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Transcript of HO #2: ELEN 384 - MOS CapacitorsPage 1S. Saha Basic Equations for Device Operation Transport...
HO #2: ELEN 384 - MOS Capacitors Page 1S. Saha
Basic Equations for Device Operation
Transport equations: The total e- and hole current densities are the sum of the drift due to electric field, and diffusion due to concentration gradient components:
where Jn and Jp are e- and hole current densities, respectively
n and p are e- and hole concentrations, respectively n and p are e- and hole mobilities, respectively.
The total current density is given by: J = Jn + Jp
)(dx
dnDnqJ nnn (1)
(2))(dx
dpDpqJ ppp
HO #2: ELEN 384 - MOS Capacitors Page 2S. Saha
Continuity Equations
Continuity equations: based on the conservation of mobile charge, e- and hole continuity equations are given by:
where Gn = e- generation rate
Rn = e- recombination rate
Gp = hole generation rate
Rp = hole recombination rate
pp
p
nnn
GRdx
dJ
qdt
dp
GRdx
dJ
qdt
dn
1
1(3)
(4)
HO #2: ELEN 384 - MOS Capacitors Page 3S. Saha
Poisson’s Equation
Poisson’s equation:
where = electric field (x) = space charge density (charge/cm3)
K = dielectric constant
o = permittivity of free space = 8.854x10-14 F/cm
oK
x
dx
d
)(
:form General (5)
oK
x
dx
d
dx
d
dx
d
)(2
2
:as Equation sPoisson' writecan weTherefore,
potential,tic electrosta if Again,
(6)
HO #2: ELEN 384 - MOS Capacitors Page 4S. Saha
Poisson’s Equation
Another form of Poisson’s Equation (Gauss’s Law):
where QS = integrated charge density
In silicon, consists of:
mobile charges: e- and holes with concentrations n and p, respectively.
fixed charges: ionized acceptors and donors with concentrations Na
and Nd+, respectively.
)()()()()(
2
2
xNxNxnxpK
q
K
x
dx
dad
oo
:as Equation sPoisson' writecan weTherefore,
(8)
0
)(
K
Q
K
dxx S
o (7)
HO #2: ELEN 384 - MOS Capacitors Page 5S. Saha
Carrier Concentration - Boltzmann’s Relations
potential intrinsic level;energy intrinsic
potential Fermi level; Fermi
ionconcentrat carrierintrinsic
where
:by given is ionconcentrat the tor,semiconduc type-p a In
:by given is ionconcentrat the tor,semiconduc type-n an In
q
EE
q
EE
n
enenp
hole
enenn
e-
fii
fff
i
kT
q
ikT
EE
i
kT
q
ikT
EE
i
iffi
fiif
)(
)(
(9)
(10)
HO #2: ELEN 384 - MOS Capacitors Page 6S. Saha
Carrier Concentration
type-p for
type-n for
midgap the from potential Fermi of separation where
: writecan we(10) and (9) From
ionconcentrat acceptor
ionconcentrat donor
where
:torsemiconduc type-p a In
:torsemiconduc type-n an In
ab
db
B
i
bifB
a
d
a
ipa
d
ind
NN
NN
n
N
q
kT
N
N
N
nnNp
N
npNn
ln
;
;
2
2
(11)
(12)
(13)
HO #2: ELEN 384 - MOS Capacitors Page 7S. Saha
MOS Capacitors - 1. Band Diagrams
Note:• three materials in contact, EF
= constant at equilibrium.
Metal(Al)
Oxide(SiO2)
Semiconductor(p-Si)
E0
qm= 4.1eV
EFM
qox=0.95eV
Eg 8eV
Ec
Ev
EF
E0
Ec
EFEv
qS=4.9eVqS= 4.05eV
Eg=1.12eVMOS structure
p-Si
SiO2
• currents through SiO2 are very small.• holes flow from semiconductor metal on contact.• e flow from metal semiconductor on contact.• bands will bend downwards in silicon at the interface (m < s) where m = metal work function
s = p-type semiconductor work function = S + Eg/2 + B.
