1. MOS Capacitor Low Frequency Characteristics

2. The field across the oxide is
D^ is continuous

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

4. Also
Taylor Series to First order
We write:
However

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

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

8. Expand the sinh and cosh
For n-type substrates

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

10. That is
where

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

12. 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.

13. We can proceed similarly to calculate C:

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

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

17. Use expansion to obtain result:

18. Note that:

19. Strong accumulation
In this case:
In this case,
and

20. i.e.,

21. Depletion
In this case:

22. Inversion
In this case:

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

24. usmin or vmin can be calculated numerically

25. 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

26. 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.

27. MOS Capacitor High Frequency C-V Characteristics

28. 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.

29. Now

30. Therefore, we have
Add the equation for d v:

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

32. 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

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

34. 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.

35. 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.

36. Again, with
we get:
With the gate voltage of

37. Since

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

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

40. With
We can get the relationship between

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

42. Close Form Application of High Frequency C-V
Assume that

43. 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.

44. 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:

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

46. 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.

47. 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 …

48. 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

49. 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:

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

51. 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!

52. 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.

53. 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.

54. 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

55. 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

56. Now
i.e. or
where

57. In case of low frequencyit S ox V G
sub
In other words,

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

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

60. 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.

61. We can also obtain YS via

62. 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.

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

64. 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

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

66. 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:

67. 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.

68. 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.

69. 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:

71. This characteristics is exploited in charge storage devices such as CCD