32 Ebers-Moll Model - nanoHUB

Lundstrom ECE 305 S15

ECE-305: Spring 2015

BJTs: Ebers-Moll Model

Professor Mark Lundstrom

Electrical and Computer Engineering Purdue University, West Lafayette, IN USA

[email protected]

4/27/15

Pierret, Semiconductor Device Fundamentals (SDF) pp. 403-407

1

last lecture

Lundstrom ECE 305 S15 2

1)  Emitter injection efficiency 2)  Base transport factor 3)  Early effect 4)  Speed (base transit time) 5)  Effects of saturation 6)  Gummel plots 7)  Transconductance 8)  HBTs 9)  Emitter crowding

2D effects

3

n+ emitter

p base

n collector

n+

p base n-collector

n+

n+

double

diffused

BJT

emitter current crowding

4

p base

n-collector

n+

n+

IBIB



5

p base

n-collector

n+

n+

IBIB

Vbase+

−



6

n-collector

n+

n+

IBIB


emitter and collector areas

7

p base n-collector

n+

n+

IBIB

AE

AC

AC >> AE


Q1) Which is the base transport factor?

8

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEp

ICn

IB

a) b) c) d) e)

IEn IEn + IEp( )

ICn IEp

ICn IEn IEp

ICn IEn + IEp( )

Q2) Which is beta_dc?

9

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEp

ICn

IB

a) b) c) d) e)

IEn IEn + IEp( )

ICn IEp

ICn IEn IEp

ICn IEn + IEp( )

Q3) Which is alpha_dc?

10

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEp

ICn

IB

a) b) c) d) e)

IEn IEn + IEp( )

ICn IEp

ICn IEn IEp

ICn IEn + IEp( )

Q4) Which is the base current?

11

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEp

ICn

IB

a) b) c) d) e)

IEn IEn + IEp( )

ICn IEp

ICn IEn IEp

ICn IEn + IEp( )

Q5) Which is the emitter injection efficiency?

12

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEp

ICn

IB

a) b) c) d) e)

IEn IEn + IEp( )

ICn IEp

ICn IEn IEp

ICn IEn + IEp( )

forward active region

13

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEpIE = IEn + IEp

ICn ≈ IEn

IC ≈ IEn

IB = IEp

emitter current: forward active region

14

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEpIE = IEn + IEp

IEn

IC ≈ IEn

IB = IEp

IEn = qAEni2

NAB

⎛⎝⎜

⎞⎠⎟Dn

WB

eqVBE /kBT

IEp = qAEni2

NDE

⎛⎝⎜

⎞⎠⎟Dp

WE

eqVBE /kBT

IE = IEn + IEp

IE = IF0 eqVBE kBT −1( )

IF0 = qAEni2

NAB

⎛⎝⎜

⎞⎠⎟Dn

WB

+ qAEni2

NDE

⎛⎝⎜

⎞⎠⎟Dp

WE

collector current: forward active region

15

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEpIE = IEn + IEp

αT IEn

IC ≈ IEn

IB = IEp

IEn = qAEni2

NAB

⎛⎝⎜

⎞⎠⎟Dn

WB

eqVBE /kBT

IEp = qAEni2

NDE

⎛⎝⎜

⎞⎠⎟Dp

WE

eqVBE /kBT

IC =αT IEn

IC =αF IF0 eqVBE kBT −1( )

IC =αTγ F IE

αF =αTγ F


base current: forward active region

16

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEpIE = IEn + IEp

IEn

IC ≈ IEn

IB = IEp

IB = IE − IC


αF =αTγ F


IB = IE − IC

IB = 1−αF( ) IF0 eqVBE kBT −1( )Lundstrom ECE 305 S15

summary: forward active region

17

n+ emitter

p base

n collector

n+

FB RB IE ICIEn

IEpIE = IEn + IEp

IEn

IC ≈ IEn

IB = IEp



IB = 1−αF( ) IF0 eqVBE kBT −1( )IF0 = qAE

ni2

NAB

⎛⎝⎜

⎞⎠⎟Dn

WB

+ qAEni2

NDE

⎛⎝⎜

⎞⎠⎟Dp

WE

αF =αTγ F

Ebers-Moll model

18 18


Question: How do we describe the BJT in any region of operation?

emitter-base junction (the forward diode)

19 19

n+ emitter

p base

n collector

n+ IE IC

IB

IEn ICn

IEp


ICp

IEn = −qA Dn

WB

ni2

NAB

eqVBE kBT −1( )

