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BJT - Mode of Operations

HO #6: ELEN 251 - BJT EM Models Page 1S. Saha

N+ PE B C

N- N+E C

B

BE C

• BJTs can be modeled by two back-to-back diodes.

• BJTs are operated in four modes.

HO #6: ELEN 251 - BJT EM Models Page 2S. Saha

Mode of Operations

1) Forward active / normal– BE junction forward biased– BC junction reversed biased

here,,BFC

BC

IIII

β=∴∝

where βF = forward gain

+ VBC

+ VBE

−

−

Saturation

NormalCut-off

Inverse

2) Reverse active– BE junction reverse biased and BC junction forward biased

here, reverse gain, βR = IE/IB ≈ 1

3) Saturation region– BE and BC are forward biased

4) Cut-off region– BE and BC are reversed biased

Basic BJT Model

HO #6: ELEN 251 - BJT EM Models Page 3S. Saha

Basic BJT model can be derived considering two back-to-back diodes as npn-BJT. When:

1) B-E junction is forward biased:– forward current, IF flows through E-B diode.

− αFIF flows in the collectorhere, αF = forward gain ≅ IC/IE if VBE is +ve

2) B-C junction is reversed biased:– reverse current, IR flows through B-C diode.

− αRIR flows in the emitterhere, αR = reverse gain ≅ IE/IC if VBE is +ve

n p

E B C

n

E C

IE IC

B IB

IF αFIF

aRIR IR

EM1 BJT Model: Injection Version

HO #6: ELEN 251 - BJT EM Models Page 4S. Saha

• The basic Ebers-Moll model (EM1):

E O

IE

O CIC

IB

αRIR αFIF

OB

IF IR

• Terminal currents: IE = − IF + αRIR (1)

IC = αFIF − IR (2)

IB = IF − αRIR − αFIF + IR

= (1 − αF) IF + (1 − αR) IR

∴ IB = (1 − αF) IF + (1 − αR) IR (3)

EM1 BJT Model: Injection Version

HO #6: ELEN 251 - BJT EM Models Page 5S. Saha

• We know from current flow analysis:

Here IES = E-B saturation current; VBE = B-E voltage ICS = C-B saturation current; VBC = B-C voltage

• The terminal currents from (1), (2), (4), (5):

⎟⎠⎞⎜

⎝⎛ −≅=

⎟⎠⎞⎜

⎝⎛ −≅=

1

1

eIII

eIII

kTV BCq

CSnCR

kTV BEq

ESnEF (4)

(5)

⎟⎠⎞⎜

⎝⎛ −−⎟

⎠⎞⎜

⎝⎛ −=

⎟⎠⎞⎜

⎝⎛ −+⎟

⎠⎞⎜

⎝⎛ −−=

11

11

eIeII

eIeII

kTV BCq

CSkTV BEq

ESFC

kTV BCq

CSRkTV BEq

ESE

α

α (6)

(7)

EM1 BJT Model: Injection Version

HO #6: ELEN 251 - BJT EM Models Page 6S. Saha

• From reciprocity property: αFIES = αRICS ≡ IS

• Therefore,

• Again, βF = CE forward current gain = αF/(1 − αF) βR = CE reverse current gain = αR/(1 − αR)

• Since IS = f(ni2) = f(T)

⎟⎠⎞⎜

⎝⎛ −−⎟

⎠⎞⎜

⎝⎛ −=

⎟⎠⎞⎜

⎝⎛ −+⎟

⎠⎞⎜

⎝⎛ −−=

11

11

eI

eII

eIeII

kTV BCq

R

SkT

V BEq

SC

kTV BCq

SkTV BEq

F

SE

α

α(8)

(9)

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=∴ ref

g

TTkE

refrefSS e

TTTITI

113

)()( (10)

EM1 BJT Model: Injection Version

HO #6: ELEN 251 - BJT EM Models Page 7S. Saha

• Model equations:

Where αF = βF/(1 + βF) αR = βR/(1 + βR)

∴ Total five model parameters: βF, βR, IS, Tref, and Eg can be used to describe basic BJT device characteristics without parasitics.

