Dr. Ismail Saad Lecturer Electrical & Electronics Engineering Program School of Engineering & IT Universiti Malaysia Sabah (UMS) E‐mail: [email protected] MOSFET Modeling & Simulation with MATLAB
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Dr. Ismail SaadLecturer
Electrical & Electronics Engineering ProgramSchool of Engineering & IT
Universiti Malaysia Sabah (UMS)
E‐mail: [email protected]
MOSFET Modeling & Simulation with MATLAB
The MOS capacitor(a) Physical structure of an n+‐
Si/SiO2/p‐Si MOS capacitor(b) cross section(c) the energy band diagram under
charge neutrality(d) the energy band diagram at
equilibrium (note that the surface of the p‐type substrate near the oxide interface has become weakly inverted).
Energy band diagrams for the MOS capacitor with the charge distributions for 3 bias conditions.
(a) Equilibrium : Electrons from the n+ gate transfer to the p‐Si substrate, resulting in a positive gate and a negative depletion region in the substrate.
(b) Accumulation :A negative voltage is applied to the gate with respect to the substrate such that holes accumulate at the silicon–to–silicon dioxide interface.
(c) Applied +ve 2V(d) With time, electrons generated in
the transition region are trapped in the potential well at the interface until steady state is reach
(b) Capacitance‐voltage (CV) characteristic for a MOS capacitor at low and high frequencies.
fCvi
ac
ac π2=
sox CCC111
+=
(a) Circuit for measuring the capacitance of a MOS capacitor.
Schematic diagram of n‐channel silicon‐based MOSFET structure
• The channel width W, length L, and oxide thickness tox are shown• The symbols S, G, D, and B represent the source, gate, drain, and substrate (body) respectively.
(a)Cross section of an n‐channel FET(b)Energy band diagram at equilibrium
Accumulation : The channel charge accumulates in the bulk near the oxide interface. In this case the channel charge consists of electrons.
Inversion : The band bending at the surface of the semiconductor. At threshold, the Fermi level is as far above the intrinsic level at the surface (left‐hand edge) as it is below the intrinsic level in the bulk.
The ID‐VDS characteristics of a typical MOSFET. The threshold voltage is 0.5 V
Parallel‐Plate Capacitor Vs. MOS Capacitor
ddACCVCQ
dA
dACCVQ
osAAA
os
κεε
κεε
====
=== ( )
( )
( ))( yVVt
VVVt
qn
ttACC
VVVCQ
GTox
ox
chTGSox
oxs
ox
oxo
ox
oxA
chTGSAA
−=
−−=
===
−−=
ε
ε
κεε
Quantum Channel(a) Square well vs MOSFET
triangular channel(b) Classical vs quantum
channel(c) Lifting of conduction
band by quantum energy
(d) The wave function peaks at a distance away from the interface
(e) Classical and quantum electron concentration profile (ns =1012 cm‐2)
Effective Capacitance due to Quantum Effect (QE)
T
ooQ
Q
SiQ q
Extt
CE3
2=≈=
ε
ox
Q
Si
oxox
G
Q
ox
ox
QoxG
ttC
C
CC
C
CCC
εε
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=
1
1
11
111
10 15 20 25 30 35 40 45 500.65
0.7
0.75
0.8
0.85
0.9
ELECTRIC FIELD (MV/m)
REL
ATI
VE C
APA
CIT
AN
CE
10 nm
5 nm
- -Gate‐Field Induced Mobility Degradation Due to QE
( )( )chGT VV
ochGT
of e
VV−≈
−+= θμ
θμμ
1l
)(1 chTGS
of VVV −−+
=θ
μμl
sVcm
VVsV
cm
VVVV
sVcm
VVsV
cm
VVVV
sVcmVVV
mF
mmF
tC
sVcmVVnmtVExample
fGTGS
fGTGS
fGTTGS
ox
oxox
Tox
.316
)4(13.01.
48045
.380
)2(13.01.