IEp = −qADp

WE

ni2

NDE

eqVBE kBT −1( )

IE VBE( ) = −IEn VBE( )− IEp VBE( )

IE VBE( ) = IF0 eqVBE kBT −1( )

Base-collector junction (the reverse diode)

20 20

n+ emitter

p base

n collector

n+ IE IC

IB

IEn ICn

IEp


ICp

ICn VBC( ) = qA Dn

WB

ni2

NAB

eqVBC kBT −1( )

ICp VBC( ) = qA Dp

WC

ni2

NDC

eqVBC kBT −1( )

IC VBC( ) = − ICn VBC( ) + ICp VBC( )⎡⎣ ⎤⎦

IC VBC( ) = −IR0 eqVBC kBT −1( )

Both junctions….

21 21

n+ emitter

p base

n collector

n+ IE IC

IB

IEn ICn

IEp


ICp

IC VBC( ) = −IR0 eqVBC kBT −1( )

IE VBE( ) = IF0 eqVBE kBT −1( )

But…. The two junctions are coupled!

Ebers-Moll model

22 22

n+ emitter

p base

n collector

n+ IE IC

IB

IEn ICn

IEp


ICp

IC VBE ,VBC( ) =αF IF0 eqVBE kBT −1( )− IR0 eqVBC kBT −1( )

IE VBE ,VBC( ) = IF0 eqVBE kBT −1( )−α RIR0 eqVBC kBT −1( )

Ebers-Moll model

23 23




IB VBE ,VBC( ) = IE VBE ,VBC( )− IC VBE ,VBC( )

See Pierret SDF, Chapter 11, sec. 11.1.4

Ebers-Moll model

24 Lundstrom ECE 305 S15



αF IF0 =α RIR0

IF0 = qAEni2

NAB

⎛⎝⎜

⎞⎠⎟Dn

WB

+ qAEni2

NDE

⎛⎝⎜

⎞⎠⎟Dp

WE

αF =αTγ F

α R =αTγ R IR0 = qAEni2

NAB

⎛⎝⎜

⎞⎠⎟Dn

WB

+ qAEni2

NDC

⎛⎝⎜

⎞⎠⎟Dp

WC

Ebers-Moll model

25 25






αF IF0 =α RIR0

Check active region

Lundstrom ECE 305 S15 26

IC VBE ,VBC( ) =αF IF0 eqVBE kBT −1( )− IR0 eqVBC kBT −1( )→αF IF0e

qVBE kBT + IR0

IE VBE ,VBC( ) = IF0 eqVBE kBT −1( )−α RIR0 eqVBC kBT −1( )→ IF0e

qVBE kBT +α RIR0

IB VBE ,VBC( ) = IE VBE ,VBC( )− IC VBE ,VBC( )→ 1−αF( ) IF0eqVBE kBT − 1−α R( ) IR0

IC ≈αF IF0eqVBE kBT

IB ≈ 1−αF( ) IF0eqVBE kBT =1−αF( )αF

IC = 1βdc

IC✔

What is ID?

27 IE = ID

IB

VBC

VBE

IE = ID

VCE =VD VD

ID

What is ID?

28 IE = ID

IBIC =αF IF0 e

qVBE kBT −1( )− IR0 eqVBC kBT −1( )

IE = IF0 eqVBE kBT −1( )−α RIR0 e

qVBC kBT −1( )

VBC = 0

VBC

VBE

IE = ID

VCE =VD


ID = IF0 eqVD kBT −1( )

IC

What is IC?

29 IE

IB = 0

IC =αF IF0 eqVBE kBT −1( )− IR0 eqVBC kBT −1( )

IE = IF0 eqVBE kBT −1( )−α RIR0 e

qVBC kBT −1( )VBC

VBE

IC

VCE

Ebers-Moll model

30 30

Lundstrom 12.5.14




αF IF0 =α RIR0


Ebers-Moll model (ii)

31 31 Lundstrom 12.5.14

IR0 = qADnB

WB

ni2

NAB

+DpC

WC

ni2

NDC

⎛⎝⎜

⎞⎠⎟

IF0 = qADnB

WB

ni2

NAB

+DpE

WE

ni2

NDE

⎛⎝⎜

⎞⎠⎟

αF = γ FαT

αF IF0 =α RIR0 →α R =αFIF0IR0

α R = γ RαT

32 Ebers-Moll Model - nanoHUB

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