Post on 30-Jun-2020
Berkeley
MOS High Frequency Distortion
Prof. Ali M. Niknejad
U.C. BerkeleyCopyright c© 2014 by Ali M. Niknejad
Niknejad Advanced IC’s for Comm
Compact Model Physics vs. Reality
Even the most advanced compact models have very humblephysical originsEssentially a 1D core transistor model due to GCA (gradualchannel approximation)Quantum effects necessarily ignored in core modelSmall dimension and 2D effects treated in a perturbationalmanner as corrections to 1D coreConclusion: Core model is important but any claims of“physical accuracy” should be taken with a grain of salt!
Niknejad Advanced IC’s for Comm
Modules in an Industrial Compact Model
Technolog:
Short/Narrow Channel Effects on Threshold Voltage
Non-Uniform Vertical and Lateral Doping Effects
Mobility Reduction Due to Vertical Field
Quantum Mechanic Effective Gate Oxide Thickness Model
Saturation:
Carrier Velocity Saturation
Channel Length Modulation (CLM)
Substrate Current Induced Body Effect (SCBE)
Unified current saturation model (velocity saturation, velocityovershoot, source end velocity limit)
Leakage:Gate Dielectric Tunneling Current Model
Gate Induced Drain Current Model (GIDL)
Trap assisted tunneling and recombination current model
RF:
RF Model (Gate & Substrate Resistance Model)
Unified Flicker Noise Model
Holistic Thermal Noise Model
Parasitic ModelNiknejad Advanced IC’s for Comm
Important MOS Non-Linearity
Square law model must be modified account for short channeleffects
Mobility degradation
Universal mobility curveVelocity saturation
Body effect in situations where body is not grounded(relatively rare)
Output impedance non-linearity (less of a concern at RF ifimpedances are low)
Cgs non-linearity for high input swings (only a concern forvery high input power, such as PAs).
Niknejad Advanced IC’s for Comm
Velocity Saturation
Electric Field (V/cm)
Car
rier
Dri
ft V
elo
city
(cm
/s)
10 101010 102 654310 5
108
10 7
10 6
Si
Ge
InPGaAs
Si
In triode, the average electric field across the channel is:E = VDS
L
The carrier velocity is proportional to the field for low fieldconditions: v = µE = µVDS
L
Niknejad Advanced IC’s for Comm
Mobility Reduction
As we decrease the channel length from say 10µm to 0.1µ andkeep VDS = 1V, we see that the average electric fieldincreases from 105V/m to 107V/m.
For large field strengths, the carrier velocity approaches thescattering-limited velocity.
This behavior follows the following curve-fit approximation:
vd =µnE
1 + EEc
For E � Ec , the linear behavior dominates. But for E � Ec ,vd = µnEc .
Niknejad Advanced IC’s for Comm
Vertical Mobility Degradation
µeff =µn
1 + θ (Vgs − Vt)
θ ∝ 1
Tox
For Tox ∼ 100A,θ ∼ 0.1− 0.4V−1
Experimentally, it is observed that mobility is degraded due tothe presence of a vertical field. A physical explanation for thisis that a stronger Vgs , vertical forces more of the carriers inthe channel towards the surface where imperfections in theSi-SiO2 interface impede their movement.
Niknejad Advanced IC’s for Comm
Unified Mobility Degradation
id =µCox
2
(W
L
)(Vgs − Vt)2
=µLFCox
2
(W
L
)1
1 + EEc
(Vgs − Vt)2
E =VDS
L≈ Vgs − Vt
L
id =µCox
2
1
1 + θ (Vgs − Vt)
(W
L
)(Vgs − Vt)2
Model horizontal mobility degradation in saturation, modifylow field equation to include velocity saturation.
Result looks like vertical field degradation.
Niknejad Advanced IC’s for Comm
Source Degeneration
id =µCox
2
(W
L
)(V ′gs − Vt
)2Vgs = V ′gs + idRS
id =µCox
2
(W
L
)(Vgs − idRS − Vt)2
=µCox
2
(W
L
)((Vgs − Vt)2 − 2idRS (Vgs − Vt) + ����(idRS )2
)
=µCox
2
(W
L
)(Vgs − Vt)2
1
1 + µCox
(WL
)RS (Vgs − Vt)
S
GS
R
V
Equation has approximately the same form and provides auseful intuition on mobility degradation.
