Berkeley
MOS High Frequency Distortion
Prof. Ali M. Niknejad
Copyright
U.C. Berkeley c 2014 by Ali M. Niknejad
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Compact Model Physics vs. Reality
Even the most advanced compact models have very humble physical origins
Essentially a 1D core transistor model due to GCA (gradual channel approximation)
Quantum effects necessarily ignored in core model
Small dimension and 2D effects treated in a perturbational manner as corrections to 1D core
Conclusion: Core model is important but any claims of
“physical accuracy” should be taken with a grain of salt!
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Modules in an Industrial Compact Model
Technolog:
Saturation:
Leakage:
RF:
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
Carrier Velocity Saturation
Channel Length Modulation (CLM)
Substrate Current Induced Body Effect (SCBE)
Unified current saturation model (velocity saturation, velocity overshoot, source end velocity limit)
Gate Dielectric Tunneling Current Model
Gate Induced Drain Current Model (GIDL)
Trap assisted tunneling and recombination current model
RF Model (Gate & Substrate Resistance Model)
Unified Flicker Noise Model
Holistic Thermal Noise Model
Parasitic Model
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Important MOS Non-Linearity
Square law model must be modified account for short channel effects
Mobility degradation
Universal mobility curve
Velocity saturation
Body effect in situations where body is not grounded
(relatively rare)
Output impedance non-linearity (less of a concern at RF if impedances are low)
C gs non-linearity for high input swings (only a concern for very high input power, such as PAs).
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Velocity Saturation
10
8
GaAs
InP
10 7
Ge
Si
10 6
Si
10 5
10 2 10 3 10 4
Electric Field (V/cm)
10 5 10 6
In triode, the average electric field across the channel is:
E =
V
DS
L
The carrier velocity is proportional to the field for low field conditions: v = µ E = µ
V
DS
L
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Mobility Reduction
As we decrease the channel length from say 10 µ m to 0.1
µ and keep V
DS
= 1V, we see that the average electric field increases from 10
5
V/m to 10
7
V/m.
For large field strengths, the carrier velocity approaches the scattering-limited velocity.
This behavior follows the following curve-fit approximation: v d
=
µ n
E
1 +
E
E c
For E E c
, the linear behavior dominates. But for E E c
, v d
= µ n
E c
.
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Vertical Mobility Degradation
µ eff
=
µ n
1 + θ ( V gs
− V t
) For T ox
∼ 100˚
θ ∼ 0 .
1 − 0 .
4 V
− 1
1
θ ∝
T ox
Experimentally, it is observed that mobility is degraded due to the presence of a vertical field. A physical explanation for this is that a stronger V gs
, vertical forces more of the carriers in the channel towards the surface where imperfections in the
Si-SiO
2 interface impede their movement.
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Unified Mobility Degradation i d
=
µ C ox
2
=
µ
LF
C ox
2
W
L
W
L
( V gs
− V t
)
2
1
1 +
E
E c
( V gs
− V t
)
2
E =
V
DS
L
≈
V gs
− V t
L i d
=
µ C ox
2
1
1 + θ ( V gs
− V t
)
W
L
( V gs
− V t
)
2
Model horizontal mobility degradation in saturation, modify low field equation to include velocity saturation.
Result looks like vertical field degradation.
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Source Degeneration i d
=
µ C ox
2
W
L
V
0 gs
− V t
2
V gs
= V
0 gs
+ i d
R
S i d
=
µ C ox
2
W
L
( V gs
− i d
R
S
− V t
)
2
V
GS
=
µ C ox
2
W
L
( V gs
− V t
)
2
− 2 i d
R
S
( V gs
− V t
) + ( i d
R
S
)
2
=
µ C ox
2
W
L
( V gs
− V t
)
2
1 + µ C ox
W
L
1
R
S
( V gs
− V t
)
Equation has approximately the same form and provides a useful intuition on mobility degradation.
