Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

voltage and cutoff frequency right away. The essential procedure in device simulation is to start with the input of doping profiles, specified either analytically or from the SUPREM output, the user specifies the device effects to be analyzed and the appropriate bias specifications. SEDAN jPISCES then solve the full set of equations describing the device operation - namely the Poisson's and carrier continuity Eqs. (3.4- 3.8) - simultaneously over all bias conditions and outputs all desired variables from these solutions, including electrostatic and quasi-Fermi potentials, current, and charge. Using utility programs with SEDAN or plotting capabilities embedded in PISCES, a wide range of output printing and plotting features are available so that the output can be viewed from many perspectives.

3.3Field-Effect Structures

In the previous section we introduced the SEDAN program and used the MOS threshold voltage as an example. To make the discussion tractable we introduced the first-order threshold Eq. (3.1) as well as the equation for the the current-voltage relationship in the channel (Eq. (3.3)). The SEDAN results focused on emulating the curve shown in Figure 3.1 by computing charge in the inversion layer versus gate bias as a reflection of the measured drain current, with mobility being the proportionality constant. In this section we will exploit the output features of SEDAN more fully to understand the internal operating principles and boundary conditions of the MOS capacitor device and pn junction structures. The same methodology applies to PISCES as well. In Chapters 4 and 5 we will review the theory from an analytical point of view. The objective here is to use SEDAN as an experimental tool to demonstrate device results.

3.3.1Components of Charge

The MOS capacitor structure is again drawn schematically in Figure 3.4. The purpose of this drawing now is to serve as the spatial and voltage bias reference for figures which follow. Three electrodes are shown. VG is the bias applied to the "gate" electrode which has a work function "Wgate." VB is that to the bulk electrode which will take on a computed work function based on the material doping at that point. The Vs bias represents the channel bias, assumed to be imposed by the source

p -type s1llcon

(a)

(b)

x

(c)

x

PI

Figure 3.4: MOS capacitor structure with various bias and charge configurations, (a) basic device structure (n-channel) with applied bias including surface potential, (b) Hatband bias with zero field at semiconductor interface, (c) bias near threshold with depletion charge shown,

(d) strong inversion with both maximum depletion and surface charge shown.

100 CHAPTER 3. DEVICE CAD

electrode (not shown) in a direction perpendicular to the cross section. The main purpose of Vs is to impose an additional boundary condition between the channel and bulk, and this bias will in turn affect the mobile channel charge. Assume for the moment that VB = Vs = O. As the gate voltage is made increasingly positive for an n-channel (Le. with p-substrate) MOS structure, the charge situation as depicted in Figures 3.4 (c) and (d) will evolve. For a normal n+ poly gate process with low interface charge density "Qint" there is a negative value of gate voltage where the electric field at the Si02 - Si interface is zero. We will define this voltage as the "flatband" condition. For a uniformly doped substrate this bias does in fact give "flat energy bands" for x > 0 (see Chapter 5 for further details). In general the energy bands are not flat due to nonuniform doping, but the electric field across the interface is exactly zero at the VPB bias condition. Bias conditions in Figures 3.4 (c) and (d) correspond to increasing amounts of negative charge built up in the semiconductor. In Figure 3.4 (c) the case is depicted with a uniform distribution of charge to a depth Xd, called the depletion depth. In this case we assume that the substrate is uniformly doped to a concentration NA and that the product XdNA satisfies the following relationship:

(3.9)

where Qa is the gate charge and QB is the bulk charge.

Note that the units for the charge are coulombs/cm2 • In Figure 3.4

(d) the gate voltage now exceeds the so-called threshold voltage. We see two components of negative charge - the depletion and inversion layer charges. As shown, the inversion charge is a sheet charge which we will call PI (for inversion). Now the charge balance is given by

(3.10)

The distance Xdmax corresponds to the bias point beyond which the majority (not exactly, but nearly all) of the additional gate charge is reflected by PI. At this bias point we also have a certain surface potential 'lj;s which defines the strong inversion condition. Namely, as the bias increases further, the surface potential increases only slightly with gate bias. In the following discussion we will use SEDAN results to illustrate and help to explain their meanings. Figure 3.2 (b) shows the output files created by SEDAN after running the input deck shown in Figure 3.2 (a). These output files, in terms of specific variable names, are in turn used

Vgs (V)

(a)

4.0

3.5

- 3.0

E

~ 2.5

>

Lt'lo- 2.0

?""