HO #2: ELEN 384 - MOS Capacitors Page 8S. Saha
MOS Capacitor at Equilibrium: Applied V = 0
• Abrupt transition in Ec and Ev levels at the material interfaces.
• For m < s, band bends downward.
• A typical potential drop ~ 0.4 eV across SiO2. This depends on EF in Si. This potential can be supported because no current flows through SiO2.
• Substantial barriers exist to current flow from:
S M (~ 3.10 eV)
M S (~ 3.15 eV).
Ec
Ev
EF
Al SiO2 p-Si
3.8 eV
3.10 eV3.15 eV
0.4 eVEc
Ev
• Depletion region exists near the surface because EF near the surface is further from Ev than the bulk region.
• An applied voltage, Vfb = m - s = ms can make the band flat.
HO #2: ELEN 384 - MOS Capacitors Page 9S. Saha
MOS Capacitor - Flat Band, V = Vfb
• EF is constant in Si because SiO2 prevents any current flow (i.e., Si is at equilibrium).
• Ec and Ev are flat.
• hole concentration is uniform in Si and SiO2 (as shown by flat Ec and Ev).
FLAT-BAND.
• The voltage required to achieve flat band condition is Vfb = ms.
fsgsmms
fsgsmms
E
E
2/
2/
:type)-/Si(n Al/SiOIn
:type)-/Si(p Al/SiOIn
2
2
HO #2: ELEN 384 - MOS Capacitors Page 10S. Saha
MOS Capacitor – Accumulation, V < VfB
• EF is still constant in the Si since SiO2 prevents any current flow.
• EF is closer to Ev at the surface.
more holes accumulated near the surface than the bulk.
ACCUMULATION.
qVEFMEc
EF
Ev
Al SiO2 p-Si
ve
HO #2: ELEN 384 - MOS Capacitors Page 11S. Saha
MOS Capacitor – Depletion, V > VFB
• EF is still constant in Si (I = 0).
• EF is farther from Ev at the surface.
The surface is depleted of holes.
DEPLETION.
qVEFM
Ec
EFEv
Al SiO2 p-Si
ve
EiB
Qg
Qd
HO #2: ELEN 384 - MOS Capacitors Page 12S. Saha
• EF is still constant in Si (I = 0).
• EF is closer to Ec at the surface than it is to Ev.
more e- than holes at the surface.
• If s = 0, the surface becomes intrinsic
• If 0 < s < B, the surface will become slightly n-type:
WEAK INVERSION.
• If s > B, the surface will have an e- concentration > hole concentration in the bulk STRONG INVERSION.
MOS Capacitor – Inversion, V >> VFB
B
s
HO #2: ELEN 384 - MOS Capacitors Page 13S. Saha
We now derive the relation among the surface potential, electric field, and charge (Qs) by solving Poisson’s eqn in the surface region of silicon.
For a metal/oxide/p-type silicon system, we define potential at any point x:
2. Electrostatic Potential and Charge Distribution
(x) i(x) i(x ) = amount of band bending at any point x.
where (x = 0) s = surface potential; i(x ) =f = Fermi potential
Thus, (x) < 0 when bands bend upward in accumulation (x) > 0 when bands bend downward in depletion and inversion
HO #2: ELEN 384 - MOS Capacitors Page 14S. Saha
Band Bending
)()()()()(
:isEquation sPoisson' The
2
2
xNxNxnxpK
q
K
x
dx
dad
oo
(14)
)()(
and silicon,-pbulk in the (12) Eq. from Also,
2
2
aa
iad
da
ia
NNnxNxN
NNnnNp
qkTveNn
xpnxn
eNenenxp
ttv
x
a
ii
tv
x
atv
x
itv
x
i
Bif
/ where
silicon, type-pa for get we(10) and (9) Eq. From
)(22
)()()(
)()(
)(
(15)
(16)
HO #2: ELEN 384 - MOS Capacitors Page 15S. Saha
Band Bending
Thus, the charge density at a depth x in the substrate is given by:
Substituting n(x), p(x), Na, and Nd in Poisson’s equation we can determine surface charge density in silicon substrate.
