436
charge distribution !(x) in the oxide is given by
! t
x!(x)
1
_
'VFB # $ _
dx
Cox 0 tox
ox
10.26
Sem
Using 1
the2023
results of Problem 10.25, calculate the shift in flELL211
at-band voltage for
HW6 (20 pts)
tox # 20 nm # 200 Å for the following oxide charge distributions:
(a) Q %ss # 8 & 1010 cm$2 is entirely located at the oxide–semiconductor interface,
$2
Q ss% # 8 & 1010a cm
is capacitor
uniformly distributed
the oxide,
andand n-type silicon substrate. Assume Na
++ polysilicon
1.(b)(3)Consider
MOS
with an nthroughout
gate
10
$2
(c) Q ss% # 8 & 10 cm forms a triangular distribution with the peak at the oxide–
= 1016 cm-3
and let
- Ecat=the
0.2metal–oxide
eV in theinterface.
n++ polysilicon. Assume the oxide has a thickness of
F zero
semiconductor
interface
andEis
!
= 300
Also assume
thatbyχusing
(polysilicon)
=χ!and
(singlecrystal silicon). (a) Sketch the energy10.27
Antox
ideal
MOSÅ.
capacitor
is fabricated
intrinsic silicon
an n! polysilicon
gate.
(a)
Sketch
the
energy-band
diagram
through
the
MOS
structure
under
band diagrams (i) for VG = 0 and (ii) at flat band. (b) Calculate the metal–semiconductor work
flat-band
conditions.
(b) Sketch
the low-frequency
C–V characteristics
from
function
difference.
(c) Calculate
the threshold
voltage for
thenegaideal case of zero fixed oxide charge
tive to positive gate voltage.
and zero interface states
10.28
Consider a MOS capacitor with a p-type substrate. Assume that donor-type
onlycapacitor
at midgap (i.e.,
EFin).+Sketch
the high-frequency
2.interface
(3)An traps
idealexist
MOS
withatan
polysilicon
gate has aC–V
silicon dioxide thickness of tox = 12
Hu_ch07v3.fm
287 Friday, February 13, 2009 4:55 PM
curve from accumulation to inversion. Compare this sketch to the ideal
C–VPageplot.
16
-3
nm
on
a
p-type
silicon
substrate
doped
at
N
=10
cm
.
Determine
the capacitance Cox, C’FB
a
10.29 Consider an SOS capacitor as shown in Figure P10.29. Assume the SiO2 is ideal
,C’
and
C’(inv)
at(
a)f=1Hz
and
(b)
f=1MHz
min
(no trapped
charge) and has a thickness of tox # 500 Å. The doping concentraProblems
tions are Nd # 1016 cm$3 and Na # 1016 cm$3. (a) Sketch the energy-band dia3.gram
(3)Consider
an
SOS
capacitor
as
shown
in
Figure.
Assume
the
SiO
is
ideal
(no
trapped
charge)
and
–3
2
through the device for (i) flat band, (ii) VG # !3 V, and (iii) VG # $3(c)V.L = 1 µm, Na = 1E15 cm .
–3
µm, N16
a = 1E17 cm
(b)has
Calculate
the flat-band
the voltage
across the oxide
for
a thickness
of toxvoltage.
= 500 (c)
Å.Estimate
The doping
concentrations
are(d)
NLd= 1=10
cm-3. and Na = 1016 cm-3.
Please pay attention to the positions of the curves relative to each other and label all
(i)(a)
VG Sketch
# !3 V the
and energy-band
(ii) VG # $3 V. diagram
(d) Sketchthrough
the high-frequency
C–V
the device
forcharacter(i)curves.
flat band, (ii) VG =+3 V, and (iii) VG
istic curve.
