Transcript No Slide Title

Basic Equations for Device Operation
Transport equations: The total e- and hole current densities
are the sum of the drift due to electric field, e and diffusion
due to concentration gradient components:
dn
Jn  q (nmne  Dn )
dx
(1)
dp
Jp  q ( pmpe  Dp )
dx
(2)
where
Jn and Jp are e- and hole current densities, respectively
n and p are e- and hole concentrations, respectively
mn and mp are e- and hole mobilities, respectively.
The total current density is given by: J = Jn + Jp
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 1
Continuity Equations
Continuity equations: based on the conservation of mobile
charge, e- and hole continuity equations are given by:
dn 1 dJn

 Rn  Gn
dt q dx
dp
1 dJp

 Rp  Gp
dt
q dx
(3)
(4)
where
Gn = e- generation rate
Rn = e- recombination rate
Gp = hole generation rate
Rp = hole recombination rate
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 2
Poisson’s Equation
Poisson’s equation:
General form :
de r ( x)

dx
Keo
(5)
where e = electric field
r(x) = space charge density (charge/cm3)
K = dielectric constant
eo = permittivity of free space = 8.854x10-14 F/cm
d
dx
Theref ore, w ecan w rite Poisson' s Equation as :
Again, if   electrostatic potential, e  
d 2
de
r ( x)




dx2
dx
Keo
S. Saha
HO #2: ELEN 384 - MOS Capacitors
(6)
Page 3
Poisson’s Equation
Another form of Poisson’s Equation (Gauss’s Law):
r ( x)dx QS
e 

Keo
Ke 0
(7)
where QS = integrated charge density
In silicon, r 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.
Therefore, w ecan w rite Poisson' s Equation as :
d 2
r ( x)
q




p( x)  n( x)  N d ( x)  N a ( x)
2
dx
Keo
Keo

S. Saha
HO #2: ELEN 384 - MOS Capacitors

(8)
Page 4
Carrier Concentration - Boltzmann’s Relations
In an n - type semiconductor, the e- concentration is given by :
n  ni e
E f  Ei
kT
 ni e
q (i  f )
kT
(9)
In a p - type semiconductor, the hole concentration is given by :
p  ni e
Ei  E f
kT
 ni e
q ( f i )
kT
(10)
w here
ni  intrinsic carrier concentration
E f  Fermi level;  f  
Ef
q
 Fermi potential
Ei  intrinsic energy level; i  
S. Saha
Ef
q
 intrinsic potential
HO #2: ELEN 384 - MOS Capacitors
Page 5
Carrier Concentration
ni2
In an n - type semiconductor : n  N d ; pn 
Nd
(11)
ni2
In a p - type semiconductor : p  N a ; n p 
Na
(12)
w here
N d  donor concentration
N a  acceptor concentration
kT  N b 

From (9) and (10) w ecan w rite: B   f  i 
ln
q  ni 
(13)
w hereB  separation of Fermi potential from the midgap
N b  N d for n - type
N b  N a for p - type
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 6
MOS Capacitors - 1. Band Diagrams
SiO2
E0
qcox=0.95eV
qFm= 4.1eV
p-Si
E0
Ec
qcS= 4.05eV
qFS=4.9eV
Ec
EFM
MOS structure
EF
Metal
(Al)
Note:
Eg 8eV
Ev
Eg=1.12eV
Semiconductor
(p-Si)
•
three materials in contact,
EF = constant at equilibrium.
•
•
•
•
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 (Fm < Fs)
where Fm = metal work function
Oxide
(SiO2)
EF
Ev
Fs = p-type semiconductor work function = cS + Eg/2 + B.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 7
MOS Capacitor at Equilibrium: Applied V = 0
0.4 eV
Ec
•
Abrupt transition in Ec and Ev levels at
the material interfaces.
3.10 eV
•
For Fm < Fs, band bends downward.
3.15 eV
Ec
EF •
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:
Ev
3.8 eV
Ev
Al
SiO2
p-Si
S  M (~ 3.10 eV)
M  S (~ 3.15 eV).
•
Depletion region exists near the surface because EF near the surface
is further from Ev than the bulk region.
•
An applied voltage, DVfb = Fm - Fs = Fms can make the band flat.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 8
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.
In Al/SiO2 /Si(p- type):
ms  m  c s  E g / 2   fs 
In Al/SiO2 /Si(n- type):
ms  m  c s  E g / 2   fs 
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 9
MOS Capacitor – Accumulation, V < VfB
EFM
Ec
EF
Ev
qV
 ve
Al
SiO2
p-Si
• 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.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 10
MOS Capacitor – Depletion, V > VFB
B
qV
EFM
Ec
Ei
EF
Ev
ve
Al
SiO2
p-Si
Qg
Qd
•
•
EF is still constant in Si (I = 0).
EF is farther from Ev at the surface.
 The surface is depleted of holes.
 DEPLETION.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 11
MOS Capacitor – Inversion, V >> VFB
B
Ys
•
•
EF is still constant in Si (I = 0).
•
•
If Ys = 0, the surface becomes intrinsic
•
If s >  B, the surface will have an e- concentration > hole
concentration in the bulk  STRONG INVERSION.
S. Saha
EF is closer to Ec at the surface than it is to Ev.
 more e- than holes at the surface.
If 0 < Ys <  B, the surface will become slightly n-type:
 WEAK INVERSION.
HO #2: ELEN 384 - MOS Capacitors
Page 12
2. Electrostatic Potential and Charge Distribution
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:
(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
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 13
Band Bending
T heP oisson's Equat ionis :
d 2
r ( x)
q