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R
S

Simple Short Channel Model
The short channel I V model can be approximated by i d
=
µ C
2 ox
W
L
( V gs
− V t
)
2
1 + θ ( V gs
− V t
)
θ models:
R sx parasitic
Vertical Field Mobility Degradation
Horizontal Field Mobility Degradation
θ = f ( T ox
, L , R sx
)
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MOS Memoryless Non-Linearity
Device is no longer square law · · · i d
=
1
2
µ
0
C ox
W
L
( V gs
− V t
)
2
1 + θ ( V gs
− V t
) i d
= g
1
( V gs
− V t
) + g
2
( V gs
− V t
)
2
+ g
3
( V gs
− V t
)
3
+ ...
g
1
= d i d d V gs
=
µ C ox
2
W
L
(1 + θ ( V gs
− V t
)) 2 ( V gs
(1 + θ ( V gs
− V t
) − θ ( V gs
− V t
))
2
− V t
)
2
=
µ C ox
2
W
L
2 + θ ( V gs
− V t
)
( V gs
− V t
)
(1 + θ ( V gs
− V t
))
2 g
2
=
1
2!
d i
2 d d V 2 gs
=
µ C ox
2
W
L
2
(1 + θ ( V gs
− V t
))
3 g
3
=
1
3!
d i
3 d d V 3 gs
=
µ C ox
2
W
L (1 + θ (
− 6 θ
V gs
− V t
))
4
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3
2
1
MOS Memoryless IIP
3
V
IIP 3
5
4
V
IIP 3
= s
4 g
1
3 g
3
0.2
0.4
V gs
0.6
− V t
0.8
1
Taking the ratio of g
3 and g
1
, we can derive the MOS transconductance V
IIP 3
= √
3 r
(1 + θ ( V gs
− V t
)) 2 (2 + θ ( V gs
θ
− V t
))( V gs
− V t
)
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Coulomb Scattering
A more complete model should include the effects of low field mobility.
At low fields, the low level of inversion exposes “Coulomb
Scattering” 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
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Mobility with Coulomb Scattering
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Need for Single Equation I V
V
V
V ds ds
=50mV
=250mV ds
=650mV
V ds
=1.25V
0 0.8
V
V ds
=50mV ds
=250mV
V ds
V ds
V ds
V
=450mV
=750mV
=850mV ds
=1.25V
1 0.2
0.4
0.6
V gs
0.8
1 1.2
0 0.2
0.4
0.6
V gs
Our models up to now only include strong inversion. In many applications, we would like a model to capture the entire I V range from weak inversion to moderate inversion to strong inversion
Hard switching transistor
Power amplifiers, mixers, VCOs
Surface potential models do this in a natural way but are implicit equations and more appropriate for numerical techniques
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1.2

Smoothing Equation
Since we have good models for weak/strong inversion, but missing moderate inversion. We can “smooth” between these regions and hope we capture the region in between (BSIM 3/4 do this too) i ds
X
2
= f ( V gs
− V
T
) = K
1 + θ X
X = 2 η kT q ln 1 + e q ( Vgs − VT
2 η kT
)
As before θ models short-channel effects and η models the weak-inversion slope
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Limiting Behavior i ds
= f ( V gs
X
2
− V
T
) = K
1 + θ X
In strong inversion, the exponential dominates giving us square law
X = 2 η kT q ln 1 + e q ( Vgs − VT )
2 η kT
≈ V gs
− V
T
In weak inversion, we expand the natural log function ln (1 + x ) ≈ x
X ≈ 2 η kT q e q ( Vgs − VT
2 η kT
) i ds
X
2
= K
1 + θ X
≈ K X
2
= K 2 η kT q
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2 e q ( Vgs − VT )
η kT

I V Derivatives
Use chain rule to differentiate the I V relation: i ds
X
2
= f ( V ) = K
1 + θ X f
V
= f f
VV
= f
XX
· X
2
V
+ f
X
· X
VV
X
· X
V f
VVV
= f
XXX
· X
3
V
+ 3 ( f
XX
· X
V
· X
VV
) + f
X
· X
VVV f
X
= K
X · (2 + θ X )
(1 + θ X )
2 f
XX
2
= K
(1 + θ X )
3 f
XXX
− 6 θ
= K
(1 + θ X )
4
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X
V
=
1
1 + s
− 2
X
VV q
=
2 η kT
1
( s + s
− 1 )
2
X
VVV q
2
= −
(2 η kT )
2 s − s
− 1
( s + s − 1 )
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Single Equation Distortion
Source: Analysis and Design of Current-Commutating CMOS Mixers,
Manolis Terrovitis Ph.D. dissertation, U.C. Berkeley 2001
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IP
3
Notice “sweet spot” where g m 3
= 0.
Notice the sign change in the third derivative. Can we exploit this?
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Bias Point for High Linearity
Assume g m nonlinearity dominates (current-mode operation and low output impedance).
V
IP 3 s
8 g m g
00 m
Sweet IIP
3 point, g
00 m
= 0, exists in the region where the transistor transits from weak inversion (exponential law) to moderate inversion
(square law) to velocity saturation
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“Multiple Gated Transistors”
Notice that we can use two parallel
MOS devices, one biased in weak inversion (positive g m 3
) and one in strong inversion (negative g m 3
).