● Trade-off between I
and I ●
off the
on oxide for (i) VG =3 V
=-3 V. (b) Calculate the flat-band voltage. (c) Estimate the voltage across
7.4 Does each of the following changes increase or decrease Ioff and Ion? A larger Vt. A
and (ii) VG =-3 V.
larger L. A shallower junction. A smaller Vdd. A smaller Tox. Which of these changes
contribute to leakage reduction without reducing the precious Ion?
tox
7.5 There is a lot of concern that we will soon be unable to extend Moore’s Law. In your
own words, explain this concern and the difficulties of achieving high Ion and low Ioff.
(a) Answer this question in one paragraph of less than 50 words.
VG
n type
SiO2
p type
(b) Support your description in (a) with three hand-drawn sketches of your choice.
(c) Why is it not possible to maximize Ion and minimize Ioff by simply picking the right
values of Tox, Xj, and Wdep? Please explain in your own words.
(d) Provide three equations that help to quantify the issues discussed in (c).
7.6 (a) Rewrite Eq. (7.3.4) in a form that does not
but contains Vt. Do so by
4. (3)An
hasProblem
the following
tox = 400 Å, Na =1015contain
cm-3W,depand
FigureNMOS
P10.29 device
| Figure for
10.29. parameters: n+ poly gate,
using Eqs. (5.5.1) and (5.4.3) assuming that Vt is given.
’
10
-2
Q ss= 5 x 10 cm . (a) Determine VT. (b) Is it possible to apply
a VSB
voltage
that
If taken
so, to reduce the
T = 0?
(b) Based
on the
answer tosuch
(a), state
what V
actions
can be
minimum acceptable channel length.
what is the value of VSB?
7.7
(a) What is the advantage of having a small Wdep?
given L and Vt, what is the impact of reducing Wdep on Idsat and gate? (Hint:
5. (3) Suppose you have a MOSFET whose gate width changes (b)as For
a function
of distance along the
consider the “m” in Chapter 6)
channel as: W(x) = W0 + x where x = 0 at the source and x =Discussion:
L at the
drain.
forbecause
its gate
it is more important to be able
Overall,
smallerExcept
Wdep is desirable
to suppress Vt roll-off so that L can be scaled.
width, assume that this MOSFET is like the typical MOSFET. (Do not consider velocity saturation.)
● MOSFET with Ideal Retrograde Doping Profile ●
(a) Find an expression for Id for this device. Ignore the bulk
chargeN-channel
effectMOSFET
(mPM
=with
1).an(b)
Derive an
12:40
7.8 Assume an 12/11/10
N+ poly gate and a substrate with an idealized
retrograde substrate doping profile as shown in Fig. 7–23.
expression for Idsat for this device
6. (5)Assume an N-channel MOSFET with an N+ poly gate
and a substrate with an idealized retrograde substrate doping
profile as shown in Figure.
Nsub
Gate
Oxide
Substrate
P!
a) Draw the energy band diagram of the MOSFET along the
Very light P type
x direction from the gate through the oxide and the
substrate, when the gate is biased at threshold voltage.
T
X
(Hint: Since the P region is very lightly doped you may
FIGURE 7–23
assume that the field in this region is constant or dE/dx =0) 0). Assume that the Fermi level in the P+
region coincides with Ev and the Fermi level in the N+ gate coincides with Ec. Label Ec, Ev, and EF
ox
rg
b)Find an expression for Vt of this ideal retrograde device in terms of Vox. Assume Vox is known.
(Hint: Use the diagram from (a) and remember that Vt is the difference between the Fermi levels in
the gate and in the substrate. At threshold, Ec of Si coincides with the Fermi level at the Si–SiO2
interface).
c) Now write an expression for Vt in terms of Xrg, Tox, εox, εsi and any other common parameters
you see fit, but not in terms of Vox. Hint: Remember Nsub in the lightly doped region is almost 0, so
if your answer is in terms of Nsub, you might want to rethink your strategy.
d) Show that the depletion layer width, Wdep in an ideal retrograde MOSFET can be about half the
Xdep of a uniformly doped device and still yield the same Vt.
x
287