p
(
x
)

n
(
x
)

N
(
x
)

N
( x)
d
a
2
dx
Keo
Keo


(14)
From Eq.(9) and (10) we get for a p - type silicon,
 f i ( x )
p( x)  ni e

n( x ) 
ni2
p( x)
vt

 B  ( x )
 ni e
ni2
Na
vt

 Nae
 ( x)
vt
(15)
 ( x)
e
where
vt
vt  kT /q 
(16)
2
n
Also, fromEq. (12)in thebulk p - silicon, p  N and n  i  N d
Na

a

S. Saha
2
n
N ( x)  N ( x)  i  N a
Na

d

a
HO #2: ELEN 384 - MOS Capacitors
Page 14
Band Bending
Thus, the charge density at a depth x in the substrate is
given by:

r ( x)  q p( x)  n( x)  N d  N a

 ( x)
   ( x) 

2 
n




 q  N a  e vt  1  i  e vt  1
 
 Na 

 



(17)
Substituting n(x), p(x), Na, and Nd in Poisson’s equation we
can determine surface charge density in silicon substrate.
 ( x)
2   ( x)





n
d
q
v
v
 e t  1  i  e t  1



N
a
 Na 

dx2
Ke 0  




2
S. Saha
HO #2: ELEN 384 - MOS Capacitors
(18)
Page 15
Band Bending
Poisson’s equation:
 ( x)
 n 2   v( x ) 
d 2
q    vt
  i  e t  1



N
e

1
a
 Na 

dx2
Ke 0  




2
d  du 
du d 2u
We know :
  2
dx  dx 
dx dx2
 Multiplying both sides of (19) by 2d((x))/dx, we get:
(19)
(20)
 ( x)
2   ( x)




  d
n
d d 
2q
v
v
i
t
t
e
Na  e
2

 1 
 1
2
 Na 
 dx
dx dx
Ke 0  




2
 ( x)
2   ( x)




  d
n
d  d 
2q
v
v
i
 e t  1
 N a  e t  1 
or,    
(21)




dx  dx 
Ke 0  
 Na 
 dx

2
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 16
Band Bending
Integrate from the bulk ( = 0, d/dx = 0) toward the surface
at any point x to get:
d
dx
0

2q
 d 
d   
Ke 0
 dx 
2
 ( x)

0

[ N a (e
 ( x)
vt
 the electric field at x, E ( x)  
2qNa vt
 d 
  
[(e
Ke 0
 dx 
2

 ( x)
vt

 ( x)
vt
 1) 
2
i
n
(e
Na
 ( x)
vt
 1)]d
(22)
d
dx
 1) 
2
i
2
a
n
(e
N
 ( x)
vt

 ( x)
vt
 1)]
(23)
 E 2 ( x)
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 =  Ke0Es
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 17
Band Bending
Qs   Ke 0 Es
  2qKe 0 N a vt [(e

s
vt

s
vt
 1) 
s
n

(e vt  s  1)]1/ 2 (24)
N
vt
2
i
2
a
The charge expression above is valid in all regions of MOS
capacitor operation - accumulation, depletion, and inversion.
The Eq can, also, be expressed as:
2vt Ke 0
Qs  
[(e
Ld

s
vt

s
vt
 1) 
np0
pp0
s
(e 
vt
w hereL d  Debye length defined as : Ld 
S. Saha
HO #2: ELEN 384 - MOS Capacitors
s
vt
 1)]1/ 2 (25)
Ke 0 kT
q2 Na
(26)
Page 18
Surface Charge Density vs. Surface Potential
Plot of (24) showing the variation of the induced surface charge
density as a function of surface potential for p-type substrate.
Note from (24):
• for s < 0: Qs > 0
 ACCUMULATION
Qs ~ exp(-s/2vt)
• for s = 0: Qs = 0
 FLAT-BAND
~ exp(s / 2vt )
~ exp(s / 2vt )
• for s > 0: Qs < 0
 DEPLETION and
WEAK-INVERSION
Qs ~ SQRT(s)
~ s
• for s > 2f
 INVERSION
Qs ~ exp(s/2vt)
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 19
Depletion Approximation
• For the simplicity of analytical integration, we can assume
that in the depletion region B < s < 2B, we can
approximate (23) as:
2qNa
2qNa
d
1 d

