Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

108 CHAPTER 3. DEVICE CAD

approximation can be described as follows. For x < Xd the charge density which satisfies the Poisson's equation is dominantly N(x) - that is IN(x) I >> p or n. The use of this approximation will allow us to solve the Poisson equation exactly over this region. Near the point x = Xd it is clear that the above approximation starts to fail. The width of the transition region from complete depletion to charge neutral (or quasi neutral for a pn diode) occurs over a distance determined by the doping. For higher doping levels the transition width is smaller according to the relationship (Nsub)-1/2. This characteristic length is called a Debye length and it will be discussed further in later chapters.

Figure 3.7 (b) also shows the variation in the depletion edge for two increments of gate voltage. It can be observed that the edge moves nearly as a vertical line, indicating that for increased bias a new sheet of mobile charge has now been removed at a distance Xd from the surface. Two interesting observations can be made with respect to this change in charge. First, for the increment of bulk charge (Q B) there is an equal and opposite polarity gate charge (QG) which occurs for the given increment of voltage. The interpretation of this result, following immediately from classical electromagnetic theory, is that we can define

Obviously the second part of this equation only applies for VG's prior to the strong-inversion condition and for the one-dimensional case. The second observation is that the capacitance is nonlinear. From the above discussions we see clearly that Xd varies with VG. Thus for any VG, the small-signal variation of QB will be different depending on the value of Xd. Using the classical parallel-plate capacitor analogy from physics, roughly speaking,

(3.23)

where Cd corresponds to the semiconductor component of capacitance only, and A is the device area. In Chapter 4 we will derive the full dependence of CG on VG. For purposes of this section it is sufficient to say that Xd will obviously vary with VG and in turn this variation will be reflected in a variation of CG.

The above discussion has considered the voltage and spatial dependences of the inversion and depletion charges in a MOS capacitor in

greater detail. The approach to this point has not been rigorous analytically. It is the intent of this section to try to stimulate your thinking and understanding of the device operation from the very broadest (physics oriented) point of view. Grasping the concepts that PI and Q B both work to satisfy the Poisson's equation and that each one has a domain in which it dominates the solution, can substantially facilitate the more analytical results presented in Chapter 5. For those readers who have already had the material in its analytical form, it is hoped that the results presented here can open new areas of questions - for example, interest in understanding exactly the meaning of strong inversion and the Debye length. In the next section we move on to consider the pn junction device.

3.4Bipolar Junction Structures

3.4.1Introduction

In the previous section we discussed the MOS or capacitor structure. The MOS was considered first because of its conceptual simplicity. Since no current flows vertically in the one-dimensional device we need to consider the solution of the Poisson's equation only. We now turn to the pn junction diode and again consider the use of SEDAN for analysis because of the lD nature of the problem under consideration. The bipolar device poses several fundamental challenges. First, the nature of current flow in such devices is bipolar - both holes and electrons participate in transport - so that under the worst case conditions the complete equation set Eqs. (3.4-3.6) needs to be solved. Second, the physical phenomena that occur in bipolar processes are more complex since carrier recombination as well as the Poisson's equation couples the two carriers. Finally, the techniques used to create heavily doped emitters (the region assumed to be the dominant carrier type in controlling device operation) alter recombination parameters significantly. These physical effects, often referred to as "heavy doping" effects, are still not completely resolved even after more than two decades of research. This problem is an open avenue for needed additional information.

3.4.2Bipolar Device Operation - Equilibrium

To introduce the bipolar device without going into the details of device physics such as band theory or defining Fermi and quasi-Fermi levels (for equilibrium and non-equilibrium, respectively) is a difficult task.

