Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

by integrating u(x) across the complete device. The results shown in Figure 3.15 (b) clearly show how the hole and electron components of current change with position. The analytic models to be discussed in Chapter 4 are quite difficult to analyze and interpret without having detailed pictures such as those shown in Figure 3.15.

3.5Summary

This chapter introduces key concepts of both MOS and bipolar device operation. The approach has been to use equations to only qualitatively understand the behavior and dependence. Based on both SEDAN and PISCES simulations, many of the detailed device effects are revealed. In most cases, the observed effects follow the first-order equation dependencies. For example, in the MOS case the voltage dependence of depletion and inversion charge follows the expected behavior. Similarly, in the bipolar case, the depletion layer and its bias dependence are easily observed.

However, one key point raised in this chapter is the fact that real devices show many complexities that go beyond our first-order models. Especially the bipolar device shows both space-charge effects and recombination current behavior which is quite complex at first glance. The reader should not look at such complexities as unusual. Rather they should stimulate thinking and the realization that real devices are indeed much more challenging than "textbook examples".

This last statement brings us to the crux of this text and a conceptual point which deserves some further thought by the reader. Analytic equations which approximate device behavior are without question the most efficient way to express device physics. Unfortunately, the approximations used to derive such equations have two limitations. First, the simplifying assumptions such as constant doping in MOS devices or neglecting recombination in the space-charge region in bipolar devices are conditions that are easily violated. Second, the methodology used to achieve many approximations - for example, complete depletion and strong-inversion conditions - are not always clear or fail safe. In that regard, the methodology used here is to explore many of these assumptions by means of complete numerical analysis. While at times this can be frustrating because of the added complexity, there are many features of semiconductor device operation which should become much clearer in

the process. Finally, such a hands-on approach to learning now opens the door for the reader to freely explore far beyond the limited equations and examples presented here. Hence this chapter in a major way signals the beginning of an open-classroom approach to learning device principles.

3.6Exercises

3-1 Using the data given in Figure 3.1 and Eq. (3.1), determine values for VTO, I and ¢s. If the value of Cor is 7.84 x 1O-8 F/cm2 , what then is the average substrate doping?

3-2 Using the data given in Figure 3.2 and Eq. (3.34) with 'f/ == 3, determine the depletion edge at the gate bias given. By what percentage does this distance change for a ±50% change in 'I]?

3-3 Using the peak-concentration point in Figure 3.7 (b) and Eq. (3.23), compute the capacitance per unit area (A == lcm2 ). Estimate the area between the curves for VGS == 0.25 and 0.5V and use Eq. (3.22) to also estimate the capacitance.

3-4 Consider Figure 3.6 (a) and especially the point where Q B = PI. In reference to Figure 3.4 (d), this should be the strong-inversion potential. How does the value compare with that calculated in Exercise 3-1?

3-5 Extrapolate the zero-bias field in Figure 3.13 (b) to find the depletion edge on the p-side. Use the distance from the peak field to this edge as the distance Xd and calculate Cd (per unit area) based on Eq. (3.23). How does the value compare with that from 3-3?

3-6 Approximate the peak region (shown with a dashed curve) in Figure 3.15 (a) with a simple Gaussian expression. Using Eq. (3.33) and integrating from 0 to 00, how does this b.Jp compare with the total "measured" hole current density shown in Figure 3.15 (b)?

3.7References

[3.1] Z. Yu and R. W. Dutton, "SEDAN III-A generalized electronic material device analysis program," Stanford University Electronics Laboratories Technical Report, July 1985.

[3.2] M. R. Pinto, C. S. Rafferty, H. R. Yeager, and R. W. Dutton, "PISCES - lIB," Stanford University Integrated Circuits Laboratory Technical Report, 1985.

[3.3] R. B. Marcus, "Chapter 12: Diagnostic Techniques," VLSI Technology, edited by S. M. Sze, McGraw-Hill Book Co., New York, 1983.