48023
.4800
109.6105
1085.89.3
.4801513.0:
2
1
2
2
1
2
2
0
23
9
12
2
01
=+
===
=+
===
====
×=×
××==
====
−
−
−−
−
−
l
l
l
μ
μ
μμ
ε
μθ
Gate‐Field Induced Mobility Degradation Due to QE
Channel electron mobility and velocity (v = μ E) as a function of lateral field for VGS = 1.42 V
dydV
v
v
L
o
satc
c
L
Lf
≈
=
+=
E
E
EEE
μ
μ
1
l
c
L
feff
EE
+=
1
lμμ
Modeling nanoMOSFET ID‐VD Characteristics
Cross‐section viewed of a typical N‐channel MOSFET which shows the transverse ET and longitudinal EL electric field and oxide thickness tOX along
the channel length L
In a nanoscale channel the validity of Ohm’s law that predict a linear driftvelocity response to the applied electric field, ν = μ0E is revealed to lost itssupremacy as the applied electric field is now much higher than the channelcritical electric field Ec. The drift velocity is now can be represented as below:
c
L
Lf
εε
εμν
+=
1l (1)
where μlf is the low‐field Ohmic mobility, the Ec was related to the saturation velocity νsat and μlf and the longitudinal electric field EL can be approximate as the gradient of applied voltage V along the negative y‐direction of the channel:
f
satc
lμν
ε =dy
ydVL
)(≈ε
y=0
V=0
y=L
V=VDEL
ET
(2)
The current in the nanoscale channel is a function of carrier concentration n, the charge q and the velocity ν of carrier in the inversion channel per unit area A:
nqvAI = (3)
In a 2D channel, the area A of the inversion channel is a function of the channelwidth W and its thickness xi and the n is replaced by carrier sheet concentrationns that are the number of carrier per meter square given by:
)/(# 2mnxn is = (4)
Equation (3) is now written as:
qvWnI sD = (5)
The density of inversion charge of a current to flow in the 2D channel is given by:
))(( yVVCqnQ GTGsi −== (6)
where VGT = VGS – VT the difference of an applied gate voltage VGS and threshold voltage VT , CG is the gate capacitance and V(y) is the channel voltage in y‐direction. By invoking eq. (1) and (6) into eq.(5), the drain current in the 2D inverted channel is represented as:
( ) WyVVCI
c
L
LfGTGD
εε
εμ+
−=1
)( l(7)
This drain current has to be integrate along y‐direction from y = 0 to y = L or from applied voltage V=0 to V = VDS the drain voltage in the inverted channel as shown in fig. below:
y=0
V=0
y=L
V=VD
EL
ET
By rearrange eq. (7) and integrate it:
( ) WdyyVVCdyI LfGT
L
Gc
LL
D εμεε
l)()1(00
−=+ ∫∫ (8)
By replacing EL = dV/dy as in eq.(2) into eq.(8):
( ) dydydVWyVVCdy
dydVI fGT
L
Gc
L
D lμε
)()11(00
−=+ ∫∫ (9)
Thus, eq.(9) simplify to:
( )dVyVVWCdydydVI
DV
GTfGc
L
D ∫∫ −=+00
)()11( lμε
( )dVyVVWCdydydVI
DD V
GTfG
VL
cD ∫∫ −=+
0
/
0
)()11( lμε
⎟⎟⎠
⎞⎜⎜⎝
⎛−=+
2)1(
2D
DGTfGDc
DVVVWCVLI lμ
ε
⎟⎠⎞
⎜⎝⎛ −=+ 2
21)
.11(. DDGTfGDc
D VVVWCVL
LI lμε
(10)
Rearrange eq.(10) and defined the critical voltage Vc as:
LLVf
satcc
lμνε == . (11)
Therefore, the drain current equation of a nanoscale channel when VDS < VDsat is given as:
c
D
DDGTfG
D
VV
VVV
LWC
I+
⎟⎠⎞
⎜⎝⎛ −
=1
21 2
lμDsatD VV ≤
(12)
The drain saturation current IDsat on the onset of the velocity saturation νsatwhen VDS = VDsat is now given by:
( ) WVVCI satDsatGTGDsat ν−= DsatD VV ≥ (13)