11
)()()(
)(2)(
tt v
x
a
iv
x
a
ad
eN
neNq
NNxnxpqx
(17)
11)(2)(
02
2tt v
x
a
iv
x
a eN
neN
K
q
dx
d
(18)
HO #2: ELEN 384 - MOS Capacitors Page 16S. Saha
Band Bending
Poisson’s equation:
Multiplying both sides of (19) by 2d((x))/dx, we get:
11)(2)(
02
2tt v
x
a
iv
x
a eN
neN
K
q
dx
d
(19)
2
22
2dx
ud
dx
du
dx
du
dx
d
:know We (20)
dx
de
N
neN
K
q
dx
d
dx
d
dx
de
N
neN
K
q
dx
d
dx
d
tt
tt
v
x
a
iv
x
a
v
x
a
iv
x
a
112
112
2
)(2)(
0
2
)(2)(
02
2
or, (21)
HO #2: ELEN 384 - MOS Capacitors Page 17S. Saha
Band Bending
Integrate from the bulk ( = 0, d/dx = 0) toward the surface at any point x to get:
At x = 0, = s and E = Es, then from Gauss’s law (7), the total charge per unit area induced in silicon (equal and opposite to the charge on the metal gate is: Qs = K0Es
)(
)]1)(
()1)(
[(2
)(
)]1()1([2
2
)(
2
2)(
0
2
)(
0
)(2)(
00
2
xE
v
xe
N
n
v
xe
K
vqN
dx
d
dx
dxEx
deN
neN
K
q
dx
dd
t
v
x
a
i
t
v
x
ta
x v
x
a
iv
x
adx
d
tt
tt
, at fieldelectric the
(22)
(23)
HO #2: ELEN 384 - MOS Capacitors Page 18S. Saha
Band Bending
The charge expression above is valid in all regions of MOS capacitor operation - accumulation, depletion, and inversion. The Eq can, also, be expressed as:
2/12
2
0
0
)]1()1[(2
t
sv
a
i
t
svta
ss
ve
N
n
vevNqK
EKQ
t
s
t
s
(24)
2/1
0
00 )]1()1[(2
t
sv
p
p
t
sv
d
ts v
ep
n
ve
L
KvQ t
s
t
s
(25)
add Nq
kTK LL
20
:as defined length Debye where (26)
HO #2: ELEN 384 - MOS Capacitors Page 19S. Saha
Surface Charge Density vs. Surface Potential
Note from (24):• for s < 0: Qs > 0 ACCUMULATION
Qs ~ exp(-s/2vt)
• for s = 0: Qs = 0 FLAT-BAND
• for s > 0: Qs < 0 DEPLETION and
WEAK-INVERSION
Qs ~ SQRT(s)
• fors > 2f
INVERSION
Qs ~ exp(s/2vt)
)2/exp(~ ts v
)2/exp(~ ts v
s~
Plot of (24) showing the variation of the induced surface charge density as a function of surface potential for p-type substrate.
HO #2: ELEN 384 - MOS Capacitors Page 20S. Saha
Depletion Approximation
• For the simplicity of analytical integration, we can assume that in the depletion region B < s < 2B, we can approximate (23) as:
• We know: at x = 0, (x) = s; Therefore,
00
21 or, ;
2
K
qN
dx
d
K
qN
dx
d aa (27)
22
0
0 0
12
1)(
2)(
ds
s
as
a
s
W
xx
K
qNx
dxx
K
qNx
d
(28)
(29)
HO #2: ELEN 384 - MOS Capacitors Page 21S. Saha
Depletion Approximation
• (29) is a parabolic Eq with the vertex at (x) = 0, x = Wd. Thus, Wd = distance to which band-bending extends = width of the depletion region.
