EM1 BJT Model: Deficiencies - SCU · device characterization ⇒ EM3 BJT model. 29-Jan-04 HO #7:...

29-Jan-04 HO #7: ELEN 251 - EM BJT Models Saha #1

EM1 BJT Model: Deficiencies

• EM1 BJT model:– advantages

♦ dc model

♦ fewest model parameters - βF, βR, IS, Tref, Eg

♦ can predict dc characteristics for most applications

– deficiencies♦ no ohmic bulk resistors to the terminals (r'e, r'b, r'c)♦ no charge storage in the devices (CDE, CDC, CjE, CjC, Csub).

E

B

C

r'c

CjE CjC

r'e r'b


EM2 Model to Improve DC Characterization

Addition of three ohmic resistors to the basic EM1 model.

C IC

E IE

IB

ICTB

IEC/βR

ICC/βF

C

B

E

N

P

N

npn-BJT EM1 model Include 3-ohmicresistors to EM1

E

r'c

r'e

r'b EM1model

B

C

B'

C'

E'


Effect of Ohmic Bulk Resistors - r'c

• Collector series resistance, r'c:– typical value ~ 200 ohm– decreases the slope of IC vs. VCE characteristics in the saturation

region of operation– improves modeling of dc device characteristics.

VCE

IC

IB3

IB2

IB1

EM1 Effect of r'c


Effect of Ohmic Bulk Resistors - r'e and r'b

• Emitter series resistance, r'e:– typical value ≈ 5 ohm due to polysilicon contact– reduces EB junction potential by a factor of r'eIE

∆VBE = IEr'e = (IC + IB)r'e = IB(1 + βF) r'e∴ r'e is equivalent to a base resistance of (1 + βF) r'e– effects IC and IB of the device.

• Base series resistance, r'b:– effects small-signal and

transient response– difficult to measure

accurately due to thedependence on r'e andoperating point. VBE

ln(I

C, I

B)

IC

IB

∆VBE = IBr'b + IEr'e


EM2 Model: Modeling Charge Storage Effect

Addition of capacitors in EM1 model to account for chargestorage effects in BJTs:

E

r'c

r'e

r'b EM1model

B

C

B'C'

E'

Csub

CjC

CjE

CDC

CDE

• CDE = EB-junction diffusioncapacitance.

• CDC = CB-junction diffusioncapacitance.

• CjE = EB-junction capacitors.

• CjC = CB-junction capacitors.Csub = C-substrate junctioncapacitance.


Complete EM2 BJT Model

The complete model to account for terminal resistors and chargestorage effects in BJTs.

E

r'c

r'e

r'bB

C

B'

C'

E'

Csub

CjC

CjE

CDC

CDE

ICT

IEC/βR

ICC/βF

−=

−=

′′

′′

1

1

Where

eII

eII

kTV CBq

R

S

R

EC

kTV EBq

F

S

F

CC

ββ

ββ

==

′′

′′

CB

EB

VVHere

BE internal voltage

BC internal voltage


EM2 Model: Junction Capacitors

CjE and CjC model the incremental fixed charges stored in the BEand BC space charge layers of BJT due to applied bias VB'E' andVB'C', respectively. From PN-junction theory (HO #4, slide-20) weget:

( )Em

E

EB

jEEBjE

V

CVC

−

=′′

′′

φ1

0(1)

( )Cm

C

CB

jCCBjE

V

CVC

−

=′′

′′

φ1

0(2)

CjE0 = EB-junction capacitance at VB'E' = 0 φE = EB built-in potential = (kT/q)ln(NABNDE/ni

2)

CjC0 = CB-junction capacitance at VB'C' = 0 φC = CB built-in potential = (kT/q)ln(NABNDC/ni

2)


EM2 Model: Diffusion Charge, QDE

Total minority carrier charge due to forward current ICC: QDE = QE + QBE + QB + QBC

= τFICC

where

τF = total forward delay time consisting of emitter delay τE, EB-space charge layer transit time τEB, base transit time τB, and CB-space charge layer transit time τCB.

E Cnp(0)

npo

B

QBQE

QB

CQBE

W


Base Transit Time, τB

In the absence of ε fields in the base (NA = constant, low levelinjection), then the injected e- concentration varies linearly acrossthe base.

Therefore, the total e- charge in the base is simply: QB = (1/2)qnpWBAE (3)

The transit time across the base is: τB = QB/ICC (4)

Using our Eq. for ICC and substituting (3) into (4), we get:

Dn = average e- diffusion constant in the base region.D

Wn

BB 2

2

≅τ (5)

npE B C

~ 0


Base Transit Time

Example:

If WB = 1 µm and the base is lightly doped so that Dn ≅ 38 cm2/sec,then τB ≅ 132 psec.