For long channel (LC) the drain voltage VD is smaller than the critical voltage Vc(VD << Vc), therefore the drain current equation is given by:
)21( 2
DDGTfG
D VVVL
WCI −= lμ (14)
Determination of VDsat and IDsat
To determined VDsat , the point on which the velocity saturate, eq.(12) and (13) must be reconcile by making ID the same and make VD = Vdsat:
( ) WVVC
VV
VVV
LWC
satDsatGTG
c
Dsat
DsatDsatGTfG ν
μ−=
+
⎟⎠⎞
⎜⎝⎛ −
1
21 2
l
(15)
Rearrange eq. (15) and solving for VDsat:
( ) )1)((21 2
c
Dsat
f
satDsatGTDsatDsatGT V
VLVVVVV +−=−lμ
ν
( ) )1)((21 2
c
DsatcDsatGTDsatDsatGT V
VVVVVVV +−=−
( ) )1(21 2
c
DsatcDsatcGTDsatDsatGT V
VVVVVVVV +−=−
22
21
DsatcDsatDsatGTcGTDsatDsatGT VVVVVVVVVV −−+=−
021 22 =−++− cGTcDsatDsatDsat VVVVVV
0222 =−+ cGTDsatcDsat VVVVV (16)
Equation (16) is a quadratic and solving it will give the VDsat as:
( )2
2422 2cGTcc
DsatVVVV
V×+±−
=
⎥⎦
⎤⎢⎣
⎡++−=
c
GTcDsat V
VVV 211
⎥⎦
⎤⎢⎣
⎡−+= 121
c
GTcDsat V
VVV (17)
The IDsat can be determined by using eq. (16) and solving it using eq.(13):
0222 =−+ cGTDsatcDsat VVVVV
( ) 22 DsatDsatGTc VVVV =− (18)
Then by inserting eq.(18) into eq.(13):
( ) WVVCI satDsatGTGDsat ν−=
c
DsatsatGDsat V
VWCI2
21 ν=
L
VWCI
f
sat
DsatsatGDsat
lμνν
2
21
=
Thus the IDsat is simplifying as:
2
21
DsatfG
Dsat VL
WCI lμ
= (19)
The ID‐VDS characteristics of the NFET : results from the simple model. For this device W/L= 5, tox = 4 nm, CG = 8.63 × 10‐3 F/m2, and mn = 500 cm2/V×s.
Comparison of ID‐VDS characteristics computed by using the constant mobility model (q = 0, dashed lines) and taking into account the effect of the transverse field (solid line) for q = 0.13 V‐1. The transverse field tends to reduce the currents
The calculated IV characteristics for the simple model and for NMOS and PMOS with carrier velocity saturation accounted for.
The saturation voltage as a function of channel length for |(VGS ‐ VT)| = 2.6 V.
Saturation current |IDsat| as a function of L for n‐ and p‐channel silicon MOSFETs. Here |(VGS ‐ VT)| = 2.6 V, the width‐to‐length ratio is W/L = 10, and Wp = 2.5 Wn.
Comparison of the simple long‐channel model, the model including velocity saturation, the model including both velocity saturation and the series resistances RS and RD, and the actual measured data. For this NFETdevice, L = 0.25 mm, W = 9.9 mm, and VT = 0.3 V. The gate‐source voltage is 1.8 V.
For short‐channel devices, the reduction of the effective channel length with increasing VDSresults in an increase in ID
(a) For long channels the drain voltage has negligible effect on the barrier at the source‐channel interface. (b) For short‐channel devices the drain voltage tends to reduce the barrier at the source end. (c) The result is that the threshold voltage is decreased. The effect is more pronounced as the channels get shorter.
The difference between long‐ and short‐channel behavior depends on the oxide thickness tox, the source and drain junction depth xj, and the depletion region thickness at the source, wS, and drain, wD.
Illustration of MOSFET scaling. To reduce short‐channel effects, when the channel length is reduced, the dimensions and doping levels of the device (a) are adjusted to reduce the oxide thickness and junction depth (b).