Niknejad Advanced IC’s for Comm
Simple Short Channel Model
The short channel I -V model can be approximated by
id =µCox
2
(W
L
)(Vgs − Vt)2
1 + θ (Vgs − Vt)
θ models:
Rsx parasiticVertical Field Mobility DegradationHorizontal Field Mobility Degradation
θ = f (Tox , L,Rsx )
Niknejad Advanced IC’s for Comm
MOS Memoryless Non-Linearity
Device is no longer square law· · ·id =
1
2µ0Cox
W
L
(Vgs − Vt)2
1 + θ(Vgs − Vt)
id = g1 (Vgs − Vt) + g2(Vgs − Vt)2 + g3(Vgs − Vt)
3 + ...
g1 =did
dVgs=µCox
2
(W
L
)(1 + θ (Vgs − Vt)) 2 (Vgs − Vt)− θ(Vgs − Vt)
2
(1 + θ (Vgs − Vt))2
=µCox
2
(W
L
)(Vgs − Vt)
2 + θ (Vgs − Vt)
(1 + θ (Vgs − Vt))2
g2 =1
2!
di2ddV 2
gs
=µCox
2
(W
L
)2
(1 + θ (Vgs − Vt))3
g3 =1
3!
di3ddV 3
gs
=µCox
2
(W
L
)−6θ
(1 + θ (Vgs − Vt))4
Niknejad Advanced IC’s for Comm
MOS Memoryless IIP3
0.2 0.4 0.6 0.8 1
1
2
3
4
5
Vgs − Vt
VIIP3
VIIP3 =
√∣∣∣∣4g13g3
∣∣∣∣
Taking the ratio of g3 and g1, we can derive the MOStransconductance VIIP3
=2√3
√(1 + θ(Vgs − Vt))2(2 + θ(Vgs − Vt))(Vgs − Vt)
θ
Niknejad Advanced IC’s for Comm
Coulomb Scattering
A more complete model should include the effects of low fieldmobility.
At low fields, the low level of inversion exposes “CoulombScattering” sites (due to shielding effect of inversion layer)
Source: Physical Background of MOS Model 11 (Level 1101), R. van
Langevelde, A.J. Scholten and D.B.M. Klaassen
Niknejad Advanced IC’s for Comm
Mobility with Coulomb Scattering
Niknejad Advanced IC’s for Comm
Need for Single Equation I -V
0 0.2 0.4 0.6 0.8 1 1.2
I ds
Vgs
Vds=50mVVds=250mVVds=650mVVds=1.25V
0 0.2 0.4 0.6 0.8 1 1.2
log(
I ds)
Vgs
Vds=50mVVds=250mVVds=450mVVds=750mVVds=850mVVds=1.25V
Our models up to now only include strong inversion. In manyapplications, we would like a model to capture the entire I -Vrange from weak inversion to moderate inversion to stronginversion
Hard switching transistorPower amplifiers, mixers, VCOs
Surface potential models do this in a natural way but areimplicit equations and more appropriate for numericaltechniques
Niknejad Advanced IC’s for Comm
Smoothing Equation
Since we have good models for weak/strong inversion, butmissing moderate inversion. We can “smooth” between theseregions and hope we capture the region in between (BSIM 3/4do this too)
ids = f (Vgs − VT ) = KX 2
1 + θX
X = 2ηkT
qln
(1 + e
q(Vgs−VT )
2η kT
)As before θ models short-channel effects and ηmodels theweak-inversion slope
Niknejad Advanced IC’s for Comm
Limiting Behavior
ids = f (Vgs − VT ) = KX 2
1 + θX
In strong inversion, the exponential dominates giving ussquare law
X = 2ηkT
qln
(1 + e
q(Vgs−VT )
2η kT
)≈ Vgs − VT
In weak inversion, we expand the natural log function
ln (1 + x) ≈ x
X ≈ 2ηkT
qe
q(Vgs−VT )
2η kT
ids = KX 2
1 + θX≈ KX 2 = K
(2η
kT
q
)2
eq(Vgs−VT )
η kT
Niknejad Advanced IC’s for Comm
I -V Derivatives
Use chain rule to differentiate the I -V relation:
ids = f (V ) = KX 2
1 + θXfV = fX · XV
fVV = fXX · X 2V + fX · XVV
fVVV = fXXX · X 3V + 3 (fXX · XV · XVV ) + fX · XVVV
fX = KX · (2 + θX )
(1 + θX )2
fXX = K2
(1 + θX )3
fXXX = K−6θ
(1 + θX )4
XV =1
1 + s−2
XVV =q
2η kT
1
(s + s−1)2
XVVV = −q2
(2η kT )2
(s − s−1
)(s + s−1)3
Niknejad Advanced IC’s for Comm
Single Equation Distortion
Source: Analysis and Design of Current-Commutating CMOS Mixers,
Manolis Terrovitis Ph.D. dissertation, U.C. Berkeley 2001
Niknejad Advanced IC’s for Comm
IP3
Notice “sweet spot” where gm3 = 0.
