07 Bipolar Junction Transistors

Chapter 7

BIPOLAR JUNCTION TRANSISTORS

Professor Hisham Z. Massoud

Department of Electrical and Computer EngineeringFitzpatrick Center, Room 3521

Duke University, Durham, NC 27708–0291

[email protected]

ECE216 Chapter 7 – Bipolar Junction Transistors 7.1

Chapter 7 – BIPOLAR JUNCTION TRANSISTORS

7.1. Introduction7.2. Integrated-Circuit Bipolar Junction Transistor Structures7.3. Bipolar Junction Transistor in Thermal Equilibrium7.4. BJT Bias Conditions and Modes of Operation7.5. Basic BJT Operation in the Forward-Active Mode7.6. BJT Static I(V ) Characteristics: Ebers-Moll Model7.7. BJT Capacitance-Voltage C(V ) Characteristics7.8. BJT Dynamic I(V ) Characteristics: Charge-Control Model7.9. BJT Small-Signal Equivalent Circuit

7.10. Temperature Effects7.11. Frequency Effects7.12. Switching Analysis7.13. BJT Breakdown7.14. BJT SPICE Model7.15. Summary


7.1. Introduction

• William Shockley at Bell Labs submitted his patent for the junction transistoron June 26, 1948, and the junction transistor Patent No. 2,569,347 was issuedon September 25, 1951.

• Semiconductor devices in which both electrons and holes participate in the con-duction are termed bipolar devices and for this reason the junction transistoris now more commonly called the bipolar-junction transistor (BJT), or, simply,the bipolar transistor.

• Shockley’s junction transistor patent included heavy doping near the contactseven though no junction transistor had yet been fabricated and demonstrated.The “existence proof” for the junction transistor was made on April 7, 1949, atBell Labs with a Ge structure fabricated by Bob Mikulyak.


7.1. Introduction

Representation of the junction (bipolar) transistor

in Shockley’s patent (2,569,347).


7.2. Integrated-Circuit BJT Structures

Junction Isolation



Junction-Isolated Bipolar Junction Transistor

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Trench-Isolated Bipolar Junction Transistor



Dopant Profiles in the Bipolar Junction Transistor

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Bipolar Junction Transistor Types and Their Symbols

NPN Bipolar Transistor PNP Bipolar Transistor


7.3. Bipolar Junction Transistor in Thermal Equilibrium

Energy-Band Diagram in Thermal Equilibrium

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Depletion Widths in Thermal Equilibrium



Charge Concentration, Field, and Potential Distributions

E B C

p+ n p

%(x)

Ex(x)

ψ(x)

x

x

x

−qN−

a,E

qN+

d,B

−qN−

a,C


7.4. BJT Bias Conditions and Modes of Operation

Bias Region Base/Emitter Junction Base/Collector Junction

Reverse-biased Reverse-biased

Cut-off NPN: VBE < 0 NPN: VBC < 0

PNP: VEB < 0 PNP: VCB < 0

Forward-biased Reverse-biased

Forward-Active NPN: VBE > 0 NPN: VBC < 0

PNP: VEB > 0 PNP: VCB < 0

Forward-biased Forward-biased

Saturation NPN: VBE > 0 NPN: VBC > 0

PNP: VEB > 0 PNP: VCB > 0

Reverse-biased Forward-biased

Reverse-Active NPN: VBE < 0 NPN: VBC > 0

PNP: VEB < 0 PNP: VCB > 0


7.4. BJT Bias Conditions and Modes of Operation

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7.5. PNP BJT Basic Operation in the Forward-Active Region

p+

E

n

B

p

CIE IC

IB

VEB VCB

vIN vOUTRL



p+

E

n

B

p

CIE IC

IB

VEB VCB



Variable Emitter Region Base Region Collector Region

Region Type p+ n p

Evac(x) Evac,E(x) Evac,B(x) Evac,C(x)

Ec(x) Ec,E(x) Ec,B(x) Ec,C(x)

Ei(x) Ei,E(x) Ei,B(x) Ei,C(x)

Ev(x) Ev,E(x) Ev,B(x) Ev,C(x)

EFn(x) EFn,E(x) EFn,B(x) EFn,C(x)

EFp(x) EFp,E(x) EFp,B(x) EFp,C(x)

Ex(x) Ex,E(x) Ex,B(x) Ex,C(x)

ψ(x) ψE(x) ψB(x) ψC(x)

Built-in Voltages Vbi,E/B Vbi,C/B



Variable Emitter Region Base Region Collector Region

n(x) nE(x) nB(x) nC(x)

p(x) pE(x) pB(x) pC(x)