MATLAB CODING1. Define Constant Data
hbar=1.0545887e-34;kb=1.380662e-23;q=1.6021892e-19;T=300;eox=3.9*8.8541878e-12;esi=11.9*8.8541878e-12;m3=0.98*9.109534e-31;m1=0.19*9.109534e-31;
tox=1.59*10^-9;L=80*10^-9;W=1.3*10^-6;VT=0.3542;VGS=0.7;
1. Velocity Saturation Model
vth2=sqrt(pi*kb*T/(2*m1)); %non-degenerate 2D velocity vth=sqrt(2*kb*T/m1); % general thermal velocity
*22 mTkB
mπν =
*2 B
tht
k Tv
m=
%The following component calculates the vsat value for ElecFDrain infinityfor k=1:6
VGS(k)=0.7+(k-1)*0.1;VGT(k)=VGS(k)-VT;Et(k)=(VGS(k)+VT)/(6*tox);Eo(k)=(hbar^2/(2*m3))^(1/3)*(9*pi*q*Et(k)/8)^(2/3);zo(k)=Eo(k)/(q*Et(k));zQM(k)=2*zo(k)/3;toxeff(k)=tox+(eox/esi)*zQM(k);CG(k)=eox/toxeff(k);uo(k)=0.0695*exp(-0.7568*VGS(k)); % exponential fit to exp
x
TGSteff t
VVE06+
≈31
*3
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛=
mEq
E teffo
h
teff
oo qE
Ez =
( )teff
oQM qE
Ez 2.33832
=QM
si
oxoxoxeff ztt
εε
+=
( )33.1.07.0 GSVf e−=lμ
Quantum Confinement & Mobility Model
teff
oo qE
Ez =113
ox oxG
QMoxeff
ox
CC
ztt
ε= =
+
13
oxoxeff ox QM ox QM
Sit t z t z
εε
= + ≈ +
( )teff
oQM qE
Ez 2.33832
=
31
*3
222
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛=
mEq
E teffo
h
x
TGSteff t
VVE06+
≈
( )33.1.07.0 GSVf e−=lμ
Continue…. Iterative solution for vsat when α=1vsat2(k)=vth2;
for j=1:10Vc(k)=vsat2(k)*L/uo(k);VDsat1(k)=Vc(k)*(sqrt(1+(2*VGT(k)/Vc(k)))-1);n2(k)=CG(k)*(VGT(k)-VDsat1(k))/q; % 2D Carrier sheet concentrationu(k)=n2(k)/NcT; %find fermi(0)eta2eta(k)= log(exp(u(k))-1); %eta2t1=fermi(eta(k),0.5); %find fermi 0.5 eta2vi2(k)=vth2*(t1/u(k));
end
n2cm(k)=n2(k)/1e4;K(k)=CG(k)*uo(k)*W/L;IDsat1(k)=K(k)*VDsat1(k)^2/2; % Idsat for full saturation end
⎥⎦
⎤⎢⎣
⎡−+= 1211
c
GTcDsat V
VVVLVf
satc
lμν
=
1/ 2 22 2
2
( )( )
Fi th
o Fv v
ηη
ℑ=
ℑ
211 2
1Dsat
fGDsat V
LWC
I lμ=
Ballistic Saturation Velocity Model
IEEE International Conference on Semiconductor Electronics (ICSE2008)25th – 27th November 2008 (Johor Bahru)
E = 0
E = LARGE
νi νi
2νi
1/ 2 22 2
2
( )( )
Fi th
o Fv v
ηη
ℑ=
ℑ
( )( )2 *
3/ 21 2 2
Bth th th
t
k Tv v v
mππΓ
= = =Γ
*2 B
tht
k Tv
m=
0
1( )( 1) 1
j
j xx dx
j e ηη∞
−ℑ =
Γ + +∫
( )22
22 .2
..F
CBi n
Nm
Tkv ηπ
21F×= ∗
22*hπ
TkmN BC =
)( 2022 ηFcNn =
*)(2
mEE CF
F−
=ν
∗=m
Tkv BNDi 22
π2*2 2
3
2 nm
v Di πh=
NanoMOSFET : I – V Characteristics
IEEE International Conference on Semiconductor Electronics (ICSE2008)25th – 27th November 2008 (Johor Bahru)
y=0
V=0
y=L
V=VDEL
ET
c
D
DDGTfG
D
VV
VVV
LWC
I+
⎟⎠⎞
⎜⎝⎛ −
=1
21 2
lμDsatD VV ≤≤0
( ) WVVCI satDsatGTGDsat να −= DsatD VV ≥
LLVf
satcc
lμνε == . satD ννα =
⎥⎦
⎤⎢⎣
⎡−+= 1211
c
GTcDsat V
VVV
211 2
1Dsat
fGDsat V
LWC
I lμ=
1=α(Full Saturation)
1 12