⎟⎠⎞⎜

⎝⎛ −−⎟

⎠⎞⎜

⎝⎛ −=

⎟⎠⎞⎜

⎝⎛ −+⎟

⎠⎞⎜

⎝⎛ −−=

1)(1)(

1)(1)(

eTI

eTII

eTIeTII

kTV BCq

R

SkT

V BEq

SC

kTV BCq

SkTV BEq

F

SE

α

α(8')

(9')

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

⎟⎟⎠

⎞⎜⎜⎝

⎛= ref

g

TTkE

refrefSS e

TTTITI

113

)()( (11)

EM1 BJT Model: Transport Version

HO #6: ELEN 251 - BJT EM Models Page 8S. Saha

Model equations (8) and (9) can be written as:

Where the reference source currents:

ECR

CCC

ECCCF

E

III

III

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−=

α

α

1

1(12)

(13)

⎟⎠⎞⎜

⎝⎛ −=

⎟⎠⎞⎜

⎝⎛ −=

1

1

eII

eII

kTV BCq

SEC

kTV BEq

SCC

E O

IE

O CIC

IB

IEC ICC

OB

ICC/αF IEC/αR

ECR

CCF

B III ⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=∴ 1111

αα

EM1 BJT Model: Nonlinear Hybrid-π

HO #6: ELEN 251 - BJT EM Models Page 9S. Saha

• From transform model (12) and (13), we get:

• Where the reference source current is:

• The diode currents are:

( )

( )R

ECCTEC

RECCCC

CTF

CCECCCCC

FE

IIIIII

IIIIII

βα

βα

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

−−=−−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

11

11 (14)

(15)

⎭⎬⎫

⎩⎨⎧ ⎟

⎠⎞⎜

⎝⎛ −−⎟

⎠⎞⎜

⎝⎛ −=−= 11 eeIIII kT

V BCqkT

V BEq

SECCCCT (16)

⎟⎠⎞⎜

⎝⎛ −=

⎟⎠⎞⎜

⎝⎛ −=

1

1

eII

eII

kTV BCq

R

S

R

EC

kTV BEq

F

S

F

CC

ββ

ββ(17)

(18)

EM1 BJT Model: Nonlinear Hybrid-π

HO #6: ELEN 251 - BJT EM Models Page 10S. Saha

• The model:

• The terminal currents:

E O

IE

O CIC

IB

ICT = ICC - IEC

OB

ICC/βF IEC/βR

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

R

EC

F

CCB

R

ECCTC

CTF

CCE

III

III

III

ββ

β

β(19)

(20)

(21)

EM1 BJT Small-Signal Model: Linear Hybrid-π

HO #6: ELEN 251 - BJT EM Models Page 11S. Saha

circuitopen model, BJT basic EM1For

:resistanceOutput

:resistanceInput

:ctanceTranscondu

1

),(

→

=∂∂

=

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

=

=∂∂

=

−

µ

µ

β

r

IV

vir

gvir

IkTq

vig

C

A

QCE

C

m

F

QBE

Bπ

CIVBE

Cm

CBE

OE

iE

OC

iC

iB

gmvBE

OB

− vBE + + vBC −

rµrπ

OE

IE

OC

IC

IB

ICT = ICC - IEC

OB

ICC/βF IEC/βR

Non-linear hybrid-π Linear hybrid-π

EM1 BJT Model: Deficiencies

HO #6: ELEN 251 - BJT EM Models Page 12S. Saha

• 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

HO #6: ELEN 251 - BJT EM Models Page 13S. Saha

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

HO #6: ELEN 251 - BJT EM Models Page 14S. Saha

• 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

HO #6: ELEN 251 - BJT EM Models Page 15S. Saha

• 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.

VBE

ln(I

C, I

B)IC

IB

∆VBE = IBr'b + IEr'e

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

transient response– difficult to measure

accurately due to the dependence on r'e and operating point.

EM2 Model: Modeling Charge Storage Effect

HO #6: ELEN 251 - BJT EM Models Page 16S. Saha

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

E

r'c

r'e

r'b EM1model

B

C

B'C'

E'

Csub

CjC

CjE

CDC

CDE

• CDE = EB-junction diffusion capacitance.