Dopant Concentration N−a,E N+

d,B N−a,C

Equi. Maj. Carrier Conc. pE = N−a,E nB = N+

d,B pC = N−a,C

Equi. Min. Carrier Conc. nE = n2

i /N−a,E pB = n2

i /N+d,B nC = n2

i /N−a,C

Depletion Width Wd Wd,E(VEB) Wd,B/E(VEB) Wd,C(VCB)

Wd,B/C(VCB)

Minority Carrier Mobility µn,E µp,B µn,C

Minority Carrier Diffusivity Dn,E Dp,B Dn,C

Minority Diffusion Length Ln,E Lp,B Ln,C

Minority Carrier Lifetime τn,E τp,B τn,C



Energy-Band Diagram

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Depletion Widths



Minority-Carrier Distributions

E B C

p+ n p

nE(x) pB(x)

nC(x)n

E

pB

n

C

0−WE WB WB+WC



Terminal Currents

E B C

p+ n p

↓IB

→ IE

→ IC



Electron and Hole Transport, Generation, and Recombination

E B C

p+

n p

Ip,C

Irec,B

Iscr,E/BIn,E

Ip,C

Iscg,C/BIn,C

In

,E

Iscr,E

/B

Irec,B

Iscg,C

/B

In

,C



Detailed Current Components

E B C

p+

n p

Ip,C

Irec,B

Iscr,E/BIn,E

Ip,CIp,E

Iscg,C/BIn,C

In

,E

Iscr,E

/B

Irec,B

Iscg,C

/B

In

,C



Detailed Current Components

Current Description of Current Component Voltage Dependence

Ip,E Hole diffusion current injected from emitter intobase

exp(qVEB/kBT )

In,E Electron diffusion current injected from base intoemitter

exp(qVEB/kBT )

Iscr,E/B Emitter-base space-charge recombination current exp(qVEB/2kBT )

Irec,B Current due to recombination of injected minority-carrier holes with majority-carrier electrons in theneutral base region

exp(qVEB/kBT )

Ip,C Portion of Ip,E injected from the emitter into thebase which diffuses across the base and flows in thecollector and is given by Ip,C = Ip,E − Irec,B

exp(qVEB/kBT )

Iscg,C/B Collector-base space-charge generation current exp(qVCB/2kBT )

In,C Current in the C/B junction due to the diffusionof electrons from collector to base and due to thediffusion of holes form collector to base

exp(qVCB/kBT )



IE = Ip,E + In,E + Iscr,E/B ,

IC = Ip,C + In,C + Iscg,C/B ,

and

IB = IE − IC ,

=(Ip,E + In,E + Iscr,E/B

)−

(Ip,C + In,C + Iscg,C/B

),

=(Ip,C + Irec,B + In,E + Iscr,E/B

)−

(Ip,C + In,C + Iscg,C/B

),

= In,E + Iscr,E/B + Irec,B − Iscg,C/B − In,C .



Current Gain

The base transport factor αT is defined as the ratio of the hole current that reaches thecollector to the hole current that enters the base region from the emitter. It is definedas

αT ≡Ip,C

Ip,E=Ip,E − Irec,B

Ip,E= 1 −

Irec,B

Ip,E.

The emitter efficiency γE is defined as the ratio of the emitter current injected into thebase region to the total emitter current. It is given by

γE ≡Ip,E

IE=

Ip,E

Ip,E + In,E + Iscr,E/B.

The common-base current gain αF is defined as the ratio of the hole collector currentto the total emitter current. It is given by

αF ≡Ip,C

IE=

Ip,C

Ip,E + In,E + Iscr,E/B=

Ip,E

Ip,E + In,E + Iscr,E/B·Ip,C

Ip,E= γE αT .



The common-base current gain is less than one and is generally found to be near 0.998.Although the current gain in the common-base configuration is less than unity, theoutput voltage is very large compared to the input voltage.

In good transistor design, all base-current components are made as small as possible.Since the electron diffusion current injected into the emitter In,E does not give anycollector current, it will be seen to be minimized by doping the emitter with a very largeacceptor concentration for the pnp BJT. Space-charge recombination current Iscr,E/B

is emitter-base current which does not contribute to the collector current so that stepsare taken to minize space-charge recombination in the emitter-base junction. Also, thebase-region recombination current Irec,B represents the loss of holes which are injectedinto the base by the emitter, and the base region is made thin to minimize this current.Therefore, bipolar transistors are designed to make these currents, which flow in thebase lead, small as compared to Ip,E , the hole diffusion current injected from the emitterinto the base.



The collector current IC is given by

IC = Ip,C + In,C + Iscg,C/B ,

= αF IE + In,C + Iscg,C/B ,

= αF IE + ICBO ,

where ICBO is the collector-base reverse saturation current when the emitter is open(so that IE = 0).