1
DsatVE
L xL
=−
l
Continue…iterative solution α<1%The following component calculates I-V characterisitcs and alpha for finite ElecFDrain
for k=1:6alphanew=1;
for i=1:20alpha1=alphanew;s=sqrt((alpha1+((1-alpha1)*VGT(k)/Vc(k)))^2+2*alpha1*(2*alpha1-1)*VGT(k)/Vc(k));VDsat(k)=(1/(2*alpha1-1))*((s-alpha1)*Vc(k)-(1-alpha1)*VGT(k)); alphanew=((1/alpha1)*VDsat(k)/Vc(k))/(1+((1/alpha1)*VDsat(k)/Vc(k)));end
alpha(k)=alphanew;vdrain(k)=alpha(k)*vi2(k);
%alpha not equal to 1s=sqrt((alpha(k)+((1-alpha(k))*VGT(k)/Vc(k)))^2+2*alpha(k)*(2*alpha(k)-1)*VGT(k)/Vc(k));VDsat=(1/(2*alpha(k)-1))*((s-alpha(k))*Vc(k)-(1-alpha(k))*VGT(k));IDsat=(alpha(k)/(2*alpha(k)-1))*K(k)*Vc(k)*(alpha(k)*VGT(k)-(s-alpha(k))*Vc(k));VDsatP(k)=VDsat;IDsatP(k)=IDsat;end
( ) ( )[ ]cGTcfG
Dsat VsVVL
WCI αα
μαα
−−−
= l
12
( ) ( ) ( )[ ]GTcDsat VVsV ααα
−−−−
= 112
1
( ) ( )c
GT
c
GT
VV
VV
s 12212
−+⎥⎦
⎤⎢⎣
⎡−+= αααα
IEEE International Conference on Semiconductor Electronics (ICSE2008)25th – 27th November 2008 (Johor Bahru)
NanoMOSFET : I – V Characteristics
( ) ( ) ( )[ ]GTcDsat VVsV ααα
−−−−
= 112
1
( ) ( )[ ]cGTcfG
Dsat VsVVL
WCI αα
μαα
−−−
= l
12
( ) ( )c
GT
c
GT
VV
VV
s 12212
−+⎥⎦
⎤⎢⎣
⎡−+= αααα
1<α
satD ανν =cD
cD
EEEE
+=
1α
0μν satcE = LVE DsatD α=
Iterative
solution
Continue..for k=1:6
VD=0:0.01:VDsatP(k);ID=(K(k)/2)*(2*VGT(k)*VD-VD.^2)./(1+(VD./Vc(k))); % for
VDM=VDsatP(k):0.01:1.0; % for alphaprime=((1/alpha(k))*VDM/Vc(k))./(1+((1/alpha(k))*VDM/Vc(k)));IDMprime=alphaprime*W*CG(k)*vsat2(k)*(VGT(k)-VDsatP(k));
VD1=VDsatP(k):0.01:VDsat1(k);ID1=(K(k)/2)*(2*VGT(k)*VD1-VD1.^2)./(1+(VD1./Vc(k)));
VD1M=VDsat1(k):0.01:1.0;ID1M=ones(1,length(VD1M))*IDsat1(k);
end
c
D
DDGTfG
D
VV
VVV
LWC
I+
⎟⎠⎞
⎜⎝⎛ −
=1
21 2
lμ
DsatD VV ≤≤0
( ) WVVCI satDsatGTGDsat να −=
DsatD VV ≥
211 2
1Dsat
fGDsat V
LWC
I lμ=
Plotting Graph% Plotting Intercept Linex=[0 0.2667 0.3311 0.3932 0.4537 0.5131 0.5716]; %VDsat1y=[0 0.0003 0.0005 0.0007 0.0009 0.0011 0.0013]/1e-3; %IDsat1xi=0:0.01:0.7;yi=interp1(x,y,xi,'cubic')
%For VDsatP & IDsatP (Alpha < 1)x1=[0 0.1880 0.2661 0.3029 0.3387 0.3739]; %VDsatPy1=[0 0.0003 0.0006 0.0008 0.0010 0.0012]/1e-3; %IDsatPxi1=0:0.01:0.5;yi1=interp1(x1,y1,xi1,'cubic')
Figure (1)
plot(xi,yi,'--b',xi1,yi1,'--r',VD,ID/1e-3,'-r',VDM,IDMprime/1e-3,':r',VD1,ID1/1e-3,'-b',VD1M,ID1M/1e-3, '--b','Linewidth', 3.0)
hold on
Continue..%This block plots the experimental results on the same plot as previous%plot has "hold on"
VDexp=0.1:0.1:1.0;IDexp12=[0.000487727 0.000874322 0.00114401 0.00132208 0.00144098... 0.00152452 0.0015872 0.00163731 0.00167961 0.00171686 ]/1e-3;IDexp11=[0.000455394, 0.000800693, 0.00102774, 0.00117086, 0.00126455...