Chapter 4

PN Junctions

4.1Introduction

In the previous chapters we have introduced process and device simulation - both data structures and their application in modeling semiconductor devices. In the next few chapters we consider in greater depth the physical effects in semiconductor devices based on analytical formulations as well as using the SUPREM and SEDAN jPISCES tools. In Chapter 3 the basic formulation of decoupling the device equations was introduced. Tracing the literature of semiconductor device analysis one finds this approach used again and again to simplify and to separate the problem into a set of smaller problems, each with independent analytic solutions. For a large number of problems this approach has worked quite well. Most introductory textbooks in the field use this approach and undoubtedly the reader has had exposure to such material. Starting with the next few sections of this chapter there will be a brief summary of many of the key results. Although it is helpful if the reader has seen this material before, it is sufficiently self-contained that all the basic points are presented and discussed. The motivation for this review is two-fold. First, as stated above, it provides a self-contained glossary of equations and modeling concepts. Second, as simulation results are presented it will be apparent how the "exact" results fit into the fabric of the analytical results. More precisely, to validate the assumptions imposed in order to separate and solve the device equations, the simulation results can directly be compared.

The flow of this chapter begins with consideration of basic semicon-

ductor properties under equilibrium conditions in Section 4.2. In Section 4.3 non-equilibrium conditions are considered, including generation and recombination. In Section 4.4 the basic current flow and continuity equations are discussed. Section 4.5 considers equilibrium and quasiequilibrium conditions in the pn junction and Section 4.6 considers the non-equilibrium cases of steady-state current flow. Finally, Section 4.7 gives numerical examples based on SEDAN and PISCES results to both validate and contrast the analytic results with the more exact numerical solutions. Especially in Section 4.7.3 we begin to see the technology dependencies of the diode properties. Section 4.8 gives a brief summary of the chapter.

4.2Carrier Densities: Equilibrium Case

The energy band diagram forms the basis for our discussion of semiconductor devices. In our simplified model we assume that there are two energy bands in which carriers can move freely. The conduction band at energy Ec (see Figure 4.1 (a)) is assumed to be the favored location of mobile electrons, and the valence band at energy Ev is the energy location for mobile holes.

In a chemically pure semiconductor, the number of mobile carriers of each type (electrons or holes) is equal. For each mobile electron, a lattice bond must be broken, which leaves behind a mobile electron vacancy or hole (within a "sea" of other bound electrons which can interchange locations freely with the holes). The number of free electrons and holes in a semiconductor is primarily a function of the temperature and the energy gap between the valence and conduction bands. In fact, the word semiconductor implies that at zero degree Kelvin the valence band has no vacancies (all electrons are locally bound to their host atoms) and no electrons are in the conduction band. As the lattice is heated, if sufficient excess energy can be imparted to the bound electrons they can be excited into the conduction energy band where they behave as free electrons (see Figure 4.1 (b)). The average number of electrons per unit volume which can surmount the energy bandgap to become "free" conduction electrons is given by

(4.1) where E9 IS the energy "bandgap" between valence and conduction

(d)

Figure 4.1: Energy band diagrams which illustrate (a) the fact that mobile holes and electrons are confined to the valence and conduction bands, (b) for an intrinsic (or chemically pure) semiconductor holeelectron pairs are generated by lattice vibrations which impart energy in excess of E9 , (c) the addition of donor impurities add localized energy levels for electrons at ED which are more easily ionized than in (b), and

(d) the addition of acceptor impurities add localized energy levels to hold electrons at EA and electrons are more easily excited into these states than to Eo.

bands, T is the absolute temperature, and No and Nv are the effective densities of available states for electrons in the conduction and holes in valence bands, respectively [4.2]. Note that No and Nv are also functions of ternperature [4.1]. The intrinsic carrier concentration, ni, is thus equal to both the hole and electron concentrations for an undoped material and naturally

It can be shown that the product of the hole and electron concentrations (numbers per unit volume) equals this constant, n~, even when the two concentrations do not equal each other. The basis for this behavior is the so-called law of mass action. The chemical rate equation for creation of hole-electron pairs is proportional to the product of the concentration of

