage, wafer thickness, and surface and background doping levels, there are generally two diffusion lengths λ that produce the same breakdown voltage. Generally, it is the λ that lies above curve "B" that is chosen, since a narrower low-concentration region yields a lower forward voltage, for devices with similar carrier lifetimes [Benda67]. The other possible λ will lie beneath the "B" curve, and will have a much wider low-concentration region, leading to a higher forward voltage. However, if forward voltage is not the main concern, the smaller λ does offer the advantage of shorter processing times, since from simple diffusion theory [Jaeg88]

(4.19)

where D_d is the diffusion coefficient of the dopant and t is the diffusion time. This can be a considerable advantage for high-voltage devices, which can have junction depths many tens of microns deep, requiring several days of high temperature diffusion.

For each of the higher-voltage contours, however, there is generally a point where only one λ will yield the desired voltage. This occurs at the minimum L that can be used to obtain this voltage. This is useful to know when the structure is to be fabricated in an epitaxial layer, and the dopants to be diffused are introduced on one side from the wafer surface and on the other side of the epi-layer from the base substrate itself. Narrower epitaxial widths are easier and faster to fabricate.

The low-voltage contours centered about the "A" curve show that for a given breakdown voltage, diffusion length, and surface and background doping levels, there are generally two diode thicknesses L that produce the same breakdown voltage. The smaller value of L may allow the diode structure to be formed in a thick epitaxial layer, or choosing the larger value of L may allow the diode to be diffused from the opposite sides of a thin uniform wafer.