• Then the depletion layer charge density is given by:
• In an MOS system, Wd reaches a maximum, Wd,max at the onset of inversion defined by s 2B = (2kT/q)ln(Na/ni).
a
sd
ds qN
KW
W
xx
0
22
and ;1)( Thus,
(30)
sadad NqKWqNQ 02 (31)
a
iad Nq
nNkTKW
20
max,
)/ln(4 (32)
HO #2: ELEN 384 - MOS Capacitors Page 22S. Saha
Strong Inversion
• Beyond strong inversion, the concentration of inversion charge n(x) becomes significant. Therefore, from (23) we get:
• This equation must be solved numerically with boundary condition, = s at x = 0. From the solution of (x), the inversion carriers, n(x) can be calculated from (16).
0.0E+00
2.0E+18
4.0E+18
6.0E+18
8.0E+18
1.0E+19
1.2E+19
0 50 100 150 200
Distance from surface, x (A)
Inv
ers
ion
lay
er
co
nc
en
tra
tio
n (
cm
-3)
N(sub) = 1016 cm-3
s = 0.88 V
s = 0.85 V
t
s
a
i
t
sta ve
N
n
vK
vqN
dx
d
2
2
0
2
• The numerically calculated n(x) Vs. depth plot is shown in Fig. It is seen that inversion charge distribution is extremely close to the surface with an inversion layer width < 50 A.
HO #2: ELEN 384 - MOS Capacitors Page 23S. Saha
Strong Inversion
• Generally, inversion-carriers must be treated quantum-mechanically (QM) as a 2-D gas. According to QM model:
– inversion layer carriers occupy discrete energy bands
– peak distribution is 10-30 A away from the surface.
• When the inversion charge density Qi > Qd, we get from (24):
t
s
a
iti
ve
N
nvqKQ
22
02
(33)
itiinv
ti
t
s
a
i
QvKqnQW
nvqKQ
ve
N
nne-
/2)0(/
)0(2
0
0
0
2
thickness, layerinversionClassical
)( :issurface the at ionconcentrat Since,
(34)
HO #2: ELEN 384 - MOS Capacitors Page 24S. Saha
3. Low Frequency C – V Characteristics
Assuming silicon at equilibrium. The total capacitance of an MOS structure is: C = d(-Qs)/dVg
where the applied gate bias, Vg of an MOS system is:
Vg = Vfb + Vox + s
or, dVg = dVox + ds
where Vox = voltage drop in the oxide
Thus, the total capacitance equals the oxide capacitance, Cox and the silicon capacitance, Cs connected in series.
sox
s
s
s
ox
s
g
CCC
V
Q
V
111
or,
(35)
HO #2: ELEN 384 - MOS Capacitors Page 25S. Saha
Low Frequency C – V Characteristics
From (24), we get:
From (36), we can calculate C - V characteristics in different regions of MOS operation.
2/1
2
2
2
2
0
)1()1(2
)1(12
)(
t
sv
a
i
t
sv
v
a
iv
d
s
ss
ve
Nn
ve
eNn
e
L
K
d
QdC
t
s
t
s
t
s
t
s
(36)
HO #2: ELEN 384 - MOS Capacitors Page 26S. Saha
Low Frequency C – V: Accumulation, 0 << s
In strong accumulation, 0 << s . Then from (36), we can show:
Since 0 << s, for large s, Cs becomes very large.