• CDC = CB-junction diffusion capacitance.

• CjE = EB-junction capacitors.• CjC = CB-junction capacitors.

Csub = C-substrate junction capacitance.

Complete EM2 BJT Model

HO #6: ELEN 251 - BJT EM Models Page 17S. Saha

The complete model to account for terminal resistors and charge storage 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 voltageBC internal voltage

EM2 Model: Junction Capacitors

HO #6: ELEN 251 - BJT EM Models Page 18S. Saha

CjE and CjC model the incremental fixed charges stored in the BE and BC space charge layers of BJT due to applied bias VB'E' and VB'C', respectively. From PN-junction theory (HO #3, page 19) we get:

( )Em

E

EB

jEEBjE

V

CVC

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=′′

′′

φ1

0(22)

( )Cm

C

CB

jCCBjC

V

CVC

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=′′

′′

φ1

0(23)

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

HO #6: ELEN 251 - BJT EM Models Page 19S. Saha

E Cnp(0)

npo

B

QBQE

QB

CQBE

W

Total minority carrier charge due to forward current ICC: QDE = QE + QBE + QB + QBC = (τE + τEB + τB + τCB) ICC

= 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.

Base Transit Time, τB

HO #6: ELEN 251 - BJT EM Models Page 20S. Saha

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

Therefore, the total e- charge in the base is simply given by: QB = (1/2)qnpWBAE (24)

where AE = emitter area The transit time across the base is:

τB = QB/ICC (25)

Since E = 0, ∴ ICC = qAEDn(dn/dx) ≅ qAEDnnp/WB

where Dn = average e- diffusivity in the base region.

npE B C

~ 0

Base Transit Time

HO #6: ELEN 251 - BJT EM Models Page 21S. Saha

Substituting for QB and ICC in (25), we get,

Example: If WB = 1 µm and the base is lightly doped so that Dn ≅38 cm2/sec, then from (26) τB ≅ 132 psec.

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

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

Usually, τB is not the dominant frequency limitation in modern BJTs.

DW

n

BB 2

2

≅τ (26)

EM2 Model: Diffusion Charge, QDC

HO #6: ELEN 251 - BJT EM Models Page 22S. Saha

E CB

QBRQC

QB

CRQBER

W

Total minority carrier charge due to reverse current IEC: QDC = QC + QBER + QBR + QBCR = (τC + τCB + τBR + τEB) IEC

= τRIEC

where τR = total reverse delay time consisting of collector delay τC, CB-

reverse space charge layer transit time τCB, reverse base transit time τBR, and reverse EB-space charge layer transit time τEB.

EM2 Model: Diffusion Capacitors CDE and CDC

HO #6: ELEN 251 - BJT EM Models Page 23S. Saha

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

( )

( )CB

ECR

CB

DCCBDC

EB

CCF

EB

DEEBDE

VI

VQVC

VI

VQVC

′′′′′′

′′′′′′

=≡

=≡

τ

τ(27)

(28)

( )

( ) RmRRVCB

EC

VCB

DCsignalsmallDC

FmFFVEB

CC

VEB

DEsignalsmallDE

gdVdI

dVdQC

gdVdI

dVdQC

EBEB

CBCB

ττ

ττ

==≡

==≡

=′′=′′−

=′′=′′−

′′′′

′′′′

00

00

(29)

(30)

The model parameters: τF and τR.

EM2 Model: Model Parameters

HO #6: ELEN 251 - BJT EM Models Page 24S. Saha

Substrate Capacitance (Csub) is considered as a constant in EM3 model.

∴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

HO #6: ELEN 251 - BJT EM Models Page 25S. Saha

• Advantages of EM2 model is the improvement in DC and AC device 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 improve device characterization ⇒ EM3 BJT model.

EM3 Model: Base Width Modulation

HO #6: ELEN 251 - BJT EM Models Page 26S. Saha

BE C

W(VBC)

N+ P N-xdbcxdeb

x = 0 x = W

• Base-width modulation describes the change in the quasi-neutral base-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.

Base Width Modulation by VBC − Early Effect

HO #6: ELEN 251 - BJT EM Models Page 27S. Saha

• 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 = forward Early voltage.

