MOS Capacitor Low Frequency Characteristics

The field across the oxide is D^ is continuous

Now where (unit: F/cm2) But linear response theory

Also Taylor Series to First order We write: However

i.e., Terminal capacitance (AC, per unit area): with

Recall the E-field at the surface: (Gauss law) But That is

Expand the sinh and cosh For n-type substrates

In other words, we have: When nS and pS are small, ND dominates inside the | |. In such a case, the MOS is in depletion.

That is where

Now, using the depletion approximation, the depletion width is given by: where ybending is the band-bending and More accurately

The term ns/ND is important in the case of accumulation The term ns/ND is important in the case of accumulation. In the case where there is inversion, ps/ND is important and: We can approximate QI by: The approximation is good to ~5% in weak inversion; it is very good at strong inversion.

We can proceed similarly to calculate C:

For n-type substrate Note that with Cox and NA (or ND) known, we can calculate yS easily. Calculated C-VG relationship

Special C-V Cases Flat-band: In this case

Use expansion to obtain result:

Note that:

Strong accumulation In this case: In this case, and

i.e.,

Depletion In this case:

Inversion In this case:

Capacitance minimum To find the value of minimum C (i.e., Csmin), take the derivative of C and set to zero:

usmin or vmin can be calculated numerically

Note that csmin ­ with increasing doping. This is due to smaller WD. 2. The spread of C over vs is large, due to starting with a larger ub NA uB

Band-bending We need to know the band-bending,yS , as afunction of VG in order to calculate C after obtaining CS(yS). First, notice that: where CLF is the measured low frequency capacitance In an actual program, one start by looking at Cmin and Cox (i.e., C at strong accumulation) to obtain NA (ND). VFB can then be determined, and integration is via numerical methods.

MOS Capacitor High Frequency C-V Characteristics

First, we will loomk at Majority Response Time Assume n-type substrate, the equations for the electrons are: n is normalized band-bending and mFn is the normalized Fermi potential for electrons. With we are dealing with the case of zero DC current.

Now

Therefore, we have Add the equation for d v:

If the frequency of the system is w, it has little t-dependence if i.e., with

Minority Response Time The source of minority carriers are: SRH generation / recombination in the depletion region Diffusion from the bulk substrate External source (such as p+ guard ring, gated diode, etc.) At very high frequency (1) or (2) cannot really follow the A-C variation. In that case, we have

Note that the C-V characteristics are different from the low-frequency case. [Caution: the capacitance shown is with p-substrate.]

High frequency C-V formulation assumptions: The majority carrier Fermi-leve is flat. This approximation is clearly justified since the response time is small. The minority carrier quasi-fermi level is flat. Note that this approximation is good at the region of importance, i.e. where the inversion charge (“minority carriers”) is large.

With these approximations we have (p-type substrate): uFn is normalized by kT/q and references to Ei(¥) To get high frequency C-V we solve the Poisson’s equation with: . i.e.

Again, with we get: With the gate voltage of

Since

To solve we use Now i.e., the extra e- due to band-bending

Since under steady-state Changing the variable of integration from dx to dv by noticing that: we have:

With We can get the relationship between

\ For weak inversion, we can ignore the minority carriers For strong inversion,

Close Form Application of High Frequency C-V Assume that

with The match vm can be made by minimizing the overall error. The optimal fit is found to be: where is the onset of strong inversion derived previously.

When VS=VL, WD = WDmax . WDmax with depletion approximation is given by for n-substrate for p-substrate where ln and lp are the extrinsic Derby lengths:

Interface Charges, Fixed Charges, and Flat-Band Voltage for MOSCAP A. Work function difference Note that the work function of silicon is given by:

If the EV, EC and Ei are flat, then the VG applied to the MOSCAP is EFM EC EFS EV The effect of the work function difference is to shift the C-V curves (both low and high frequency) by an amount =fms horizontally. In other words, the VG is shifted by fms when compared to the characteristics derived earlier.