Notice the sign change in the third derivative. Can we exploitthis?
Niknejad Advanced IC’s for Comm
Bias Point for High Linearity
Assume gm nonlinearitydominates (current-modeoperation and low outputimpedance).
VIP3∼=√
8gm
g ′′m
Sweet IIP3 point, g ′′m = 0,exists in the region wherethe transistor transits fromweak inversion (exponentiallaw) to moderate inversion(square law) to velocitysaturation
Niknejad Advanced IC’s for Comm
“Multiple Gated Transistors”
Notice that we can use two parallelMOS devices, one biased in weakinversion (positive gm3) and one instrong inversion (negative gm3).
The composite transistor has zerogm3 at bias point!
Source: “A Low-Power Highly Linear
Cascoded Multiple-Gated Transistor
CMOS RF Amplifier With 10 dB IP3
Improvement,” Tae Wook Kim et al.,
IEEE MWCL, vol. 13, no. 6, Jun. 2003.
Niknejad Advanced IC’s for Comm
MGTR (Multi-Gated Transistor)
Niknejad Advanced IC’s for Comm
MOS LNA Distortion
Vcas
LgC∞
Rs
LL RL
Ls
Vout
M1
M2
V1
Vdd
CL
+
–
Workhorse MOS LNA is a cascode with inductive degeneration
Assume that the device is square law
Neglect body effect (source tied to body)
Niknejad Advanced IC’s for Comm
Governing Differential Equations
vS = vRs + vLg + v1 + vLs
ii = Cgsdv1dt
vRs = RsCgsdv1dt
LgC∞
RsLs
M1V1
+
–
vLg = Lgd
dt
(Cgs
dv1dt
)= LgCgs
d2v1dt2
vLs = Lsd
dt(ii + id ) = LsCgs
d2v1dt2
+ Lsd
dt
(gm1v1 + gm2v
21 + · · ·
)vS = RsCgs
dv1dt
+ LgCgsd2v1dt2
+ v1 + LsCgsd2v1dt2
+
Lsd
dt
(gm1v1 + gm2v
21
+ · · ·)
Niknejad Advanced IC’s for Comm
Linear Analysis
Letv1 = A1(ω1) ◦ vs + A2(ω1, ω2) ◦ v2
s+ A3(ω1, ω2, ω3) ◦ v3
s+ · · ·
Equating linear terms
1 = CgsRs jω1A1 − (Lg + Ls)Cgsω21A1 + Ls jω1gm1A1
A1 =1
1− (Lg + Ls)Cgsω21
+ jω1(CgsRs + gm1Ls)
By design, at resonance we have
A1 =1
1− (Lg + Ls)Cgsω20︸ ︷︷ ︸
0
+jω0Cgs(Rs + ωTLs︸ ︷︷ ︸Rs
)=
1
jω02CgsRs
= −jQnet
Niknejad Advanced IC’s for Comm
Relevant Time Constants
τ1 = RsCgs
ω20 =
1
(Ls + Lg )Cgs
τ2 = Lsgm1 =gm1
CgsCgsLs = ωTLsCgs = RsCgs = τ1
Q =1
2Rsω0Cgs=
1
2τ1ω0
τ1 =1
2Qω0
Using these relations we simplify A1
A1 =1
1−(ωω0
)2+ j ωω0
1Q
=1
D(ω)
Niknejad Advanced IC’s for Comm
Second Order Terms
Equate second-order terms
0 = CgsRs j(ω1 + ω2)A2 − (Lg + Ls)Cgs(ω1 + ω2)2A2 + A2 +
Ls j(ω1 + ω2) (gm1A2 + gm2A1(ω1)A1(ω2))
A2(ω1, ω2) =−A1(ω1)A1(ω2)j(ω1 + ω2)gm2Ls
1− (ω1 + ω2)2(Ls + Lg )Cgs + j(ω1 + ω2)(Cgs Rs + Ls gm1)
Or assuming Square Law:
A2(ω1, ω2) =−A1(ω1)A1(ω2)j(ω1 + ω2)gm1Ls
1− (ω1 + ω2)2(Ls + Lg )Cgs + j(ω1 + ω2)(Cgs Rs + Ls gm1)
1
2(Vgs − Vt)
A2(ω1, ω2) =−j(ω1 + ω2)τ1
D(ω1)D(ω2)D(ω1 + ω2)
1
2(Vgs − Vt)
Niknejad Advanced IC’s for Comm
Third Order Terms
If the MOS is truly square law, then there are no A3 terms inthe core device.