Figure 3.8 (a) shows the p and n materials with uniform doping NA and ND, respectively, and the N(x), p(x), and n(x) distributions as-

suming that the two materials are not connected. In each material, the majority carrier concentration at equilibrium is equal to the net doping concentration and the minority carriers obey the law of mass action which states that at the equilibrium,

where ni is the intrinsic carrier concentration at the given temperature, or the value of p = n = ni if no dopants are present. However, in this case we have added N D donors to one material and N A acceptors to the other (details will be given in Chapter 4). The donors give up their extra electrons easily and similarly the acceptors easily capture electrons, thus creating conducting "holes." Clearly, if we put the two materials together there will be substantial fluxes of electrons into the p-type material (because nnO >> npo) and holes into the n-type material (because PnO << ppo) due to the diffusion of carriers. The motion of these carriers will in turn leave net Njj charge on the n-side and N"A charge on the p-side and thereby build-up an electric field. When the materials are connected together, there is a unique equilibrium solution for the distribution of charge and the associated electric field. The equilibrium carrier, electrical field, and potential distributions with x are shown in Figures 3.8 (b )-(d). The derivation of their exact forms is postponed until Chapter 4. However, we will now discuss the physical interpretation in light of the simple conceptual picture given in Figure 3.8.

The curves shown in Figure 3.8 (b) indicate a smooth (but exponentially dependent) transition of hole and electron concentrations in going from the n-side to p-side. The reduced values of the mobile electrons and holes compared to the uniform donor and acceptor concentrations give rise to the electric field shown in Figure 3.8 (c). The polarity of £(x) corresponds to an £-field pointing from Njj toward N;. Comparing Figures 3.8 (b)-(c), one can note that there is a positive gradient of holes (Le. op/ox > 0) while the electric field also points in the positive x-direction so that the two flux components of the current

UU

Figure 3.8: Ideal np junction with various key parameters shown versus distance, (a) log doping distributions with majority and minority concentrations shown (prior to joining the materials), (b) equilibrium distribution of holes and electrons, (c) built-in electric field due to depletion of mobile charge in the transition region (Xn and Xp denote edges of the depletion), (d) electrostatic potential (integral of electric field) referenced to the n-side.

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(Eq. (3.8)) will exactly cancel each other at every point x. At equilibrium, both hole and electron currents independently have to equal zero. Thus, by definition, the conditions of equilibrium impose the constraint that solution of the Poisson's equation is sufficient to completely specify the distributions of n(x) and p(x) for all x. The final plot shown in Figure 3.8 (d) is the integral of £(x) which shows that there is an electron potential difference between the nand p sides. Looking back at Figure 3.8 (a), we can observe that nnO >> npo. This means that the conduction band on the n-side is at lower electron energy than the conduction band on the p-side, which in turn corresponds to higher electric potential according to the fundamental relationship

to develop Eq. (3.25) more fully. The point to be emphasized here is that electron energy and potential are inversely related, so that it is not surprising to see lower potential for higher conduction band energy. A final point to emphasize with regard to Figure 3.8 (d) is that the potential difference between nand p regions cannot IDQyg charge to the external world. This point will become clearer from the next discussion. It is a potential difference due to the different concentrations of charged particles in the two materials.

3.4.3Non-Equilibrium and the Coupled Equations

Having discussed equilibrium between the nand p materials, we can now consider what happens with applied bias and how this situation differs from the pictures shown in Figure 3.8. Figure 3.9 shows a conceptually simplified representation of the np junction. Three regions are delineated in Figure 3.9 (a): two ohmic regions labelled Tn and Tp and the space-charge region (corresponding to the region in Figure 3.8 (c) where £(x) 1= 0). Shown above in Figure 3.9 (a) is a circuit schematic indicating that the "£(x) box" really represents the nonlinear device which we call a "diode." For all bias conditions other than V = 0, we have some voltage drops across Tp and Tn. Assuming that these voltages Vp and Vn are small compared to the applied voltage V, we can gain insight by sketching the change in potential for two bias conditions, VF and - YR. These voltage changes are selected as shown in Figure 3.9 (b), which

(a)

~o

Figure 3.9: Simple conceptual picture on np device to facilitate understanding of bias conditions, current flow, and potential drops: (a) schematic view as circuit elements - resistances (subscripted to indicate majority carrier) and the space-charge region; (b) I-V relationship for bias polarity shown in (a), the two points refer to forward and reverse biases; (c) plot showing where the applied potential is dropped in the device. Both forward and reverse bias drops most of the potential across the space charge. However, small increments are needed in the neutral regions to sustain majority current flow.