Minority-Carrier Distribution in the Base Region

E

p+

B

n

C

p

0 WB x

Wd,B/E Wd,B/C




We assume that the base widthWB is smaller than the minority-carrier diffusion lengthLp,B of holes in the base region of a pnp transistor, (WB < 0.1Lp,B). The minority-carrier concentration goes to zero at the edge of the depletion region at the base-collector junction because this junction is reverse-biased. We assume that

Wd,B/E < WB ,

Wd,B/C < WB ,

or that

Wd,B/E ' 0 ,

WB −Wd,B/C 'WB ,

that

pB(0) ' pB exp

(qVEB

kBT

)

,

and

pB(WB) ' 0 .




We can then write that for a linear distribution of minority carriers in the base regiongoing from pB exp(qVEB/kBT ) at x ' 0 to pB exp(qVCB/kBT ) at x 'WB in the form

pB(x) =

[

pB exp

(qVEB

kBT

)

− pB exp

(qVCB

kBT

)] (

1 −x

WB

)

+ pB exp

(qVCB

kBT

)

,

= pB exp

(qVEB

kBT

) (

1 −x

WB

)

+ pB exp

(qVCB

kBT

)x

WB,

' pB exp

(qVEB

kBT

) (

1 −x

WB

)

.



Minority-Carrier Charge Stored in the Base Region

The net charge stored in the base region Qn,B (C) is given by

Qp,B(VEB, VCB) = q A

∫ WB

0

[pB(x) − pB ] dx ,

= q A

∫ WB

0

pB exp

(qVEB

kBT

) (

1 −x

WB

)

dx

+ q A

∫ WB

0

pB exp

(qVCB

kBT

)x

WBdx− q A

∫ WB

0

pB dx ,

=q A pB WB

2pB exp

(qVEB

kBT

)

+q A pB WB

2pB exp

(qVCB

kBT

)

− q A pB WB .

and for VCB < 0, we can write that

Qp,B(VEB) 'q A pB WB

2exp

(qVEB

kBT

)

'q AWB n

2i

2N+d,B

exp

(qVEB

kBT

)

.



Emitter-Base Current

The emitter-base hole diffusion current is found from the expression for pB(x) as

Ip,E = −q ADp,BdpB(x)

dx

∣∣∣∣x=Wd,B/E

,

= −q ADp,B pB exp

(qVEB

kBT

) (

−1

WB

)

,

=q ADp,B n

2i

WB N+d,B

exp

(qVEB

kBT

)

.

The emitter-base electron diffusion current is found from the expression for

In,E = q ADn,EdnE(x)

dx

∣∣∣∣x=−Wd,E

,

= q ADn,E nE

[

exp

(qVEB

kBT

)

− 1

] (1

WE

)

,

=q ADn,E n

2i

WE N−a,E

[

exp

(qVEB

kBT

)

− 1

]

.



Base Recombination Current

The recombination of the injected minority carriers in the base region is obtained fromthe continuity equation of holes which is written as

∂pB(x)

∂t= −

1

q

∂Jp,B(x)

∂x+ (Gp,B −Rp,B) = 0 ,

in steady state. With no external source of generation, we can write that Gp,B = 0and the continuity equation becomes

−dJp,B(x)

dx= q Rp,B = q

[pB(x) − pB ]

τp,B.

Integration of this equation yields

−

Jp,B(WB)∫

Jp,B(0)

dJp,B(x) =q

τp,B

∫ WB

0

[pB(x) − pB ] dx .

The integral on the left-hand side gives Irec,B which is then given by

Irec,B = −A[Jp,B(WB)︸︷︷︸

Jp,C

− Jp,B(0)︸︷︷︸

Jp,E

] =q A

τp,B

∫ WB

0

pB exp

(qVEB

kBT

)(

1 −x

WB

)

dx .



Base Recombination Current

Integration gives

Irec,B = Ip,E − Ip,C ,

=q A

τp,B

[

pB exp

(qVEB

kBT

) (

x−x2

2WB

)]WB

0

,

=q AWB n

2i

2 τp,B N+d,B

exp

(qVEB

kBT

)

,

=Qp,B

τp,B.



Collector Current

The collector hole current is the difference between the hole current injected into thebase and the base recombination current, or

Ip,C = Ip,E − Irec,B ,

=q ADp,B n

2i

WB N+d,B

exp

(qVEB

kBT

)

−q AWB n

2i

2 τp,B N+d,B

exp

(qVEB

kBT

)

.

The collector current for the emitter open and the collector-base junction reverse-biasedis the reverse saturation current given by

In,C = −q ADn,C n

2i

WC N−a,C

.