If the base doping is graded (typically in IC transistors), an aiding εfield speeds up the carriers and τB is reduced by at least 2 times.

Also, under high level injection, to maintain base neutrality, thehole concentration in the base↑ and has a gradient similar to the e-gradient. This sets up an e field which also speeds up the e-.

Usually, τB is not the dominant frequency limitation in modernBJTs.


EM2 Model: Diffusion Charge, QDC

Total minority carrier charge due to reverse current IEC: QDC = QC + QBER + QBR + QBCR

= τRIEC

where

τR = total reverse delay time consisting of collector delay τC, CB-reverse space charge layer transit time τCB, reverse base transittime τBR, and reverse EB-space charge layer transit time τEB.

E CB

QBRQC

QB

CRQBER

W


EM2 Model: Diffusion Capacitors CDE and CDC

The diffusion charges are modeled by two non-linear capacitorsCDE and CDC given by:

The model parameters: τF and τR.

( )

( )CB

ECR

CB

DCCBDC

EB

CCF

EB

DEEBDE

VI

VQ

VC

VI

VQ

VC

′′′′′′

′′′′′′

=≡

=≡

τ

τ(6)

(7)

( )

( ) RmRR

VCB

EC

VCB

DC

signalsmallDC

FmFF

VEB

CC

VEB

DE

signalsmallDE

gdVdI

dVdQ

C

gdVdI

dVdQ

C

EBEB

CBCB

ττ

ττ

==≡

==≡

=′′=′′−

=′′=′′−

′′′′

′′′′

00

00

(8)

(9)


EM2 Model: Model Parameters

Substrate Capacitance (Csub) is considered as a constant in EM3model.

∴the complete set of model parameters is given by:

five EM1/DC - βF, βR, IS, Tref, Eg

three bulk ohmic resistors - r'c, r'e, r'b

three EB-junction capacitors - CjE0, φE, mE

three CB-junction capacitors - CjC0, φC, mC

two minority-carrier delay time - τF,τR

one C-substrate capacitor - Csub


EM2 Model: Discussions

• Advantages of EM2 model is the improvement in DC and ACdevice characterization over EM1 model by addition of:– bulk ohmic resistors– charge storage in the devices

• The limitations of EM2 model are:– base-width modulation

– variation of β with current level

– distributed collector capacitance

– variation of device parameters with temperature

– high current effect on τF.

• New model parameters are added to EM2 BJT model to improvedevice characterization ⇒ EM3 BJT model.


EM3 Model: Base Width Modulation

• Base-width modulation describes the change in the quasi-neutralbase-region W due to a change in the CB-junction voltage, VBC.

• In the normal active mode of BJT operation:– BE-junction is forward biased– BC-junction is reversed biased– depletion width, Xd = f(VBC)∴ base-width, W changes significantly with VBC.

BE C

W(VBC)

N+ P N-xd2xd1

x = 0 x = W


Base Width Modulation by VBC − Early Effect

• As the reverse bias across the base-collector junction increases:– BC-junction depletion-layer width increases– W decreases

– injected minority carrier gradient, (dn/dx)↑ in W∴ IC↑ as VCE↑.

• Base-width modulation is modeled by a parameter, VA = forwardEarly voltage.

IB5

IB4

IB3

IB2

IB1

VA VCE

IC IB5 > IB4 > IB3 > IB2 > IB1

0

• Effect of base-widthmodulation:– IS = f(VBC, VA)

− βF = f(VBC, VA)

− τF = f(VBC, VA)


Base Width Modulation

• Let us assume– uniformly doped base-region– linear region of operation– W = f(VBC)

∴ Using Taylor’s series expansion, we can show that

• We define:

+=∴

+=

=

==

=

0

)0()0(

0

)0(1)0()(

][)0()(

BC

BC

VBC

BCBC

BCVWW

VBCBCBC

dVdW

WV

WVW

dVdW

VWVW

(10)

CC

BCBEBC

CCC

C

CEA I

dIdV

VVdId

IIdI

dVV −=+−=≡ )( (11)



Now,

From (11) and (12):

Combining (10) and (13) we get:

)0(WdW

dVV

BCV

BCA −= (13)