t
s
t
s
t
s
v
dv
v
ds e
L
K
e
e
L
KC 20
2
0
22
2
(37)
ox
oxsox
CC
CCCC
:onaccumulati inThus,
1111
(38)
HO #2: ELEN 384 - MOS Capacitors Page 27S. Saha
Low Frequency C – V: Flat Band, s = 0
For s = 0: the inversion charge term in (36) can be neglected to get:
Expanding the exponential term into power series and retaining the first five terms, we get:
2/10 )]1[(2
t
sv
d
ts v
eL
vKQ t
s
(39)
2
1
2
20
2/1
4
4
3
3
2
20
12
1
3
11
124
1
6
1
2
11
2
t
s
t
s
d
s
t
s
t
s
t
s
t
s
t
s
d
ts
vvL
K
vvvvvL
vKQ
(40)
HO #2: ELEN 384 - MOS Capacitors Page 28S. Saha
Low Frequency C – V: Flat Band
2
20
24
1
6
11
211
t
s
t
s
d
ss
vvL
KQ
xx
of expansionBinomial know, We
(41)
1
0
0
1
0)(
K
L
CCC
L
K
sd
sdQC
d
oxfb
dss
band Flat
(42)
Thus, Eq. (42) shows that Cfb is somewhat less than Cox.
HO #2: ELEN 384 - MOS Capacitors Page 29S. Saha
Low Frequency C – V: Depletion, B < s < - B
In depletion, the general expression for Cd is given by (36). However, we can derive an approximate expression from depletion approximation using Eq. (30) and (31).
We know that:
Thus, C decreases with the increase in Vg
ds
dd W
K
d
QdC 0)(
0
2
0
21
2
KqN
VC
CC
C
NqK
C
Q
C
QV
a
gox
oxs
sox
sas
ox
ds
ox
sg
:show can we, gEliminatin
(43)
HO #2: ELEN 384 - MOS Capacitors Page 30S. Saha
Low Frequency C – V: Strong Inversion
In strong Inversion, s >> . Then from (36), we can show that:
Negative sign indicates that the charge has changed sign.
Since s >> 0, for large s, Cs becomes very large.
ox
oxsox
CC
CCCC
:inversion strongin Thus,
1111
t
s
t
s
t
s
v
a
i
dva
vi
ds e
N
n
L
K
eN
en
L
KC 20
2
0
22
2
(44)
HO #2: ELEN 384 - MOS Capacitors Page 31S. Saha
4. Intermediate and High Frequency C - V
• There are plenty of majority carriers which can respond to the A.C. signal.
• The minority carriers, on the other hand, are scarce and have to originate from:
– diffusion from bulk substrate
– generation in the depletion region
– external sources (e.g. n+ diffusion region etc.)
• Thus, the inversion charge can not respond to the applied A.C. signal higher than 100 Hz.
• Therefore, at any high frequency applied signal to Vg:
Qi is assumed to be constant
Qd = Cs
HO #2: ELEN 384 - MOS Capacitors Page 32S. Saha
Intermediate and High Frequency C - V
• At any high frequency applied signal to Vg:
– Qd will vary with the signal around its maximum value Qd,max
– Wd will vary with the signal around its maximum value Wd,max.
• Thus, the high frequency capacitance is given by:
a
ia
ox
d
ox
NqK
nNvt
CC
CK
W
CC
0min
min0
max,
)/ln(411
111
(45)
HO #2: ELEN 384 - MOS Capacitors Page 33S. Saha
MOS C - V Characteristics
HO #2: ELEN 384 - MOS Capacitors Page 34S. Saha
5. Oxide and Interface Charges
HO #2: ELEN 384 - MOS Capacitors Page 35S. Saha
5. Oxide and Interface Charges
• Types of origins:
– fixed interface charge, Qf: believed to be uncompensated
Si-Si bonds, on the order of +1010 cm-2.
– interface trapped-charge, Qit: due to un-terminated Si
bonds, on the order of 1010 cm-2.
– Oxide trapped-charge, Qot: due to defects in the SiO2
network, usually negligible in advanced MOSFET devices. Can be guaranteed after a large amount of charges have passed through the SiO2 and broken up the network.
– mobile charge, Qm: sue to alkaline ions (e.g., Na, K), usually
very low concentration (~ 109 cm-2) in advanced technology.