The total collector current is the sum of Ip,C and In,C given by

IC ' Ip,C + In,C = Ip,E − Irec,B + In,C ' Ip,E ,

for Ip,E > Irec,B > In,C . The collector current IC may be written as

IC ' Ip,E =q ADp,B n

2i

WB N+d,B

exp

(qVEB

kBT

)

.



Collector Current

or that

IC 'q AWB n

2i

2N+d,B

exp

(qVEB

kBT

)

︸︷︷︸

Qp,B

2Dp,B

W 2B

,

which permits writing IC as

IC ' Qp,B2Dp,B

W 2B

'Qp,B

τt,B,

where τt,B is the base transit time. This equation emphasizes that the collector currentis directly proportional to the charge stored in the base region.



Base Transit Time

The velocity of holes in the base region is given by vp,B(x). The hole flux in the baseFp,B is written as

Fp,B(x) = vp,B(x) pB(x) = vp,B(x) pB exp

(qVEB

kBT

)(

1 −x

WB

)

.

The flux is multiplied by A and q to give

Ip,B = q A vp,B(x) pB exp

(qVEB

kBT

) (

1 −x

WB

)

,

which is also given by the expression for the diffusion current

Ip,B = −q ADp,BdpB(x)

dx=q ADp,B p

B

WBexp

(qVEB

kBT

)

.



By equating these two expressions for Ip,B we find that the hole velocity can then beexpressed as

vp,B(x) =Dp,B

WB

(

1 − xWB

) =Dp,B

(WB − x).

The base transit time is given by

τt,B =

∫ WB

0

dx

vp,B(x)=

∫ WB

0

(WB − x)

Dp,Bdx =

W 2B

2Dp,B.

If WB is 1 µm and Dp,B is 1 cm2/s, then τt,B is 5 × 10−9 s.



Gummel Number

The Gummel number GN is defined as

GN ≡

∫ WB

0

N+d,B(x) .dx

It accounts for position-dependent dopant profiles in the base region. In the case of auniformly doped base region, then GN = N+

d,B WB . The collector current is rewrittenas

IC 'q ADp,B n

2i

WB N+d,B

exp

(qVEB

kBT

)

'q ADp,B n

2i

GNexp

(qVEB

kBT

)

.

In the absence of recombination in the base region, the collector current is inverselyproportional to the integrated base doping, and the smaller the Gummel number, thehigher the collector current for a given VEB. Typical values of GN in high-performancebipolar junction transistors range from 1012 to 1013 cm−3.



Evaluation of the Common-Base Current Gain αF

We derive an expression to evaluate the common-base current gain. For VEB > 3kBT/q,the −1 term in the diffusion current expressions may be neglected. By cancellation ofcommon terms qA, the emitter efficiency for the bipolar transistor becomes

γE ≡Ip,E

Ip,E + In,E + Iscr,E/B,

=1

1 +In,E

Ip,E+Iscr,E/B

Ip,E

,

=1

1 +Dn,E N

+d,B WB

Dp,B N−a,E WE

+N+

d,B WB Wd,EB exp(qVEB/2kBT )

2niDp,B τp,B exp(qVEB/kBT )

,

=1

1 +Dn,E N

+d,B WB

Dp,B N−a,E WE

+N+

d,B WB Wd,EB

2ni L2p,B

exp(

−qVEB2kBT

),

where Wd,EB = Wd,E +Wd,B/E .



Evaluation of the Common-Base Current Gain αF

The expression for γE shows that increasing the ratio of emitter doping to base doping,i.e. making N−

a,E N+d,B , decreases the electron current from the base into the emitter

so that γE approaches unity. The term due to space-charge recombination has a voltagedependence as exp(−qVEB/2kBT ) in the denominator. As VEB increases, the effect ofspace-charge recombination on γE will be reduced.

The base transport factor αT is given by

αT = 1 −Irec,B

Ip,E,

= 1 −q AWB n

2i

2 τp,B N+d,B

N+d,B WB

q ADp,B n2i

,

= 1 −W 2

B

2L2p,B

.

It is clear that the base transport factor approaches unity for WB Lp,B .



Bias-Dependence of Carrier Distributions in the Base Region

!

"

Minority-carrier hole concentration pB(x) inthe base region of a pnp transistor for

• an open emitter to give ICBO

• a shorted emitter-base junctionto give VEB = 0.




Minority-carrier hole concentration pB(x) inthe base region of a pnp transistor biased inthe forward-active region with increasing val-ues of VEB and a reverse-biased collector-basejunction.




Minority-carrier hole concentration pB(x) inthe base region of a pnp transistor biased inthe forward-active region showing the effectsof increasing the reverse bias of the collector-base junction on the width of the base region,while VEB is kept constant.




!

Minority-carrier hole concentration pB(x) inthe base region of a pnp transistor biased inthe forward-active region showing the effectsof changing VEB when VCB = 0.