For a specified temperature, the probability that a state with energy E is occupied by an electron is given by the Fermi-Dirac distribution function:

(4.4)

where the Fermi energy, EF, is the energy at which the probability of that energy state being occupied is one-half and k is the Boltzmann constant. The density of electrons in the conduction band, n, is thus the sum of probabilities of all available states in the conduction band. An alternative way to calculate n is to introduce an effective density of states at the conduction band edge such that

(4.5)

That is, to map all the available states in the conduction band to its edge. Conversely for holes, the density will be given by the probability that the state at the valence band edge is not occupied by an electron multiplied by the effective density of states in that band:

For the impurity (donor or acceptor) state, the probability distribution function for electrons changes slightly due to different applicable statistics. For donors, each level can accept one and only one electron with either spin direction (up or down) or has no carrier at all. Simple mathematical manipulation [4.4] shows that the electron occupation probability at this level is

(4.7)

where 9D = 2 for silicon is called the spin degeneracy factor and the value of fD is actually bigger than that given by the standard FermiDirac distribution function (Eq. (4.4)). Thus the density of ionized donors becomes

(4.8)

The situation for acceptors is a little bit more complicated partly because of the degeneracy of the valence bands at k = 0 where k is the

wave vector, and partly due to the fact that only one hole with either spin or no hole at all can be allowed to exist on these degenerated energy levels. For such semiconductors like silicon and germanium, there are two degenerated levels, and in terms of electrons the above rule for hole occupation can be interpreted as either three or four electrons are allowed to be on the acceptor level. This leads to the degeneracy factor being four. The above reasoning seems not that intuitive, and we recommend interested readers to refer to [4.5] for further discussion. Accepting this fact, the electron distribution function at the acceptor level is

(4.9)

where gA = 4 for Si and

(4.10)

(4.11)

By comparing Eq.s (4.8) and (4.11), one can conclude that for materials where gA > gD acceptors are less likely to be ionized than are donors given the same separation between the impurity and Fermi levels.

Now let us try to find the position of the Fermi level knowing the impurity concentration. However, there are a number of simplifying assumptions which can generally be made to avoid precise solution of the above equations for EF. First, for ED - EF ~ kT, one can assume that Njj = ND (or fD(ED, EF) =0) from Eq. (4.8). Then, substituting Eq. (4.2) into Eq. (4.3),

This equation invokes a second appropriate assumption, i.e., if ND ~ 2ni then n ::::: N D. For most practical situations this last assumption is true. Hence, the only condition which we must confirm is the first assumption. For heavily doped regions (for example the emitter in a double-diffused npn transistor) the Fermi level can approach the donor level and in fact can go above it. This means that full ionization (Nt =

ND) is a poor assumption. We must then use Eqs. (4.5-4.8). For these high doping levels, however, we assume n ~ p(= nUn». Hence using Eqs. (4.5), (4.8-4.7) in Eq. (4.3):

(4.14)

It is necessary to solve for the Fermi energy, EF, which satisfies the above equation.

We now discuss the situation for p-type semiconductor. In Eq. (4.9) EA is the acceptor impurity energy level (see Figure 4.1 (d». Condition 2 (charge neutrality) applied to this situation gives:

Again, to avoid the exact solution of the above equation for EF it is useful to assume that EF - EA ~ kT, which gives NA ~ NA. Thus Eq. (4.15) becomes, after substitution of Eq. (4.2),

Solving for p:

(4.17)

~ 2ni then p ~ NA. As for the n-type material, the assumption that EF - EA ~ kT must be confirmed. This is most easily done by assuming p = NA and calculating EF - Ev using Eqs. (4.6) and (4.11). That is, in the case that EF - EA ~ kT is violated one must solve the equation

(4.18)

When both donor and acceptor atoms are present in a solid, one generally considers the effective doping for calculation of carrier concentration. That is,