The composite transistor has zero g m 3 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.
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MGTR (Multi-Gated Transistor)
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MOS LNA Distortion
R s
C
∞
C
L
L g
V cas
+
V
1
L s
–
V dd
L
L
R
L
V out
M2
M1
Workhorse MOS LNA is a cascode with inductive degeneration
Assume that the device is square law
Neglect body effect (source tied to body)
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Governing Differential Equations
C
∞
L g v
S
= v
R s
+ v
L g
+ v
1
+ v
L s i i
= C gs d v
1 dt v
R s
= R s
C gs d v
1 dt v
L g d
= L g dt
C gs d v
1 dt
R s
= L g
C gs d
2 v
1 d t 2
+
V
1
L s
–
M1 v
L s d
= L s dt
( i i
+ i d
) = L s
C gs d 2 v
1 d t 2 d
+ L s dt g m 1 v
1
+ g m 2 v
2
1
+ · · · v
S
= R s
C gs d v
1 dt
+ L g
C gs
L s d
2 v
1 d t 2 d dt
+ v
1
+ L s
C gs g m 1 v
1
+ g m 2 v
1
2 d
2 v
1 d t
+
2
· · ·
+
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Linear Analysis
Let v
1
= A
1
( ω
1
) ◦ v s
+ A
2
( ω
1
, ω
2
) ◦ v s
2 + A
3
( ω
1
, ω
2
, ω
3
) ◦ v s
3 + · · ·
Equating linear terms
1 = C gs
R s j ω
1
A
1
− ( L g
+ L s
) C gs
ω
2
1
A
1
+ L s j ω
1 g m 1
A
1
A
1
=
1
1 − ( L g
+ L s
) C gs
ω
1
2 + j ω
1
( C gs
R s
+ g m 1
L s
)
By design, at resonance we have
A
1
=
1
1 − ( L g
+ L s
) C gs
ω
2
0
| {z
0
}
+ j ω
0
C gs
( R s
+ ω
T
L s
)
| {z }
R s
=
1 j ω
0
2 C gs
R s
= − j Q net
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Relevant Time Constants
τ
1
= R s
C gs
τ
2
= L s g m 1
=
ω
2
0
=
( L s g m 1
C gs
C gs
L s
+
1
L g
) C gs
= ω
T
L s
C gs
= R s
C gs
= τ
1
Q =
1
2 R s
ω
0
C gs
=
1
2 τ
1
ω
0
1
τ
1
=
2 Q ω
0
Using these relations we simplify A
1
A
1
=
1 − ω
ω
0
1
2
+ j
ω
ω
0
1
Q
1
=
D ( ω )
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Second Order Terms
Equate second-order terms
0 = C gs
R s j ( ω
1
+ ω
2
) A
2
− ( L g
+ L s
) C gs
( ω
1
+ ω
2
)
2
A
2
+ A
2
+
L s j ( ω
1
+ ω
2
) ( g m 1
A
2
+ g m 2
A
1
( ω
1
) A
1
( ω
2
))
A
2
( ω
1
, ω
2
) =
− A
1
( ω
1
) A
1
( ω
2
) j ( ω
1
+ ω
2
) g m 2
L s
1 − ( ω
1
+ ω
2
)
2
( L s
+ L g
) C gs
+ j ( ω
1
+ ω
2
)( C gs
R s
+ L s g m 1
)
Or assuming Square Law:
A
2
( ω
1
, ω
2
) =
1 − ( ω
1
− A
1
( ω
1
) A
1
( ω
2
) j ( ω
1
+ ω
2
) g m 1
L s
+ ω
2
)
2
( L s
+ L g
) C gs
+ j ( ω
1
+ ω
2
)( C gs
R s
+ L s g m 1
)
1
2( V gs
− V t
)
A
2
( ω
1
, ω
2
) =
− j ( ω
1
+ ω
2
) τ
1
D ( ω
1
) D ( ω
2
) D ( ω
1
+ ω
2
) 2( V gs
1
− V t
)
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Third Order Terms
If the MOS is truly square law, then there are no A
3 the core device.