!

Minority-carrier hole concentration pB(x) inthe base region of a pnp transistor biased inthe saturation region with both junctionsforward-biased.




Minority-carrier hole concentration pB(x) inthe base region of a pnp transistor biased inthe cut-off region with both the emitter-baseand collector-base junctions reverse-biased.



Common-Emitter Current Gain βF in a PNP Transistor

In a common-emitter configuration, the input current is the base current IB and theoutput current is the collector current IC and their ratio is of interest.




We can write that

IB = IE − IC = (Ip,E + In,E + Iscr,E/B) − (Ip,C + In,C) .

Neglecting the collector-base reverse-bias saturation current In,C which is very smallfor Si transistors, we can write that

IB = (Ip,E + In,E + Iscr,E/B) − (Ip,E − Irec,B) = In,E + Iscr,E/B + Irec,B .

By arraning the bipolar junction transistor in the common-emitter configuration, theoutput current IC is much larger than the input current IB . The collector current iswritten as

IC = Ip,C + In,C = αT Ip,E + In,C = αF IE + ICBO .




The emitter current is given by

IE = IB + IC .

We can then write thatIC = αF (IB + IC) + ICBO ,

or thatIC(1 − αF ) = αF IB + ICBO ,

and

IC =αF

(1 − αF )IB +

ICBO

(1 − αF ),

where ICBO IB for Si transistors. The ratio of IC to IB is known as the common-emitter current gain βF defined as

βF ≡ICIB

=αF

(1 − αF ).

The collector-emitter leakage current for IB = 0 is represented by the second term as

ICEO ≡ICBO

(1 − αF ).




In most bipolar transistors, αF approaches unity, which means that βF can be muchgreater than 1, and ICEO will be much larger than ICBO. For example, when αF =0.98, then βF = 49.0. The usefulness of the bipolar transistor in the common-emitterconfiguration is that a small base current can control a much larger collector current.

The variation of βF with VEB or the collector current indicates that at small valuesof IC , βF does not become relatively constant until the space-charge recombinationcurrent Iscr,E/B is much smaller than the diffusion current Ip,E . At larger values of IC ,βF decreases due to high-level-injection effects which give a diffusion current variationas exp(qVEB/2kBT ).



Common-Emitter Configuration and Bias with an NPN Transistor


7.6. BJT Static I(V ) Characteristics: Ebers-Moll Model

Basic Model

The Ebers-Moll model is the most commonly used static model for the bipolar junctiontransistor. It is the basis for the model used in SPICE. Space-charge generation andrecombination currents are not included in this model but these currents can be readilyadded. Carrier recombination in the base region is included.

We consider an NPN transistor in this analysis. The terminal currents IC , IE , and IBare expressed in terms of the terminal voltages VBE and VBC . We divide this problem intwo parts by applying each voltage at a time, solve for the resulting currents, and thenuse superposition to combine the two solutions into the total solution. The currentsobtained are labeled forward when VBE is applied and VBC = 0, and reverse when VBC

is applied and VBE = 0.



The forward current IF is given for the base-emitter junction as

IF = In,E + Ip,E = IBE·S

[

exp

(qVBE

kBT

)

− 1

]

,

where IBE·S is the base-emitter junction saturation current, and VBE is the base-to-emitter voltage.

The reverse current IR is given for the base-collector junction as

IR = In,C + Ip,C = IBC·S

[

exp

(qVBC

kBT

)

− 1

]

,

where IBC·S is the base-collector junction saturation current, and VBC is the base-to-collector voltage.



The interaction of the two junctions in a bipolar junction transistor results from thethin-base region which results in a fraction of the emitter current reaching the collectorand vice versa.

The collector current IC is expressed in terms of IF and IR as

IC = αF IF − IR ,

where αF IF is the part of the collector current due to the base-emitter junction and IR

the part due to the base-collector junction. The emitter current IE expressed in termsof IF and IR as

IE = IF − αRIR ,

where IF is the part of the emitter current due to the base-emitter junction and αRIRthat duet to the base-collector junction.

The base current is given by

IB = IE − IC = (1 − αF )IF + (1 − αR)IR .





Consider an NPN transistor where the diffusion lengths of minority carriers in theemitter, base, and collector regions are much larger than the physical widths of theemitter, base, and collector regions, respectively. Assume that the emitter area is AE

and that the collector area is AC .

We can write the saturation current for the base-emitter junction as

IBE·S ≡ In,E·S + Ip,E·S ,

where

In,E·S ≡q AE Dn,B n

2i

WB N−a,B

,

and

Ip,E·S ≡q AE Dp,E n

2i

WE N+d,E

.