C. 1.5

W1.0

0.5

0.0

Vgs (V)

(b)

Figure 3.5: Plots of MOS potential and electric field versus gate bias,

(a) the surface potential is shown flattening above strong inversion condition, (b) the electric field remains lined up with bias and extrapolated intercept gives the flat band condition.

(b) indicate two features of the device. First, for the condition & = 0 we have defined the flatband voltage (which is indeed negative). That is, if we extrapolate the plot to a value where & = 0, that gate voltage (the flat band voltage) has a negative value. Second, for all VG greater than VFB the increased electric field indicates a continual build-up of charge on the semiconductor side. That is, by simply considering Gauss' law for x > 0, the total semiconductor charge will be

(3.11)

Now turning to Figure 3.5 (a) we can follow the dependence of surface potential with gate bias. The applied gate potential is shown as a dashed curve. Note the nonlinear dependence of 1/Js on VGs. For VGS

1/Js tracks VGS, although there is an offset voltage corresponding to the initial surface potential at thermal equilibrium (VGS = 0). For gate voltages above VTO, 1/Js flattens out, which corresponds to the strong inversion condition. Returning to Figure 3.4, we can now interpret several features. The plot of & vs. VGS directly reflects the continuous build-up of total substrate charge (QB + PI). On the other hand, the nonlinearity of 1/Js vs. VGS indicates the transition from the regime in which the charge is due solely to the depletion layer into the regime in which 1/Js is "pinned" and further charge build-up comes from PI. It may seem to be a contradiction that 1/Js can be pinned and that the build up of PI can continue. This matter is discussed in the next section as well as in Chapter 5.

3.3.2Charge Build-up

In this section we will discuss the charge build-up in a MOS device in a microscopic sense.

We begin with the following empirical equations which accurately reflect data shown in Figures 3.5-3.7:

(3.12)

Note that for the moment we have neglected the effect of the work function difference between the gate material and the substrate. This subject will be discussed in detail in Chapter 5.

Recall that QG is the gate charge, PI is the inversion-layer charge, and

QB is the bulk charge. kl and k2 are unspecified parameters, and Xd is shown as a function of 1/Js. Note that Xd is a polynomial function of 1/Js

(as discussed in Chapter 5) while PI is exponential in 1/Js. Additionally, one can observe that QG is linear in (VGS - 1/Js). In general,

(3.15)

However, two regimes of operation are important to distinguish. For 1/Js less than that at strong inversion, the contribution of PI can be neglected and

(3.16)

For biases well above the strong inversion point, where 1/Js becomes nearly constant and Q B becomes nearly fixed. Thus

(3.17)

and, considering subsequent changes in gate and inversion charges,

(3.18)

Since PI is exponential on 1/Js, only small changes in 1/Js are necessary in this region of operation to balance linear increases in QG.

To emphasize the leverage of changes in 1/Js compared to VG, assume that k2 is q/mkT with m in the range of two to three. Then for an order of magnitude (Le., ten times) change in VG, the change in surface potential will be 2.3 x (mkT/q). At room temperature kT/q is about 26mV, the total change in 1/Js for the assumed decade change in VG will be on the order of hundreds of millivolts. Looking again at Figure 3.5

(a) we can see this reflected as the growing split between VG and 1/Js. Over this same range of biases, the continued increase in £(0) with VGS (Figure 3.5 (b)) directly shows the build-up of PI.

In the above discussion we stated certain assumptions, such as the exponential dependence of inversion charge with gate bias, which need to be examined in greater detail. A key advantage of device simulation is the ability to look inside the device in ways not possible from terminals, using electrical measurements. In this next discussion we will look in detail at the build-up of inversion charge and the simultaneous depletion of bulk charge. The exponential dependence of PIon 1/Js is evident from the semi-logarithmic plot in Figure 3.6 (a). This result confirms the

-6

-7

-8

§

~

~ -9

U

0'

~

-10

-II

-12 L..-~--+~~-+-~~+-----'--~~.-L-.-O_

0.15 0.22 0.29 0.36 043

~. volt,

(a)

r

4

'",,,,1,,,,, 1 ",,,,, ..