0.00133073,0.00138145,0.00140317, 0.00144182, 0.00147613,]/1e-3;IDexp10=[4.1575E-04 7.1329E-04 8.9562E-04 1.0055E-03 1.0770E-03... 1.1286E-03 1.1695E-03 1.2044E-03 1.2358E-03 1.2649E-03 ] /1e-3;IDexp09=[ 3.6741E-04 6.1091E-04 7.4837E-04 8.2842E-04 8.8132E-04...
9.2118E-04 9.5439E-04 9.8396E-04 1.0114E-03 1.0376E-03]/1e-3;IDexp08=[ 3.0908E-04 4.9363E-04 5.8896E-04 6.4404E-04 6.8234E-04...7.1317E-04 7.4037E-04 7.6569E-04 7.8995E-04 8.1361E-04] /1e-3;
IDexp07=[ 2.648E-04 4.065E-04 4.604E-04 4.843E-04 5.013E-04... 5.148E-04 5.258E-04 5.352E-04 5.431E-04 5.500E-04 ]/1e-3;
plot(VDexp,IDexp07, '.g',VDexp,IDexp08, '+g',VDexp,IDexp09, '*g',VDexp,IDexp10, 'xg', ...VDexp,IDexp11, 'sg' , VDexp,IDexp12, 'dg', 'MarkerSize',5.0,'Linewidth', 2.0)ylabel('\itI_{D} (mA)')xlabel('\it V_{D} (V)‘)
hold off
I-V characteristics of 80-nm MOSFET for Gate Voltage, VGS =0.7 - 1.2. Solid lines are for VD < VDsat. The dotted lines are for VD > VDsat . The dashed lines are for α =1.
Reference1. Vijay K. Arora, Michael L. P. Tan, Ismail Saad, Razali Ismail. 2007‐09‐06. ballistic
Quantum Transport In A Nanoscale Metal‐oxide‐semiconductor Field Effect Transistor.Jil. 91. American Institute Of Physics: Applied Physics Letter (apl 2007).
2. Michael L. P. Tan, Ismail Saad, Razali Ismail, Vijay K. Arora. 2007‐07‐18. enhancement InNano‐rc Switching Delay Due To The Resistance Blow‐up In Ingaas. Jil. 2. 4. WorldScientific.com: World Scientific Publishing Company (nano 2007).
3. Ismail Saad , Michael L.p Tan, Ing Hui Hii, Razali Ismail And Vijay K. Arora. 2008‐07‐26.Ballistic Mobility And Saturation Velocity In Low‐dimensional Nanostructures. Vol. 40,No. 3 (2009) pp 540‐542. www.elsevier.com: Microelectronics Journal.
4. Ismail Saad , Michael L. P. Tan, Aaron Chii Enn Lee, Razali Ismail And Vijay K. Arora.2008‐07‐25. Scattering‐limited And Ballistic Transport In Nano‐cmos Transistors. Vol. 40,No. 3 (2009) pp 581‐583. Www.elsevier.com: Microelectronics Journal.
5. Michael L. P. Tan, Vijay K. Arora, Ismail Saad, Mohammad Taghi Ahmadi And RazaliIsmail. 2009‐04‐02. The Drain Velocity Overshoot In An 80 Nm Metal‐oxide‐semiconductor. American Institute Of Physics: Journal of Applied Physics (2009).