We can write the saturation current for the base-collector junction as

IBC·S ≡ In,C·S + Ip,C·S ,

where

In,C·S ≡q AC Dn,B n

2i

WB N−a,B

,

and

Ip,C·S ≡q AC Dp,C n

2i

WC N+d,C

.

We note that, if AE = AC , then In,E·S = In,C·S .



Reciprocity Relationship

We can write the product αF IBE·S as

αF IBE·S = γE αT (In,E·S + Ip,E·S) ,

=In,E·S

(In,E·S + Ip,E·S)αT (In,E·S + Ip,E·S) = αT In,E·S ,

and

αRIBC·S = γC αT (In,C·S + Ip,C·S) ,

=In,C·S

(In,C·S + Ip,C·S)αT (In,C·S + Ip,C·S) = αT In,C·S .

When AE = AC , we can write that In,E·S = In,C·S , and consequently

αF IBE·S = αR IBC·S ≡ IS .

A more detailed proof shows that the reciprocity relationship is satisfied for a generalgeometery where AE 6= AC .



SPICE Ebers-Moll Model

We write αF IF and αRIR as

ICC ≡ αF IF ,

= αF IBE·S

[

exp

(qVBE

kBT

)

− 1

]

,

= IS

[

exp

(qVBE

kBT

)

− 1

]

,

and

IEC ≡ αR IR ,

= αR IBC·S

[

exp

(qVBC

kBT

)

− 1

]

,

= IS

[

exp

(qVBC

kBT

)

− 1

]

.

The current IF becomes ICC/αF and the current IR becomes IEC/αR.



In the transport version of the Ebers-Moll model, the terminal currents are

IC = ICC −IEC

αR,

IE =ICC

αF− IEC ,

and

IB =ICC

αF− IEC − ICC +

IEC

αR,

=

(1

αF− 1

)

ICC +

(1

αR− 1

)

IEC .

We define the common-emitter current gains βF and βR as

βF ≡αF

1 − αF,

and

βR ≡αR

1 − αR.





The terminal currents can then be written as

IC = ICC −IEC

αR,

= ICC − IEC + IEC −IEC

αR,

= (ICC − IEC) +(αR − 1)

αRIEC ,

= (ICC − IEC) −IEC

βR,

and

IE =ICC

αF− IEC ,

= ICC − IEC − ICC +ICC

αF,

= (ICC − IEC) −(αF − 1)

αFICC ,

= (ICC − IEC) +ICC

βF,



The two reference currents are replaced by a single source

ICT ≡ ICC − IEC = IS

[

exp

(qVBE

kBT

)

−

(qVBC

kBT

)]

.

The two diode currents become

ICC

βF=ISβF

[

exp

(qVBE

kBT

)

− 1

]

,

andIEC

βR=ISβR

[

exp

(qVBC

kBT

)

− 1

]

,





Limitations of the Ebers-Moll Model

1. Base-Width Modulation (Early effect)2. Punchthrough3. Base-Collector Junction Breakdown4. Space-Charge Layer Recombination5. High-Level Injection6. Emitter Crowding7. Series Resistance8. Nonuniform Doping Profiles


7.7. BJT Dynamic I(V ) Characteristics: Charge-Control Model

Under time-dependent conditions, the relationship between the instantaneous values ofterminal currents and terminal voltages is described by the differential equations thatrelate the currents and the charges stored in the emitter, base, and collectore quasi-neutral regions, and in the base-to-emitter and base-to-collector depletion regions.In a pn junction, the charges stored in the diode are Qn,P and Qp,N in the quasi-neutral regions and QDEP in the depletion region. The relationship between the totalinstantaneous value of the diode current and the diode charges is written as

iD(t) =dQp,N (t)

dt+

Qp,N (t)

τp,N+dQn,P (t)

dt+

Qn,P (t)

τn,P+dQDEP (t)

dt.

In an NPN bipolar junction transistor, the chages stored in the emitter, base, and collec-tor regions are Qp,E , Qn,B , and Qp,C , respectively. The charges stored in the depletionregion of the base-to-emitter and base-to-collector depletion regions are QDEP,B/E andQDEP,B/C , respectively.


7.7. BJT Dynamic I(V ) Characteristics: Charge-Control Model

The complete charge-control model equations for an npn transistor are as follows

iC(t) =QF (t)

τF−

(1

τR+

1

τBR

)

QR(t) −dQR(t)

dt−dQV C(t)

dt−dQV S(t)

dt,

iB(t) =QF (t)

τBF+dQF (t)

dt+QR(t)

τBR+dQV C(t)

dt+dQV E(t)

dt,

iE(t) = −QR

τR+

(1

τF+

1

τBF

)

QF (t) +dQF (t)

dt+dQV E(t)

dt,

where

QF (t) ' QF0

[

exp

(qvBE(t)

kBT

)

− 1

]

,

and

QR(t) ' QR0

[

exp

(qvBC(t)

kBT

)

− 1

]

,

See Hodges and Jackson, p.220.


