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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
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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
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2D effects
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n+ emitter
p base
n collector
n+
p base n-collector
n+
n+
double
diffused
BJT
emitter current crowding
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p base
n-collector
n+
n+
IBIB
Lundstrom ECE 305 S15
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emitter current crowding
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p base
n-collector
n+
n+
IBIB
Vbase+
−
Lundstrom ECE 305 S15
emitter current crowding
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n-collector
n+
n+
IBIB
Lundstrom ECE 305 S15
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emitter and collector areas
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p base n-collector
n+
n+
IBIB
AE
AC
AC >> AE
Lundstrom ECE 305 S15
Q1) Which is the base transport factor?
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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( )
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Q2) Which is beta_dc?
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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?
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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( )
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Q4) Which is the base current?
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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?
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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( )
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forward active region
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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
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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
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collector current: forward active region
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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
Lundstrom ECE 305 S15
base current: forward active region
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n+ emitter
p base
n collector
n+
FB RB IE ICIEn
IEpIE = IEn + IEp
IEn
IC ≈ IEn
IB = IEp
IB = IE − IC
IC =αF IF0 eqVBE kBT −1( )
αF =αTγ F
IE = IF0 eqVBE kBT −1( )
IB = IE − IC
IB = 1−αF( ) IF0 eqVBE kBT −1( )Lundstrom ECE 305 S15
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summary: forward active region
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n+ emitter
p base
n collector
n+
FB RB IE ICIEn
IEpIE = IEn + IEp
IEn
IC ≈ IEn
IB = IEp
IC =αF IF0 eqVBE kBT −1( )
IE = IF0 eqVBE kBT −1( )
IB = 1−αF( ) IF0 eqVBE kBT −1( )IF0 = qAE
ni2
NAB
⎛⎝⎜
⎞⎠⎟Dn
WB
+ qAEni2
NDE
⎛⎝⎜
⎞⎠⎟Dp
WE
αF =αTγ F
Ebers-Moll model
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Lundstrom ECE 305 S15
Question: How do we describe the BJT in any region of operation?
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emitter-base junction (the forward diode)
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n+ emitter
p base
n collector
n+ IE IC
IB
IEn ICn
IEp
Lundstrom ECE 305 S15
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)
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n+ emitter
p base
n collector
n+ IE IC
IB
IEn ICn
IEp
Lundstrom ECE 305 S15
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( )
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Both junctions….
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n+ emitter
p base
n collector
n+ IE IC
IB
IEn ICn
IEp
Lundstrom ECE 305 S15
ICp
IC VBC( ) = −IR0 eqVBC kBT −1( )
IE VBE( ) = IF0 eqVBE kBT −1( )
But…. The two junctions are coupled!
Ebers-Moll model
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n+ emitter
p base
n collector
n+ IE IC
IB
IEn ICn
IEp
Lundstrom ECE 305 S15
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( )
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Ebers-Moll model
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Lundstrom ECE 305 S15
IC VBE ,VBC( ) =αF IF0 eqVBE kBT −1( )− IR0 eqVBC kBT −1( )
IE VBE ,VBC( ) = IF0 eqVBE kBT −1( )−α RIR0 eqVBC kBT −1( )
IB VBE ,VBC( ) = IE VBE ,VBC( )− IC VBE ,VBC( )
See Pierret SDF, Chapter 11, sec. 11.1.4
Ebers-Moll model
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IC VBE ,VBC( ) =αF IF0 eqVBE kBT −1( )− IR0 eqVBC kBT −1( )
IE VBE ,VBC( ) = IF0 eqVBE kBT −1( )−α RIR0 eqVBC kBT −1( )
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
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Ebers-Moll model
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Lundstrom ECE 305 S15
IC VBE ,VBC( ) =αF IF0 eqVBE kBT −1( )− IR0 eqVBC kBT −1( )
IE VBE ,VBC( ) = IF0 eqVBE kBT −1( )−α RIR0 eqVBC kBT −1( )
IB VBE ,VBC( ) = IE VBE ,VBC( )− IC VBE ,VBC( )
See Pierret SDF, Chapter 11, sec. 11.1.4
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✔
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What is ID?
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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
IE = IF0 eqVBE kBT −1( )
ID = IF0 eqVD kBT −1( )
IC
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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
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Lundstrom 12.5.14
IC VBE ,VBC( ) =αF IF0 eqVBE kBT −1( )− IR0 eqVBC kBT −1( )
IE VBE ,VBC( ) = IF0 eqVBE kBT −1( )−α RIR0 eqVBC kBT −1( )
IB VBE ,VBC( ) = IE VBE ,VBC( )− IC VBE ,VBC( )
αF IF0 =α RIR0
See Pierret SDF, Chapter 11, sec. 11.1.4
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Ebers-Moll model (ii)
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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