HO #2: ELEN 384 - MOS Capacitors Page 36S. Saha
Effects of Qf and Qot on C - V
• The flat-band voltage change due to Qf and Qot is:
• If Qot is small, Qf can be estimated by Vfb.
ox
oxot
oxox
f
ox
oxoxox
ox
ox
oxoxfb
xdx
x
xQ
CC
Q
xdx
x
xx
K
x
xxdxx
KV
0
0
0
1
)(
)(1
(46)
HO #2: ELEN 384 - MOS Capacitors Page 37S. Saha
Effects of Qf and Qot on C - V
Figure.
HO #2: ELEN 384 - MOS Capacitors Page 38S. Saha
Effects of Qm on C - V
By bias-temperature stress, Qm can be moved from Si-SiO2 interface to the metal-SiO2 interface. The shift in VFB can be used to estimate Qm.
Figure.
HO #2: ELEN 384 - MOS Capacitors Page 39S. Saha
Effects of Qit on C - V
• Interface trapped-charge, Qit:
– the traps have energy levels distributed in the band gap
– the charge state of the traps depends on Fermi level.
thus, as the surface potential is swept from flat band towards depletion, the traps will try to pin the surface potential, i.e. it will take more gate voltage to move the surface potential through the traps.
HO #2: ELEN 384 - MOS Capacitors Page 40S. Saha
Effects of Qit on C - V
Assume acceptor type traps:
Thus, Cs is in parallel with Cit. Hence, the capacitor model is:
densitytypeacceptor)(
densitytypeacceptor)(
sdit
sait
D
D
its
s
sit
s
ss
s
total
sitsstotal
CC
QQQ
QQQ
Total
))(())((
)()(
(47)
)(
)()(
sit
sdits
aitit
qD
DqDC
(48)
HO #2: ELEN 384 - MOS Capacitors Page 41S. Saha
6. Polysilicon Gates
At equilibrium (Vg = 0).
Ec
Ev
EF
n+ poly SiO2 p-Si
Ei
B
Work-function difference:
n+poly with p-type Si: MS = Eg/2 (kT/q)ln(NA/ni) (49)
p+poly with n-type Si: MS = Eg/2 + (kT/q)ln(ND/ni) (50)
At Inversion (Vg >> 0).
Ec
Ev
EF
n+ poly SiO2 p-Si
EiS
VG
Ei
Ev
EF
p
Ec
Vox
HO #2: ELEN 384 - MOS Capacitors Page 42S. Saha
Polysilicon Depletion Effect
Note:
• Capacitance at inversion, Cinv does not return to Cox.
• Cinv shows a maximum value, Cmax < Cox.
• Cmax as the polysilicon doping concentration, Np.
• As Np, depletion width
Cmax Coxfor higher Np.
Total capacitance at strong inversion is given by:
1/C = 1/Cox + 1/Csilicon + 1/Cpoly
As VG, Csiliconbut xd(poly) Cpoly.
Low frequencyC V shows a local maximumat a certain VG.
HO #2: ELEN 384 - MOS Capacitors Page 43S. Saha
7. Inversion Layer Quantization
• Typically, near the silicon surface, the inversion layer charges are confined to a potential well formed by:– oxide barrier
– bend Si-conduction band at the surface due to the applied gate potential, VG.
• Due to the confinement of inversion layer e (in p-Si): – e- energy levels are grouped in discrete sub-bands of energy, Ej
– each Ej corresponds to a quantized level for e motion in the normal direction.
E2
E1
E0
Edge of EC
Distance from the surfaceBottom ofthe well
E
HO #2: ELEN 384 - MOS Capacitors Page 44S. Saha
Inversion Layer Quantization
• Due to Quantum Mechanical (QM) effect, the inversion layer concentration:– peaks below the SiO2/Si interface
0 at the interface determined by the boundary condition of the e- wave function.
• Solve Schrodinger and Poisson Eq self-consistently with the boundary conditions for wave function equal to:– 0 for x < 0 in oxide– 0 at x = .