7.8. BJT Small-Signal AC Model

We assume that our DC or quiescent point Q is defined by VBE and VCE . We expandiB and iC as

iB(vBE , vCE) = iB(VBE, VCE) + (vBE − VBE)∂iB∂vBE

∣∣∣∣Q

+ (vCE − VCE)∂iB∂vCE

∣∣∣∣Q

+ higher order terms ,

and

iC(vBE , vCE) = iC(VBE , VCE) + (vBE − VBE)∂iC∂vBE

∣∣∣∣Q

+ (vCE − VCE)∂iC∂vCE

∣∣∣∣Q

+ higher order terms .

We recognize that

iB(VBE , VCE) ≡ IB ,

iC(VBE , VCE) ≡ IC ,

vBE − VBE ≡ vbe ,

vCE − VCE ≡ vce ,

iB − IB ≡ ib ,

iC − IC ≡ ic .



We assume that ic and ib are small enough that we can ignore the higher-order terms,and rewrite the above equations as

ib =∂iB∂vBE

∣∣∣∣Q

vbe +∂iB∂vCE

∣∣∣∣Q

vce = gπ vbe + gr vce ,

and

ic =∂iC∂vBE

∣∣∣∣Q

vbe +∂iC∂vCE

∣∣∣∣Q

vce = gm vbe + go vce ,

where

gπ ≡∂iB∂vBE

∣∣∣∣Q

input conductance ,

gr ≡∂iB∂vCE

∣∣∣∣Q

reverse transconductance ,

gm ≡∂iC∂vBE

∣∣∣∣Q

forward transconductance ,

go ≡∂iC∂vCE

∣∣∣∣Q

output transconductance .



For an npn transistor biased in the forward-active region, we have

IC ' IS exp

(qVBE

kBT

)

.

The forward transconductance is then given by

gm ≡∂iC∂vBE

∣∣∣∣Q

=q

kBTIS exp

(qVBE

kBT

)

=IC

(kBT/q).

The input conductance is given by

gπ ≡∂iB∂vBE

∣∣∣∣Q

=∂(iC/βF )

∂vBE

∣∣∣∣Q

=gm

βF≡

1

rπ.

The output conductance is given by

go ≡∂iC∂vCE

∣∣∣∣Q

=IC|VA|

=kBT gm

q|VA|≡

1

ro.



The base-emitter capacitance Cπ consists of the B/E junction depletion and diffusioncapacitances because the junction is forward-baised. It is given by

Cπ = AE

√√√√

q εSi N+d,E N

−a,B

2Vbi,B/E (N+d,E +N−

a,B)

√

Vbi,B/E

Vbi,B/E − VBE+ τt,B

qISkBT

exp

(qVBE

kBT

)

.

The base-collector capacitance Cµ consists of only the depletion capacitance becausethe junction is reverse-baised. It is given by

Cµ = AC

√√√√

q εSiN−a,B N

+d,C

2Vbi,B/C (N−a,B +N+

d,C)

√

Vbi,B/C

Vbi,B/C − VBC.

The variation of vCE changes the collector depletion-layer width which results in achange in the base width. The variation in base width results in a change in theminority-carrier charge stored in the base and a change in the base current. A smallchange in vCE causes a change in the base current iB which is represented by a resistancerµ from the collector to the base of the equivalent circuit. This resistance is given by

rµ =∂vCE

∂iB=∂iC∂iB

∂vCE

∂iC= βF ro .



Basic Small-Signal AC Equivalent Circuit for an NPN Bipolar Transistor



Complete Hybrid-π Small-Signal Equivalent Circuit for an NPN BipolarTransistor



Cutoff Frequency fT

The cutoff frequency fT is the frequency at which the current gain in the common-emitter configuration is unity while the output is shorted for an ac signal. The shortedoutput eliminates the output resistance ro and connects Cµ in parallel with Cπ. In thissimplified model, rb, re and rc are ignored, but a more complete model would generallyinclude rb. Because rµ is a large resistance which is now in parallel with the smallresistance rπ, rµ may be neglected. For these conditions, the equivalent circuit is



Cutoff Frequency fT

The input current is given by

iin = [1/rπ + jω(Cπ + Cµ)] vbe ,

and the output current isiout = −gm vbe .

The magnitude of the input current is

|iin| =√

1/r2π + ω2(Cπ + Cµ)2 vbe .

The ratio of the magnitude of the output to the input current is

∣∣∣∣

iout

iin

∣∣∣∣=

gm√

1/r2π + ω2(Cπ + Cµ)2.