; or,

dx
Ke 0
Ke 0
 dx
(27)
• We know: at x = 0, (x) = s; Therefore,
 ( x)

s
d

x
 
0
2qNa
dx
Ke 0
(28)
2



qN
x
a



  ( x )   s 1 
x    s 1 
2 Ke 0s 
 Wd

S. Saha
HO #2: ELEN 384 - MOS Capacitors



2
(29)
Page 20
Depletion Approximation

x
T hus,  ( x)  s 1 
 Wd
2

2 Ke 0s
 ; and Wd 
qNa

(30)
• (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:
Qd  qNaWd   2qKe 0 N as
(31)
• In an MOS system, Wd reaches a maximum, Wd,max at the
onset of inversion defined by s  2B = (2kT/q)ln(Na/ni).
4 Ke 0 kT ln(N a / ni )
Wd ,max 
q2 Na
S. Saha
HO #2: ELEN 384 - MOS Capacitors
(32)
Page 21
Strong Inversion
s 

2
2qNa vt  s ni vt 
d

  2e 
dx
Ke 0  vt N a



• 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).
•
S. Saha
Inversion layer concentration (cm -3)
• Beyond strong inversion, the concentration of inversion charge
n(x) becomes significant. Therefore, from (23) we get:
1.2E+19
N(sub) = 1016 cm-3
1.0E+19
8.0E+18
s = 0.88 V
6.0E+18
s = 0.85 V
4.0E+18
2.0E+18
0.0E+00
0
50
100
150
200
Distance from surface, x (A)
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 22
Strong Inversion
• Generally, inversion-carriers must be treated quantummechanically (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):
s
2qKe v n 2vt
Qi  
e
Na
2
0 t i
Since,e- concentration at the surface is : n(0) 
 Qi  2qKe 0 vt n(0)
(33)
ni2
Na
s
e
vt
(34)
Classical inversion layer thickness, Winv  Qi / qn(0)  2 Ke 0 vt / Qi
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 23
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 + ds
where Vox = voltage drop in the oxide

Vg
 Qs 
or,

Vox
s

 Qs   Qs 
1
1
1


C Cox Cs
(35)
Thus, the total capacitance equals the oxide capacitance,
Cox and the silicon capacitance, Cs connected in series.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 24
Low Frequency C – V Characteristics
From (24), we get:
d (Qs )
Cs 
ds


s
2
 s


ni
vt
vt
1  e  2 (e  1)
Na


2 Ke 0
1/ 2
s
s
Ld
2
 v 

n

v
2(e t  s  1)  i 2 (e t  s  1)
vt
Na
vt


(36)
From (36), we can calculate C - V characteristics in different
regions of MOS operation.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 25
Low Frequency C – V: Accumulation, 0 << s
In strong accumulation, 0 << s . Then from (36), we can
show:

s
s
2 Ke 0  e
Ke 0 2vt
Cs  

e
s

Ld
2 Ld
2 vt
2e
Since 0 << s, for large s, Cs becomes very large.
vt

(37)
1
1
1
1
 


C Cox Cs Cox
T hus, in accumulation :
C  Cox
S. Saha
HO #2: ELEN 384 - MOS Capacitors
(38)
Page 26
Low Frequency C – V: Flat Band, s = 0
For s = 0: the inversion charge term in (36) can be
neglected to get:
2 Ke 0vt
Qs  
[(e
Ld

s
vt

s
vt
 1)]1/ 2
(39)
Expanding the exponential term into power series and
retaining the first five terms, we get:
2 Ke 0 vt
Qs  
Ld
Ke 
 0 s
Ld
S. Saha
 
1  s 
 vt

1 

1 s2
2 vt2

1 s3
6 vt3

1
1 s 1 s2  2


2

3 vt 12 vt 
HO #2: ELEN 384 - MOS Capacitors
1 s4
24 vt4
 s 
   1
 v

t

1/ 2
(40)
Page 27
Low Frequency C – V: Flat Band
x
We know, Binomialexpansionof 1  x  1 
2
Ke 0s  1 s 1 s2 