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represents the normal diode I-V characteristics with polarity reference from p-region to n-region shown in Figure 3.8 (a).

In comparing the two plots in Figure 3.9 (c), several features are clearly observed. First, for both Vp and - VR bias, there is a finite

dropped across the "£(x) box" region. Note that the boundary points, Xn and xp, vary with bias voltage because they depend on the potential drop across the space-charge region. The key point is that most of the voltage drops across the "£(x) box," despite the dependence of Xn and xp on voltage. This allows us to greatly simplify the complexity of pn junction analysis.

The analysis method depicted schematically in Figure 3.10 refers to solving the coupled equations, Eqs. (3.4-3.8), in a sequential manner. The number of equations and which equations to be solved at each step depend on the noted modifications. For all parts of the figure we assume steady-state analysis, namely that aniat = aplat = O. We will study this diagram further in Chapter 4, but for the moment let us see by example how this approach works for the diode case specifically.

The first step assumes that an "equilibrium-like" solution, Le., the voltage drop across the space-charge region is known from the applied bias, applies in the space-charge region, consistent with the results depicted in Figure 3.9 (c). The second step assumes that the equilibriumlike boundary conditions at the edges of the space-charge region together with the knowledge of the built-in electric field give a good estimate of what should happen to the minority carriers in the quasi-neutral region. The built-in field is calculated entirely based on the doping gradient. By solving Eqs. (3.5) and (3.7) in p-neutral region and Eqs. (3.6) and (3.8) in n-neutral region, not only the initial distribution of the excess holes in the n-material and the excess electrons in the p-material can be determined, but also the total current through the diode can be calculated by summing up the minority currents at the edges of the space-charge region. Note that at this stage the potential drop due to the built-in fields in the quasi-neutral regions is negligible. In step three the majority carrier profile is derived from that of minority carriers and dopants by sticking to the quasi-neutral condition. The updated electric field can be calculated from the carrier profile and the known total current.

The integral of £(x) outside the space charge contributes to an error in the assumption used in step one. By correcting for the error in

where the exact form of u is yet to be specified. Typically for analytical solutions we assume that u is some simple function of p and n, then Eqs. (3.30-3.31) can be solved in a closed form. This approach will be taken in Chapter 4. For purposes of the discussion to be presented in the next section, it is more useful to return to Eqs. (3.26-3.27) in their general form:

What these equations say is that the difference of the current flowing in a device from point to point can be related to the integral of net recombination occurring in the region bounded by these two points. There are several advantages to this formulation. First, an integral is the most convenient formulation for computer implementation as is used in SEDAN, since integration requires point values (and not derivatives of point values) to be known accurately. Second, given the integral form for current density, we can more easily handle complex and new physical forms for u. If the kinetics involved in u are complex it is not necessary to separate these kinetics into a differentiable form. This latter result is especially important for bipolar devices with heavily doped emitters. In this case, effects such as bandgap narrowing and polysilicon-bulk contacted emitters can be implemented more easily.

In summary, the analysis of minority carrier effects in bipolar devices generally involves the solution of Eqs. (3.26-3.27) in their appropriate device regions. For general ease of implementation the integral form as given by Eqs. (3.32-3.33) is most appropriate. This is especially true when the expressions for u become more complex. Under the assumption that we can neglect electric field effects for minority carriers, then Eqs. (3.30-3.31) are appropriate. In Chapter 4 we will return to this path to consider analytical solutions for special cases of pn junctions. While in previous sections we illustrated MOS analysis primarily using the 1D SEDAN tool, in the following section we will show the analysis of the n+p diode using PISCES. Although we do not discuss further the syntax of the input specification, details can be found in Appendix C or [3.2].

Now we will proceed to use PISCES instead of SEDAN to analyze a typical pn junction structure and to look at how the results support