Cutoff Frequency fT

At high frequencies, where ω(Cπ + Cµ) > 1/rπ, the ratio |iout|/|iin| = 1 at fT , whichis given by

fT =gm

2π (Cπ + Cµ).

Because the diffusion capacitance can be larger than the depletion capacitances, (Cπ +Cµ) → Cπ = τt,B gm. The cutoff frequency is then given by

fT '1

2π τt,B.

With τt,B = W 2B/2Dn,B, fT may be written as

fT 'Dn,B

πW 2B

.

This expression emphasizes that fT depends on the minority-carrier diffusivity in thebase region (favoring NPN over PNP), and that a thin WB is necessary for high fT .



Maximum Frequency fmax

The high-frequency behavior of transistors has also been specified by the maximumavailable power gain at high frequencies. For maximum power gain, the load resistanceRL must be matched to the output resistance ro of the transistor. At high frequencies,the power gain decreases with frequency. The power gain falls to unity at fmax whichis obtained as

fmax =

[gm

16π2 rb(Cπ + Cµ)2

]1/2

=

[fT

8π rb(Cπ + Cµ)

]1/2

.

Both fT and fmax are often used as figures of merit for comparison of high-frequencytransistors.


7.9. SPICE Model for Bipolar Transistors

Element and Model Lines

The element and model lines in SPICE for the bipolar transistor have been summarizedby Banzhaf.The general form of the element line for the bipolar transistor is

QXXXXXX NC NB NE <NS> MODNAME <AREA> <OFF> <IC=VBE,VCE>

where QXXXXXX is the name of the bipolar transistor, NC the collector node, NB the basenode, NE the emitter node, and MODNAME the model name which is used in an associated.MODEL control line. These items are required in the bipolar transistor element line.The optional parameters are the quantities in the < · · · > and an element line may becontinued by entering a + sign at the start of the next line.



The meanings of the optional parameters are:

NS The node of the substrate which defaults to 0.

AREA The area parameter specifies how many of thebipolar transistor model MODNAME are connectedin parallel to make one QXXXXXX.

OFF The initial condition of QXXXXXX for dc analysis.

IC=VBE,VCE SPICE will use VBE and VCE as the initial condi-tions for the bipolar transistor base-emitter andcollector-emitter voltages rather than the quies-cent operating point for a transient analysis.

The model form or the model line for the bipolar transistor is

.MODEL MODNAME NPN<(PAR1=PVAL1 PAR2=PVAL2 . . .)>

.MODEL MODNAME PNP<(PAR1=PVAL1 PAR2=PVAL2 . . .)>where MODNAME is the model name given to a bipolar transistor in the element line, andNPN or PNP denote that the device is an npn or pnp transistor. PAR is the parametername of one of the optional parameters listed in Table 4.2 for PSpice. PVAL is the valueof the designated parameter. Care must be taken to assign the correct units which arealso designated in the tables.



SPICE DC Model Parameters

No. Text SPICE Parameter Name Default Units

Symbol Keyword Value

1 Is IS Saturation current 1.0E-14 A

2 βF BF Ideal maximum forward current gain 100 –

3 βR BR Ideal maximum reverse current gain 1 –

4 nF NF Forward current ideality factor 1.0 –

(1.0<nF<2.0)

5 nR NR Reverse current ideality factor 1.0 –

(1.0<nR<2.0)

6 Is,src ISE Emitter-base space-charge recombination 1.0E-13 A

saturation current

7 nE NE Emitter-base ideality factor (1.0<nE<2.0) 1.0 –

8 Is,src ISC Collector-base space-charge recombination 1.0E-13 A

saturation current

9 nC NC Collector-base ideality factor (1.0<nC<2.0) 1.0 –



SPICE DC Model Parameters

No. Text SPICE Parameter Name Default Units

Symbol Keyword Value

10 VA VAF Forward Early voltage ∞ V

11 RB RB Zero-bias base resistance 0 Ω

12 RE RE Emitter resistance 0 Ω

13 RC RC Collector resistance 0 Ω

14 CjE(0) CJE Zero-bias emitter-base depletion capacitance 0 F

15 mE MJE Emitter-base grading coefficient 0.33 –

16 Vbi,EB VJE Emitter-base built-in voltage 0.75 V

17 CjC(0) CJC Zero-bias collector-base depletion capacitance 0 F

18 mC MJC Collector-base grading coefficient 0.33 –

19 Vbi,CB VJC Collector-base built-in voltage 0.75 V

20 τF TF Ideal forward transit time 0 s


7.10. Temperature Effects


7.11. Frequency Effects


7.12. Switching Analysis


7.13. BJT Breakdown


7.14. Summary


Documents

07 Bipolar Junction Transistors