B. Charges in the oxide (1) Oxide fixed charges and interface states Oxide fixed charges and interface states probably have the same origins. However, interface states are closer to the Si/ SiO2 interface, and hence can “communicate” with the charges in the semiconductor. Oxide fixed charges are farther away, and do not change the charge state with varying fermi-levels inside the semiconductors. Origins of interface states Dit and oxide fixed charges Qf: Stress/Strain induced point deects, dangling bonds, defects …

Effects of oxide fixed charges Recall: if no charges exist inside the SiO2, Eox is constant and Also With oxide fixed charges, Gauss law implies SiO2 Metal Si x x=0

Assuming that the fixed charges are right at the Si/SiO2 interface, with a density of Nf (unit cm-2) In general, with a distribution of oxide fixed charge, r(x), inside the oxide:

Again, we have a horizontal shift of the C-V curves from the “ideal case” by an amount:

Interface trapped charges (interface states) First, the charge state of an interface state depends on the location of the Fermi level. Whether the state is “donor-like” (+ve charged when empty), or “acceptor-like” (neutral when empty), as the gate voltage swaps from VG1 to VG2 andYS changes from YS1 to YS2, the change in charges is: EV Empty states Filled states ES EC For example, for “donor-like” states whose energies are below Ei+|fB|/q, they do not affect the VTH (on-set of strong inversion) for a p-substrate MOSCAP since these states are below the Fermi level at VTH and therefore not-ionized!

In the low frequency case, interface states will affect the C-V characteristics: it D I ox V G sub With Dit Ideal VG C In the high frequency case, if w is higher than the relaxation time of the interface states (which come to equilibrium by the SRH process, similar to bulk traps), their effect on C is zero. Hence, the only effect is the charge state of interface states below EF at the D.C. bias point.

Worse subthreshold slope Scattering Þ lower mobilities VTH shifts However, if Dit is high, the interface states will “smear” out the high frequency C-V curve: Ideal With Dit C VG Interface states and oxide fixed charges are important for MOS transistor because: Worse subthreshold slope Scattering Þ lower mobilities VTH shifts Therefore, it is important to eliminate these charge states as much as possible.

Interface states characterization There are two ways commonly used to extract interface states. Capacitance method fast and easy, but least accurate Conductance method complicated, but gives more information and is also more accurate. Capacitance method for extracting Dit

First, how interface states affect MOSCAP steady state characteristics From charge neutrality, we have i.e., with the inclusion of Qit, QS changes with fixed QG. r x

Now i.e. or where

In case of low frequency it S ox V G sub In other words,

To get Cit(yS), we notice that which peaks sharply when

That is Cit(yS) can be approximated by In other words, we have

This is, in case of low frequency C-V, and the difference between “with” and “without” Dit is the parallel of Cit=q Dit(fS) with CS. That is Cs can be calculated if we know the doping concentration of the substrate, which can be estimated from Cmin.

We can also obtain YS via

High Frequency C-V with Dit If the a-c signal is very high in frequency, the interface will not be able to follow it, and Cit is roughly zero. However, the interface state charge does change with the D-C bias, which, in turn, leads to a stretch-out of the C-V curves, as discussed in the figure earlier. By comparing the calculated C-V with the measured one, we can obtain Dit via: before the onset of strong inversion. That is Cs(ys) can be calculated ed from CHF if we know Cox which can be estimated from Cinv and the substrate doping concentration which can be estimated from Cmin.

However, the VG-Ys relationship follows the quasi-static equation (DC bias):

High-Low C-V for Dit Both high frequency and low frequency require the calculation of Cs(ys), and hence can introduce error(s). However, by using the combined high and low frequency C-V, one can obtain Dit directly in the depletion and weak inversion regions. In these regions With we get

(a) Is the HF C-V and (b) is the LF C-V. n-type substrate MOSCAP.

Integrating one more time, we have: Mobile Ion Charge Ionized metallic ions can exist inside the SiO2 Þ same effects as the oxide fixed charge, i.e. VTH shift. Poisson’s Equation inside the SiO2 with no(x) the density of the (charged) mobile ions: SiO2 Si xo x=0 x Integrating one more time, we have:

and the VFB will be shifted by this amount. That is, the additional gate voltage due to the mobile charge in the oxide is: and the VFB will be shifted by this amount. E The reason we call these ions “mobile ion charge” is because they drift with the gate field over time. That is, m is not zero, and the VTH changes with time. This is very bad for circuits.

They drift much faster at high temperature They drift much faster at high temperature. Therefore, to characterize a mobile ion charge, we stress the MOSCAP at high T and measure the C-V before and after stress. By observing the change in the flat-band voltage, we can roughly deduce the amount of the mobile charge and its centroid.

Deep Depletion If the gate bias sweep is too rapid, QI cannot respond fast enough due to limited minority carrier generation. Þ The capacitive charge and the voltage drop is supported by the majority carriers. Þ Depletion width expands beyond Wmax and:

This characteristics is exploited in charge storage devices such as CCD