IB5

IB4

IB3

IB2

IB1

VA VCE

IC IB5 > IB4 > IB3 > IB2 > IB1

0

• Effect of base-width modulation:

IS = f(VBC, VA)

βF = f(VBC, VA)τF = f(VBC, VA)

Base Width Modulation

HO #6: ELEN 251 - BJT EM Models Page 28S. Saha

• 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(1)0()(

)0()0( Here )0()(

BC

BC

VBC

BCBC

BCVBC

BCBC

dVdW

WVWVW

VWWdVdWVWVW

(31)

CC

BCBEBC

CCC

C

CEA I

dIdVVV

dIdII

dIdVV −=+−=≡ )( (32)

Base Width Modulation

HO #6: ELEN 251 - BJT EM Models Page 29S. Saha

Now,

From (32) and (33):

Combining (31) and (34) we get:

)0(WdW

dVVBCV

BCA −= (34)

(33)0

0

2

)0()0(

)(

=

=

=−⇒−=∴

−≅

BC

BC

kTBE

VC

CCC

V

Vq

BCAB

inBC

dWW

dIIdW

WIdI

eVWNnDqAI

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

A

BCBC V

VWVW 1)0()( (35)

Base Width Modulation - Model parameters

HO #6: ELEN 251 - BJT EM Models Page 30S. Saha

(36)⎟⎟⎠

⎞⎜⎜⎝

⎛−=∴

+==⇒

=≅

A

BCSBCS

A

BC

S

BCSBCS

BCAB

inBS

VVIVI

VV

IVW

WIVI

VWNnDqAI

1)0()(

1

)0()(

)0()0()(

)0()0(

2

Similarly, we can show that:

2

1)0()(

1)0()(

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

A

BCBBCB

A

BCBCF

VVV

VVV

ττ

ββ (37)

(38)

Base Width Modulation

HO #6: ELEN 251 - BJT EM Models Page 31S. Saha

• 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 the active region of BJT operation. slope = (dIC/dVCE)|VBE = constant = −(dIC/dVBC)|VBE = constant

⎭⎬⎫

⎩⎨⎧ ⎟

⎠⎞⎜

⎝⎛ −−⎟

⎠⎞⎜

⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛+

= 111

)0(ee

VV

II kTV BCq

kTV BEq

A

BC

SCT (39)

⎟⎠⎞⎜

⎝⎛ −+⎟

⎠⎞⎜

⎝⎛ −= 1

)0()0(1

)0()0(

eI

eII kT

BCkT

BE Vq

R

SVq

F

SB ββ

(40)

EM3 Model: βdc Variation with Current

HO #6: ELEN 251 - BJT EM Models Page 32S. Saha

• 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 observed βdc vs. IC characteristics.

βdc Variation with Current

HO #6: ELEN 251 - BJT EM Models Page 33S. Saha

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

effect of base-width modulation, VA)

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

⇒ VBE = VB'E'

VBC = VB'C'

– normal active mode i.e., forward current gain analysis only.

ln(I C

, IB)

VBE

IB

IC

βF

Reg

ion-

I

Reg

ion-

II

Region-III

VBC = 0

ln(IS)

⎟⎟⎠

⎞⎜⎜⎝

⎛

FM

SIβ

ln

βdc Variation with Current

HO #6: ELEN 251 - BJT EM Models Page 34S. Saha

• 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

, IB)

VBE

IB

IC

βFReg

ion-

I

Reg

ion-

II

Region-III

VBC = 0

ln(IS)

⎟⎟⎠

⎞⎜⎜⎝

⎛

FM

SIβ

ln⎟⎠⎞⎜

⎝⎛ −=

⎟⎠⎞⎜

⎝⎛ −=

1)0(

1)0(

eII

eII

kTBE

kTBE

Vq

F

SB

Vq

SC

β

(41)

(42)

βdc Variation with Current: Low IC Region

HO #6: ELEN 251 - BJT EM Models Page 35S. Saha

• ∆ΙB in (43) 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

(43)

⎟⎠⎞⎜

⎝⎛ −+⎟

⎠⎞⎜

⎝⎛ −=∴ 1)0(1)0(

2 eICeII kTE

BEkT

BEnVq

SVq

FM

SB β

(44)