 Qs  

1 
2
Ld  6 vt 24 vt 
dQs
 C s (Flat band)  
d s
C  C fb
 s 0
 1
Ld 




 Cox Ke 0 
(41)
Ke 0

Ld
1
(42)
Thus, Eq. (42) shows that Cfb is somewhat less than Cox.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 28
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).
d (Qd ) Ke 0
Cd 

ds
Wd
We know that:
Vg  
2qKe 0 N as
Qs
Q
 s  d  s 
 s
Cox
Cox
Cox
Cox
Eliminatin
g s , we can show : C 
1
2
2Cox
Vg
(43)
qN a Ke 0
Thus, C decreases with the increase in Vg
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 29
Low Frequency C – V: Strong Inversion
In strong Inversion, s >> 0. Then from (36), we can show
that:
2 Ke 0
Cs  
Ld
s
ni e vt
s
2 N a e 2 vt
s
Ke 0 ni 2 vt

e
2 Ld N a
(44)
Negative sign indicates that the charge has changed sign.
Since s >> 0, for large s, Cs becomes very large.
1
1
1
1



C Cox Cs Cox
T hus,in stronginversion:
C  Cox

S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 30
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:
S. Saha

Qi is assumed to be constant

Qd = C s
HO #2: ELEN 384 - MOS Capacitors
Page 31
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:
Wd ,max
1  1
 

C  Cox
Ke 0

S. Saha
1
Cmin


1
 
 Cmin
4vt ln(N a / ni )
1

Cox
qKe 0 N a
HO #2: ELEN 384 - MOS Capacitors
(45)
Page 32
MOS C - V Characteristics
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 33
5. Oxide and Interface Charges
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 34
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.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 35
Effects of Qf and Qot on C - V
• The flat-band voltage change due to Qf and Qot is:
xox
1
DV fb  
r ( x) xdx

K oxe ox 0
xox
xox
x

r ( x)
dx

K oxe ox 0
xox
xox
Qf
1
x


Qot
dx

 Cox
 0
Cox
xox
(46)
• If Qot is small, Qf can be estimated by DVfb.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 36
Effects of Qf and Qot on C - V
Figure.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 37
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.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 38
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.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 39
Effects of Qit on C - V
Assume acceptor type traps:
Qtotal  Qs (s )  Qit (s )
Qtotal (Qs (s ))
(Qit (s ))


s
s
s
 Total  Cs  Cit
(47)
Thus, Cs is in parallel with Cit. Hence, the capacitor model is:

Cit  qDita ( s )  Ditd ( s )

 qDit ( s )
(48)
Dita (s )  acceptor type density
Ditd (s )  acceptor type density
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 40
6. Polysilicon Gates
At equilibrium (Vg = 0).
At Inversion (Vg >> 0).
Vox
B
Ec
Ei
EF
Ev
Ec
Ei
EF
Ev
S
VG
Ec
EF
Ei
p
Ev
n+ poly
SiO2
p-Si
n+ poly SiO2
p-Si
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)
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 41
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  Cox for higher Np.
Total capacitance at strong inversion is given by:
1/C = 1/Cox + 1/Csilicon + 1/Cpoly
As VG, Csilicon but xd(poly)  Cpoly.
Low frequency C  V shows a local maximum at a certain VG.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 42
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.
E2
E1
E0
Bottom of
the well
E
Edge of EC
Distance from the surface
• 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.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 43
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.
n (cm-3)
Classical
QM
Depth
• 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 = .
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 44
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, tOXeff
by:
DtOX = (eox/esi)Dz
(51)
• Inversion layer quantization can be treated as bandgap
widening due to an increase in the effective bandgap by DEg
given by:
DEg

(52)
2 kT
niQM  nCL
i e
Here, DEg = EgQM  EgCL.
(52)  ni and n due to QM effect.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 45
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.
Al
SiO2
N
P
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.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 46
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 Qit.
(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).
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 47
Home Work 1: Due Oct 6, 2005
3)
MOS capacitors are fabricated on uniformly doped P-substrates with NA =
1x1017 cm3, physical gate oxide thickness TOX = 6.7 nm, and N+ polysilicon gate doping concentrations Np = 5x1018, 1x1019, and 5x1019 cm3 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 cm3 and
Vg  3.5 V.
(c) Describe the possible reasons
for C/COX < 1 even for a heavily
doped poly with Np = 5x1019
cm3 in the accumulation as
well as in the strong inversion
regions?
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 48
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 DEg = EgQM  EgCL  104
mV. Here EgQM and EgCL 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:
niQM  nCL
i e

DEg
2 kT
(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 DVth due to QM effects.
S. Saha
HO #2: ELEN 384 - MOS Capacitors
Page 49