QM
Classical
Depth
n (c
m-3)
HO #2: ELEN 384 - MOS Capacitors Page 45S. Saha
QM Effect on MOS Capacitors
• At high fields, Vth since more band bending is required to populate the lowest sub-band, which is some energy above the bottom of EC.
• Once the inversion layer forms below the surface, a higher Vg over-drive is required to produce a given level of inversion charge density. That is, the effective gate oxide thickness, tOX
eff by:
tOX = (ox/si)z (51)
• Inversion layer quantization can be treated as bandgap widening due to an increase in the effective bandgap by Eg given by:
Here, Eg = EgQM Eg
CL.
(52) niand ndue to QM effect.
kT
ECLi
QMi
g
enn 2
(52)
HO #2: ELEN 384 - MOS Capacitors Page 46S. Saha
Home Work 1: Due 0ct 6, 2005
1) An MOS capacitor is built with the structure shown below. The capacitor areas over the N and P regions are the same. The threshold voltages for the N and P substrates are -1 and +1 volts respectively. Sketch the shape of the high frequency (1 MHz) C-V curve that you would expect to measure for this structure. Label as many points as possible and explain.
2) In practical MOS systems, measurements of C - V sometimes show hysteresis effects; that is, the C - V curves look like the sketch in Fig. The sketch refers to measurements made when VG is swept with a very low frequency triangular wave (~ 1 Hz) and the A.C. measurement frequency is of the order of 1 kHz or higher. The sense of the hysteresis on such a curve can be observed experimentally to be either counterclockwise, as shown, or else be clockwise.
N P
SiO2
Al
HO #2: ELEN 384 - MOS Capacitors Page 47S. Saha
Home Work 1: Due Oct 6, 2005
The hysteresis sense allows one to differentiate between the two most common cause of non-ideal behavior.
(a) Show this by considering the following non-ideal effects: (i) field-aided movement of positive ions in the insulator and (ii) trapping of free
carriers from the channel in traps at the oxide-Si interface Q it.
(b) Using qualitative reasoning, prepare a table with sketches of the expected C - V plots for n- and p-type substrates; on each sketch indicate the sense of the hysteresis (i.e., clockwise or counterclockwise).
HO #2: ELEN 384 - MOS Capacitors Page 48S. Saha
Home Work 1: Due Oct 6, 2005
3) MOS capacitors are fabricated on uniformly doped P-substrates with NA = 1x1017 cm3, physical gate oxide thickness TOX = 6.7 nm, and N+ poly-silicon gate doping concentrations Np = 5x1018, 1x1019, and 5x1019 cm3 as shown in the following C – V plots. Here C and Cox are the gate and oxide capacitances, respectively.
(a) Explain the observed variation in C/COX vs. Vg plots for each Np in the strong inversion region.
(b) Explain the observed increase in C/COX for Np = 5x1018 cm3 and Vg 3.5 V.
(c) Describe the possible reasons for C/COX < 1 even for a heavily doped poly with Np = 5x1019 cm3 in the accumulation as well as in the strong inversion regions?
HO #2: ELEN 384 - MOS Capacitors Page 49S. Saha
Home Work 5: Due Oct 6, 2005
4) A p-type MOS-capacitor with Na = 1x1018 cm-3 and TOX = 3 nm was
fabricated to characterize van Dort’s analytical bandgap widening quantum mechanical (QM) model we discussed in class. Due to inversion layer quantization, the increase in the effective bandgap Eg = Eg
QM EgCL 104
mV. Here EgQM and Eg
CL represent the QM and classical (CL) values of
bandgap (Eg), respectively. Assume Qf = 0, VSUB = 0, and N+ poly gate.
(a) Show that the intrinsic carrier concentration due to QM effect is given by:
(b) Calculate the value of niQM at temperature T = 300oK.
(c) Calculate the value of threshold voltage Vth(CL) using the classical
approach.
(d) Calculate the value of threshold voltage Vth(QM) due to QM effects.
(e) Calculate the shift Vth due to QM effects.
kT
ECLi
QMi
g
enn 2