VGS ' volts

(b)

Figure 3.6: Dependence of charge components on both surface and gate potentials, (a) the semi-logarithmic plot with 1/;8 shows the exponential build-up of inversion charge, (b) the linear plot with gate bias shows the saturation of bulk charge.

dependence suggested by Eq. (3.13). Moreover, from the slope of the PI vs. "ps curve, we can extract a value of 60 mV/ decade, which translates to m = 2.3 (assuming kT/ q = 26 mV at T=300 K). The discussion of the physical meaning of the m-factor will be postponed until Chapter 5. The dependence can roughly be stated in terms of the division of potential between the oxide and the semiconductor. Namely, since some of the applied gate voltage drops across the oxide, it is not available to increase electron charge as rapidly as the theoretical maximum value which would simply be q"ps/kT (Le., m = 1). The linear scale plot shown in Figure 3.6 (b) indicates the rapid build up of QB with gate voltage followed by its saturation after strong inversion. This is consistent with the discussion of Eqs. (3.16-3.18).

Turning to the spatial dependence of electron concentration, in Figure 3.7 (a) we see an extremely rapid fall-off of electrons with position away from the surface. Thinking in terms of Eq. (3.13), the spatially dependent electron concentration is of the form:

(3.19)

Since we are looking at point values, kl here differs from the integrated quantity (the sheet charge) given in Eq. (3.13). If "p(x) falls off by even a few tenths of a volt, Eq. (3.19) predicts a drastic fall-off in n(x). Clearly this is the case as reflected in Figure 3.7 (a). To become more quantitative about the exact spatial dependence of n(x) is beyond the scope of the present discussion. But given an accurate representation of 'I/J(x), one can use Eq. (3.19) to see its spatial dependence.

3.3.3Bulk Charge - QB

Having considered the inversion charge and its exponential dependence, we now turn to consider the depletion charge.

As stated above in Eqs. (3.16-3.18), the charge which satisfies Gauss' law at x = 0 is composed of both depletion and inversion charges. The build-up of inversion charge is exponential with increasing "ps. The depletion charge also has a nonlinear dependence on voltage, but its

charge dominates the solution of the Poisson's equation. If we assume

(b)

Figure 3.7: Spatial distributions of charge showing both exponential dependence and linear dependence on doping, (a) logarithmic plot showing rapid fall-off of holes and exponential rise of electrons in the inversion layer, (b) linear plot of hole concentration at two gate bias conditions. The doping is Gaussian with a peak of 1.3 x 1016 cm-3 . Hence these biases produce depletion near the back edge of the implanted boron.

a constant bulk doping, then looking again at the Poisson's equation,

(3.20)

where 'IjJ is the potential and Nsub is the constant substrate doping. Without solving the equation exactly, which will be done in Chapter 5 for appropriate boundary conditions, we can observe that the form of the solution will be

(3.21)

where the constants will be determined in Chapter 41 .

From this solution form we can see that the dependence of the depletion length, Xd, on 'IjJ will be of the form Xd ex Vifj. Looking at Figure 3.6 (b) we see that QB increases with a sublinear dependence on Vas. Assuming that QB ex Xd as given in Eq. (3.9), this suggests that a fractional power-law dependence of QB on voltage is reasonable. Combining the information given in Eqs. (3.16-3.18) along with the results of Figure 3.6 we can start to become more quantitative about the asymptotic solutions associated with the depletion charge. As the surface potential increases and PI becomes dominant, the change in the depletion layer is substantially reduced. At this point, the depletion edge increases with ..,fif;, whereas the inversion charge increases exponentially with 'ljJs. Thus, at the point where IPII > IQBI, only small changes in 'l/Js are needed to produce the required charge.

Figures 3.6 emphasizes the trade-off between inversion and space charge as a function of Va (and 'ljJs). Looking at the spatial dependence of the charges as shown in Figure 3.7 we see another aspect of the interrelation of PI and Q B. Figure 3.7 (a) shows the exponential spatial dependence of both nand p. Clearly, as the one increases exponentially, the other one decreases. However, from the point of view of the space charge, if we plot the net doping N (x) and free hole concentration p(x) on a linear scale, as shown in Figure 3.7 (b), we see that a rather abrupt transition is observed. Namely, the difference N( x) - p( x) becomes almost totally dominated by Nsub for x < Xd. This observation is supportive of the approximation used in later chapters which is called the "complete depletion" approximation. The meaning of this

1 Differentiate Eq. (3.21) twice and see that it indeed can satisfy the original differential equation.