(12)0

0

2

)0()0(

)(

=

=

=−⇒−=∴

−≅

BC

BC

kTBE

VC

CCC

V

Vq

BCAB

inBC

dWW

dII

dWW

IdI

eVWNnDqA

I

+=

A

BCBC V

VWVW 1)0()( (14)


Base Width Modulation - Model parameters

(15)

−=∴

+==⇒

=≅

A

BCSBCS

A

BC

S

BCSBCS

BCAB

inBS

VV

IVI

VV

IVW

WIVI

VWNnDqA

I

1)0()(

1

)0()(

)0()0()(

)0()0(

2

Similarly, we can show that:

2

1)0()(

1)0()(

+=

−=

A

BCBBCB

A

BCBCF

VV

V

VV

V

ττ

ββ (16)

(17)



• The base-width modulation model parameter, VA:– does not change the equivalent circuits– changes the terminal current equations over EM1/EM2 models.

• The expression for current source:

• VA is measured from the slope of IC vs. VCE characteristic in theactive region of BJT operation. slope = (dIC/dVCE)|VBE = constant = -(dIC/dVBC)|VBE = constant

−−

−

+

= 111

)0(ee

VV

II kT

V BCqkT

V BEq

A

BC

SCT (18)

−+

−= 1

)0()0(

1)0()0(

eI

eI

I kTBC

kTBE Vq

R

SVq

F

SB ββ

(19)


EM3 Model: βdc Variation with Current

• Experimental βdc vs. IC data can be divided into three regions:

Region I RegionII

Region III

log(IC)

βF

βFM

VBC = 0

• βdc vs. IC plot does not clearly show the dependence of:− βdc on various component of IB in Region-I

− βdc on various factors causing degradation in IC in Region-III.

∴ IC vs. VBE and IB vs. VBE plots are required to explain the observedβdc vs. IC characteristics.


βdc Variation with Current

• For the simplicity ofanalysis, we assume:– VBC = 0 (neglect the

effect of base-widthmodulation, VA)

– neglect ohmic-bulkresistors (r'e, r'b, r'c)

⇒ VBE = VB'E'

VBC = VB'C'

– normal active modei.e., forward currentgain analysis only.

ln(I

C, I

B)

VBE

IB

IC

βF

Reg

ion-

I

Reg

ion-

II

Region-III

VBC = 0

ln(IS)

FM

SIβ

ln


βdc Variation with Current

• Region-II− βF ≡ βFM (maximum gain) = constant– EM1 model holds∴ @ VBC = 0, we get

• Region-I− βdc↓ as IB↑ due to its additional components from

♦ carrier recombination at the surface, IB(surface)♦ carrier recombination in the EB space-charge layer, IB(EB-scl)♦ EB surface channels, IB(channel).

ln(I

C, I

B)

VBE

IB

IC

βFReg

ion-

I

Reg

ion-

II

Region-III

VBC = 0

ln(IS)

FM

SIβ

ln

−=

−=

1)0(

1)0(

eI

I

eII

kTBE

kTBE

Vq

F

SB

Vq

SC

β

(20)

(21)


βdc Variation with Current: Low IC Region

• ∆ΙB in (22) can be represented by an additional non-ideal EB-diode in the basic EM1 model with diode current given by:

• Here we have two new parameters for modeling low IC region:– nE = low-current forward emission coefficient (~ 2)– C2 = models various components of IS in low IB (~ 103).

−=∆∴

+−+≡∆

1)0()total(

)channel()sclEB()surface()total(Thus,

2 eICI

IIII

kTEBE

nVq

SB

BBBB

−1)0(2 eIC kTE

BEnVq

S

(22)

−+

−=∴ 1)0(1

)0(2 eICe

II kTE

BEkT

BEnVq

S

Vq

FM

SB β

(23)


βdc Variation with Current: Low IC Region

• If VBC > 0 in the inverse region of BJT operation, two additionalparameters can be used to model the components of IB:– nC = low-current inverse emission coefficient (~ 2)– C4 = components of IS in IB in the inverse region (~ 103).

• ∆ΙB in the inverse region is represented by an additional non-idealCB-diode in the basic EM1 model with diode current given by:

• Therefore, the general expression for IB is given by:

−=∆ 1)0(4)region-inverse( eICI kTC

BCnVq

SB

−+

−+

−+

−=

1)0(1)0(

1)0(1)0(

4

2

eICeI

eICeI

I

kTcBC

kTBC

kTEBE

kTBE

nVq

S

Vq

RM

S

nVq

S

Vq

FM

SB

β

β

(24)


βdc Versus IC: Modeling Low IC Region

E

r'c

r'e

C

C'

E'

Csub

r'bB B'

CjC

CjE

CDC

CDE

ICT

IEC/βR

ICC/βF

−1)0(2 eIC kTE

BEnVq

S

−1)0(4 eIC kTC

BCnVq

S

Non-linear hybrid BJT model to model β vs. IC at low current level.