βdc Variation with Current: Low IC Region

HO #6: ELEN 251 - BJT EM Models Page 36S. Saha

• If VBC > 0 in the inverse region of BJT operation, two additional parameters 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-ideal CB-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

eICeII

kTcBC

kTBC

kTEBE

kTBE

nVq

SVq

RM

S

nVq

SVq

FM

SB

β

β

(45)

βdc Versus IC: Modeling Low IC Region

HO #6: ELEN 251 - BJT EM Models Page 37S. Saha

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

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

βdc Variation with Current: High IC Region

HO #6: ELEN 251 - BJT EM Models Page 38S. Saha

• 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 model parameter θ in EM3 model.

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

eI kTBEVq

C 2level)(high ∝−

⎟⎠⎞⎜

⎝⎛ −

⎟⎠⎞⎜

⎝⎛ +

= 11

)0(

2

ee

II kTBE

kTBE

Vq

VqS

Cθ (46)

βdc Variation with Current: Model Parameters

HO #6: ELEN 251 - BJT EM Models Page 39S. Saha

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

HO #6: ELEN 251 - BJT EM Models Page 40S. Saha

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

• To account for the ohmic resistors in the model equations, we change:– 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

HO #6: ELEN 251 - BJT EM Models Page 41S. Saha

• Improved CjC - r'b model: – CjC is distributed over rb'– split up on either side of rb'– additional model parameter RATIO

is 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

HO #6: ELEN 251 - BJT EM Models Page 42S. Saha

• 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 ,1411)0()( CC

C

CCEFL

CC

DEacF II

II

WL

dIdQI ≥

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+=≡ ττ (47)

EM3 Model: Variation of τF with IC

HO #6: ELEN 251 - BJT EM Models Page 43S. Saha

• Two additional model parameters:– IC0

– (LE/W)

• The model parameters are obtained by curve fitting τFac vs. ICplots to Eq. (47).

ICIC0

acFτ

)0(FLτ

EM3 Model: Temperature Dependence

HO #6: ELEN 251 - BJT EM Models Page 44S. Saha

• 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

TTWTWTT ττ (48)

( ){ }φε γγ TTjj mTTTCTC −−+= 2)(1)()( refrefxx (49)

{ }2ref2ref1ref )()(1)()ar( TTTCTTTCTParTP −+−+= (50)

EM3 Model: Summary

HO #6: ELEN 251 - BJT EM Models Page 45S. Saha

• 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.

Home Work 3: Due April 21, 2005

HO #6: ELEN 251 - BJT EM Models Page 46S. Saha

1) An npn-BJT is used as an open-collector diode as shown in Figure below:(a) use the injection version of EM1 BJT model to derive an expression for IE as a

function of VBE.(b) use Eq derived in part (a) to calculate VBE for IE = −1 mA.

Given that:αF = 0.98αR = 0.49IES = 10−16 AT = 300°K

2) The npn-BJT in Problem #1 is used as a shorted base-collector diode as shown in Figure below:(a) use the injection version of EM1 model to derive an expression for IE as a

function of VBE

(b) use Eq derived in part (a) to calculate VBE if IE = -1 mA.

+VBE−

IE

+VBE−

IE

Home Work 3: Due April 21, 2005

HO #6: ELEN 251 - BJT EM Models Page 47S. Saha

3) The EM1 model discussed in class can be simplified for different modes of operation. As an example, in the forward active mode, IR is negligibly small and therefore, the model can be simplified for hand calculation. Use the injection version of EM1 model to:(a) show the simplified model in the normal active mode of an npn-BJT operation.(b) show the simplified model in the saturation mode of an npn-BJT operation.

Clearly state any assumptions you make.

4) Following the procedure discussed in class, derive the models for lateral pnp-BJTs:(a) sketch the basic EM1 model(b) write Eq for terminal currents(c) include bulk-ohmic resistors and charge storage elements in EM1 model to

generate lateral pnp-BJT EM2 model.

5) Develop and sketch small-signal linear hybrid-π EM2 model for npn-BJTs. Derive the model equations.