terms in
But due to the source feedback and second-order interaction,
3rd order disto is generated any way. We’ll solve the general case including g m 3
0 = τ
1 j ( ω
1
+ ω
2
+ ω
3
) A
3
−
1
ω 2
( ω
1
+ ω
2
+ ω
3
)
2
A
3
+ A
3
+ j ( ω
1
+ ω
2
+ ω
3
) L s g m 1
A
3
+ 2 g m 2
A
1
A
2
+ g m 3
A
1
( j ω
1
) A
1
( j ω
2
) A
1
( j ω
3
))
A
3
( j ω
1
, j ω
2
, j ω
3
) =
1
D ( j ( ω
1
+ ω
2
+ ω
3
)
− 2 g m 2
A
1
A
2
+ g m 3
A
1
( j ω
1
) A
1
( j ω
2
) A
1
( j ω
3
)
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Drain Current
With an expression relating v
1 drain current to v s
, we now generate the i d
= g m 1
A
1
( ω
1
) ◦ v s
+ A
2
( ω
1
, ω
2
) ◦ v s
2
+ · · · +
2 g m 2
A
1
( ω
1
) ◦ v s
+ A
2
( ω
1
, ω
2
) ◦ v s
2
+ · · ·
3
+ g m 3
A
1
( ω
1
) ◦ v s
+ A
2
( ω
1
, ω
2
) ◦ v s
2
+ · · · i d
= B
1
( ω
1
) ◦ v s
+ B
2
( ω
1
, ω
2
) ◦ v s
2
+ · · ·
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Output Current Volterra Series
Using Volterra algebra, we have
B
1
( ω
1
) = g m 1
A
1
= g m 1
D ( ω
1
)
B
2
( ω
1
, ω
2
) = g m 1
A
2
+ g m 2
A
1
A
1
=
− j ( ω
1
+ ω
2
) τ
1
D ( ω
1
) D ( ω
2
) D ( ω
1
+ ω
2
)
+
= 1 − j ( ω
1
D ( ω
+
1
ω
+
2
ω
) τ
1
2
) D ( ω
1
D
) D ( ω
(
2
ω
1 g m 1
1
) D ( ω
2
) g
)2( V gs
− V t
) m 2
B
3
( ω
1
, ω
2
, ω
3
) = g m 1
A
3
( j ω
1
, j ω
2
, j ω
3
) + g m 2
2 A
1
( ω
1
) A
2
( ω
1
, ω
2
)+ g m 3
A
1
( j ω
1
) A
1
( j ω
2
) A
1
( j ω
3
)
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Square Law MOS HD2/IIP2
B
1
( ω
1
) = g m 1
D ( ω
1
)
B
2
( ω
1
, − ω
2
) = 1 − j ( ω
1
− ω
2
) τ
1
D ( ω
1
− ω
2
) g m 1
D ( ω
1
) D ( − ω
2
)2( V gs
− V t
)
B
2
( ω
1
, − ω
2
)
B
1
( ω
1
)
= 1 − j ( ω
1
− ω
2
) τ
1
D ( ω
1
− ω
2
)
1
D ( − ω
2
)2( V gs
− V t
)
V
IIP 2
=
B
1
( ω
1
)
B
2
( ω
1
, − ω
2
)
=
D ( − ω
2
)2( V gs
−
1 − j ( ω
1
D ( ω
− ω
2
) τ
1
1
− ω
2
)
V t
)
=
D (
D ( − ω
ω
1
2
) D ( ω
1
− ω
2
)
− ω
2
) − j ( ω
1
− ω
2
) τ
2( V gs
− V t
)
The expression for IIP
2 is now very simple. For a zero-IF system, we care about distortion at DC:
V
IIP 2
=
B
1
( ω
1
)
B
2
( ω
1
, − ω
2
)
=
D (
D ( − ω
0
) D ( ω
0
− ω
0
)
ω
0
− ω
0
) − j ( ω
0
− ω
0
) τ
= | D ( − ω
0
) | 2( V gs
− V t
) = 2( V gs
− V t
) · Q
2( V gs
− V t
)
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Distortion Due to Output Swing
Although our emphasis thus far has focused on g m or transconductance non-linearity, it reality the output stage of an amplifier can also produce distortion due to the finite headroom. As a device drain (collector) swings away from a nominal operating point, distortion is generated due to the non-linearity of the output impedance of the device.
In RF applications, the load is a small resistor (to get bandwidth), so this tempers the output stage non-linearity.
But if the swing is large enough, the device will leave the saturation region and enter triode, whereby the output will clip and produce distortion.
If we take the “linear” swing range at the output and input refer by the gain, we can estimate if the compression of an amplifier will be limited by the input transconductance or the output swing.
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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. van
Langevelde, A.J. Scholten and D.B.M. Klaassen, Philips
Electronics N.V. 2003 (Unclassified Report)
UCB EECS 242 Class Notes, Ali M Niknejad, Spring 2002,
Robert G. Meyer, Spring 1995.
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