βdc Variation with Current: High IC Region

• Region-III– at high level injection, the minority carrier concentration into base

region is significant with respect to majority carrier concentration.

– from the condition of quasi-neutrality in the base: ρ(base) = q[NA(x) + ∆n(x)]

– under these conditions it can be shown that:

– the high-level injection is modeled with an additional modelparameter θ in EM3 model.

– for high level injection, IC @ VBC = 0 is given by:

eI kTBEVq

C2level)(high ∝−

−

+

= 11

)0(

2

ee

II kT

BE

kTBE

Vq

VqS

C

θ (25)


βdc Variation with Current: Model Parameters

ln(I)

qVBE/kT

IB

IC

βF

VBC = 0

ln(IS(0))

FM

SIβ

)0(ln

ln(C2IS(0))

θ)0(

ln SI

slope = 1slope = 1

slope = 1/nE

slope = 1/2• EM3-Model parameters

to describe βF - IC data:– Region-I

♦ C2

♦ nE

– Region-II♦ βFM

– Region-III♦ θ

• The parameter set {C2, nE, βFM, θ} is obtained from above plot.

• In the inverse region a set {C4, nC, βRM, θR} is defined at VBE = 0.


βdc Versus IC: Effect of Ohmic Resistors

• The series resistances (r'e, r'b, r'c):– do not effect theoretical analysis and model equations– effects measured device characteristics.

• To account for the ohmic resistors in the model equations, wechange:– measured VBE with the internal VB'E' given by

♦ VB'E' = VBE + (IB r'b + IE r'e)

– measured VBC with the internal VB'C' given by♦ VB'C' = VBC − (IC r'c + IB r'b)


EM3 Model: Improved Charge Storage Model

• Improved CjC - r'b model:

– CjC is distributed over rb'

– split up on either side of rb'

– additional model parameter RATIOis used to model distributed CjC

– 0 < RATIO < 1. E

r'cC

C'

r'bB

B'

CjCRest ofEM3

Model

E

r'cC

C'

r'bB

B'

(RATIO)CjC(VB'C') Rest ofEM3

Model

(1−RATIO)CjC(VB'C')


EM3 Model: Variation of τF with IC

• At high level injection, τF↑ as IC↑ due to an increase in τB.

• At high current level, the increase in τB is due to:– a reduction in the low-level aiding field effect in drift transistors– an effective base-widening effect (Kirk-effect)– two-dimensional spreading effect.

• In the normal active region, it can be shown that:

where LE = emitter width W = base width IC0 = IC at which τF starts to increase.

0

2

0

2

C for,141

1)0()( CCC

CCEFL

CC

DE

acF IIII

WL

dIdQ

I ≥

−

+=≡ ττ (26)


EM3 Model: Variation of τF with IC

• Two additional model parameters:– IC0

– (LE/W)

• The model parameters are obtained by curve fitting τFac vs. ICplots to Eq. (26).

ICIC0

acFτ

)0(FLτ


EM3 Model: Temperature Dependence

• Physics-based temperature variation:– inherent temperature dependence for parameters τF, CjC, and CjE.

• Analytical temperature variation:– additional temperature coefficients for parameters βF, r'b, r'c.

Where Par = target parameter TC1 = first-order temperature coefficient TC2 = second-order temperature coefficient.

5.12

refref )(

)()()(

=

refFLF T

TTWTW

TT ττ (27)

( ){ }φε γγ TTjj mTTTCTC −−+= 2)(1)()( refrefxx (28)

{ }2ref2ref1ref )()(1)()ar( TTTCTTTCTParTP −+−+= (29)


EM3 Model: Summary

• Additional model parameters– base-width modulation:

♦ VA

− β roll-off:♦ {C2, nE, βFM, θ} (inverse region: {C4, nC, βRM, θR})

– distributed CjC:

♦ RATIO, IC0, LE/W

– temperature coefficients for βF, r'b, r'c:♦ TC1

♦ TC2.

EM1 BJT Model: Deficiencies - SCU · device characterization ⇒ EM3 BJT model. 29-Jan-04 HO #7:...

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Transcript of EM1 BJT Model: Deficiencies - SCU · device characterization ⇒ EM3 BJT model. 29-Jan-04 HO #7:...