MOS High Frequency Distortionrfic.eecs.berkeley.edu/ee242/pdf/Module_4_4_MosDisto.pdf · nE c....

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!


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).


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


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 .


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.


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.


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.


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 )


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


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


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


Mobility with Coulomb Scattering


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


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


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


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


Single Equation Distortion

Source: Analysis and Design of Current-Commutating CMOS Mixers,

Manolis Terrovitis Ph.D. dissertation, U.C. Berkeley 2001


IP3

Notice “sweet spot” where gm3 = 0.

Notice the sign change in the third derivative. Can we exploitthis?


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


“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.


MGTR (Multi-Gated Transistor)


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)


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

+ · · ·)


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


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)


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)


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)

)


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+ · · ·


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)


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.


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.


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