But due to the source feedback and second-order interaction,3rd order disto is generated any way. We’ll solve the generalcase including gm3
0 = τ1j(ω1 + ω2 + ω3)A3 −1
ω2(ω1 + ω2 + ω3)2A3 + A3+
j(ω1 + ω2 + ω3)Ls
(gm1A3 + 2gm2A1A2+
gm3A1(jω1)A1(jω2)A1(jω3))
A3(jω1, jω2, jω3) =1
D(j(ω1 + ω2 + ω3)
(−2gm2A1A2 + gm3A1(jω1)A1(jω2)A1(jω3)
)
Niknejad Advanced IC’s for Comm
Drain Current
With an expression relating v1 to vs , we now generate thedrain current
id = gm1
(A1(ω1) ◦ vs + A2(ω1, ω2) ◦ v2
s+ · · ·
)+
gm2
(A1(ω1) ◦ vs + A2(ω1, ω2) ◦ v2
s+ · · ·
)2+gm3
(A1(ω1) ◦ vs + A2(ω1, ω2) ◦ v2
s+ · · ·
)3id = B1(ω1) ◦ vs + B2(ω1, ω2) ◦ v2
s+ · · ·
Niknejad Advanced IC’s for Comm
Output Current Volterra Series
Using Volterra algebra, we have
B1(ω1) = gm1A1 =gm1
D(ω1)
B2(ω1, ω2) = gm1A2 + gm2A1A1
=
( −j(ω1 + ω2)τ1D(ω1)D(ω2)D(ω1 + ω2)
+1
D(ω1)D(ω2)
)gm2
=
(1− j(ω1 + ω2)τ1
D(ω1 + ω2)
)gm1
D(ω1)D(ω2)2(Vgs − Vt)
B3(ω1, ω2, ω3) = gm1A3(jω1, jω2, jω3) + gm22A1(ω1)A2(ω1, ω2)+
gm3A1(jω1)A1(jω2)A1(jω3)
Niknejad Advanced IC’s for Comm
Square Law MOS HD2/IIP2
B1(ω1) =gm1
D(ω1)
B2(ω1,−ω2) =
(1−
j(ω1 − ω2)τ1
D(ω1 − ω2)
)gm1
D(ω1)D(−ω2)2(Vgs − Vt)
B2(ω1,−ω2)
B1(ω1)=
(1−
j(ω1 − ω2)τ1
D(ω1 − ω2)
)1
D(−ω2)2(Vgs − Vt)
VIIP2 =B1(ω1)
B2(ω1,−ω2)=
D(−ω2)2(Vgs − Vt)(1− j(ω1−ω2)τ1
D(ω1−ω2)
) =D(−ω2)D(ω1 − ω2)
D(ω1 − ω2)− j(ω1 − ω2)τ2(Vgs−Vt)
The expression for IIP2 is now very simple. For a zero-IFsystem, we care about distortion at DC:
VIIP2 =
∣∣∣∣ B1(ω1)
B2(ω1,−ω2)
∣∣∣∣ =
∣∣∣∣ D(−ω0)D(ω0 − ω0)
D(ω0 − ω0)− j(ω0 − ω0)τ
∣∣∣∣ 2(Vgs − Vt)
= |D(−ω0)| 2(Vgs − Vt) = 2(Vgs − Vt) · QNiknejad Advanced IC’s for Comm
Distortion Due to Output Swing
Although our emphasis thus far has focused on gm ortransconductance non-linearity, it reality the output stage ofan amplifier can also produce distortion due to the finiteheadroom. As a device drain (collector) swings away from anominal operating point, distortion is generated due to thenon-linearity of the output impedance of the device.
In RF applications, the load is a small resistor (to getbandwidth), so this tempers the output stage non-linearity.But if the swing is large enough, the device will leave thesaturation region and enter triode, whereby the output willclip and produce distortion.
If we take the “linear” swing range at the output and inputrefer by the gain, we can estimate if the compression of anamplifier will be limited by the input transconductance or theoutput swing.
Niknejad Advanced IC’s for Comm
General References
Analysis and Design of Current-Commutating CMOS Mixers,Manolis Terrovitis Ph.D. dissertation, U.C. Berkeley 2001.
Physical Background of MOS Model 11 (Level 1101), R. vanLangevelde, A.J. Scholten and D.B.M. Klaassen, PhilipsElectronics N.V. 2003 (Unclassified Report)
UCB EECS 242 Class Notes, Ali M Niknejad, Spring 2002,Robert G. Meyer, Spring 1995.
Niknejad Advanced IC’s for Comm