Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

first the continuity equation for minority carriers in the n-region, i.e. holes, and use coordinates marked x' in the figures:

Considering only the diffusion component for minority carrier flow, which is reasonable for conditions of low-level injection (as discussed in Chapter 3), Eq. (4.83) simplifies to

For steady-state, ap/at = O. Changing variables so that p' ~ p - PnO, the steady-state equation becomes

p

where L~ ~ DpTp is the diffusion length for holes in the n-region and we have assumed that the doping is uniform so that PnO = const which leads to {}2PnO/ {}x z = O. This is valid regardless of subsequent boundary conditions. The solution which satisfies this equation has the following form:

(4.86)

The determination of Cl and C2 depends directly on the boundary conditions. For a pn junction under bias we know from the foregoing discussions that

The -1 at the right hand side (RRS) of the above equation results from the fact that we are considering the excess quantity p'. To specifically determine Cl and Cz we must know a second boundary condition.

CASE 1 For the case corresponding to Figure 4.8 (a), P must go to zero long before x = Wn . Rence we cannot allow a solution with a term e+x'/Lp since it would become large without bound (Le., e+Wn /

as x' increases. The result is that for case 1, C2 = O. Using the boundary condition at x = 0 to determine Cl, one obtains the solution

Again, as for case 1, the majority carrier distributions follow the minority carrier distributions (as shown in Figure 4.8 (b» in order to maintain approximate charge neutrality. The carrier distributions for this case represent the "short-base" diode solutions, which will be useful in considering the operation of bipolar transistors. Having determined the carrier distributions it is a fairly straightforward matter to calculate net current flow in the device. The usual approach is to consider the total current JT, which must be constant at every point in space (for de steady state), and find a convenient point(s) at which to compute it. If the effects of recombination or generation in the space-charge region are of negligible consequence in comparison to the magnitude of JT, then the current flow due to electrons at x" = 0 must exactly equal the current flow due to electrons at x' = O. Notice, at x" = 0 we can make the calculation based only on diffusion of minority carriers which are electrons. This is not true, however, for majority carrier which are also electrons at x' = O. To find the total current flow we need only compute its value at one point, since current is constant through a two-terminal device. If we choose x' = 0, then we say

Jo

and for case 2

(4.97)

It is important to realize that although the total current is constant with position, the individual components (hole and electron flows) vary. For example, consider the component of current flow due to holes in the n-region:

I

I

I

X'4-______________- J - J o

Figure 4.9: (a) Hole and electron components of JT, (b) drift and diffusion components of I n , (c) field necessary to achieve I n Idrift.

space-charge region edge and at the contacts. As recombination occurs the minority-carrier currents decrease. The slope reflects this change. Hence the slopes are smaller at the contacts (than at the space-charge region edge) by an amount proportional to the bulk recombination component of current. In Figure 4.10 (b), P < PnO (and n < npo) which implies that u < O. This means that thermal generation is occurring. The sense of the applied potential is such that holes must be removed from the n-region and electrons removed from the p-region (to the degree possible). Hence, the external current flow results from carriers generated within a distance sufficiently close to the space-charge region so that the minority species can diffuse to the transition region and be swept across it before recombining. For Figure 4.8 (a) this "sufficiently

This equation can now be used with Eq. (4.103) to estimate the effects of generation-recombination in the space-charge region on the terminal

The primary function of voltage is the exponential term. This term is due to space-charge recombination and depends on eqV/2kT rather than eqV/kT as for our diffusion terms in the bulk. This is a result of the fact that both species are important to recombination in the space-charge region. The approximations that p = nand pn = nleqV/ kT give rise to this voltage dependence. Experimental results for a silicon device are plotted for forward bias in Figure 4.11. The log I vs. V plot indicates, via the asymptotes, that for low currents the space-charge recombination term dominates, whereas when current increases the usual eqV/ kT term takes over.

For this case, the space-charge width, I(V), is the only function of voltage. As compared with the simplest expressions for "reverse-leakage" currents given by Eq. (4.101), Eq. (4.109) does vary with V. Basically this means that increased reverse bias increases the space-charge width and hence increases the volume within which generation can occur. The magnitude of the current given by Eqs. (4.107) and (4.109) clearly indicate that generation-recombination effects in the space-charge region are important. The range of importance can be determined best by experimental evidence, such as that shown in Figure 4.11.

c

~ 10- 6

::J

U

"2 o

~

~ 10 -8

Figure 4.11: Forward current vs. voltage for a Si pn junction at 25°C. The space-charge recombination at low V is illustrated.

4.7SEDAN Analysis

The above sections use an analytical approach to obtain and outline the essential aspects of pn junction theory in both equilibrium and nonequilibrium conditions. By partitioning the device into the space charge region (SCR) and quasi-neutral regions (QNR), we are able in some special cases to find the physical boundaries of the SCR under certain bias by integration of the Poisson's equation. The incremental junction capacitance can also be obtained using this simple approach. Furthermore, by assuming "quasi-equilibrium" for low-level current conditions, the injected minority carrier concentrations at the boundary between SCR and QNR can be obtained, which makes it possible to calculate the currents in the QNR, and hence the total current flowing through the entire device in general. In most cases, the minority carrier distribution and its current can be computed by solving the diffusion equation only in QNR. Finally it is pointed out that the recombination and generation in SCR may playa major role in determining device characteristics.

However, throughout these discussions a variety of simplifications were made. Moreover device behavior under large biases are hard to

N (x)

,/

/

/

/holes

'f

Figure 4.12: (a) Doping profile and carrier distribution at Vd = 0 V

predict using analytical formulations, for the key assumptions leading to the "quasi-equilibrium" are frequently violated. Thus arises the need of using numerical analysis to justify if the assumptions we have made to facilitate the analytical analysis are reasonable and for a more practical reason, to have a general analysis tool available. In this section we will use SEDAN (although PISCES will do equally well) to analyze the pn junctions in a more realistic way and to look specifically at many of the points presented in Sections 4.5 and 4.6.

While doping profiles such as those shown in Figure 4.6 are useful in explaining the basic physical principles, the junctions obtained in silicon technology are typically highly asymmetric and nonuniformly doped. Figure 4.12 (a) shows an n+p junction which is typical of an arsenic source/drain contact with 1020 cm-3 doping at the surface and a junction depth of 0.33/-lm into a p-substrate doped with boron at 2 x 1015 em-3. For this case the n-type doping profile is specified using a Gaussian function and the substrate is uniformly doped. Also shown in the diagram is the SEDAN input file used to generate this doping profile and for the simulation at various biases. Figure 4.12 also shows the carrier profiles which indicate the position of the depletion edge on the p-side. From the figure one can see that the n-side depletion edge is virtually impossible to define. Since the n-side is very heavily doped