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Bipolar Junction Transistor
Session 11: Solid State Physics
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Outline
� A� B
� C
� D
� E
� F� G
� H
� I
� J
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Introduction
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In recent decades, the higher layout density and low-power advantage
of CMOS technology has eroded the BJT’s dominance in integrated-
circuit products.
(higher circuit density � better system performance)
BJTs are still preferred in some integrated circuit applications because
of their high speed and superior intrinsic gain.
� faster circuit speed
� larger power dissipation
� limits device density (~104 transistors/chip)
Si (npn, pnp) � SiGe HBT � GaAs (InSb) HBT
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BJT Types and Definitions
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Emitter
��
�np+ p-
Collector
Base
The BJT is a 3-terminal device, with two types: PNP and NPN
Emitter
��
�pn+ n-
Collector
Base
Asymmetric!
��� ��
pnp npn
�� ���
��� �
����� �� � ��
�� ���
�� ��
�����
�
![Page 5: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/5.jpg)
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Review: Current Flow in a
Reverse-Biased pnJunction
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� In a reverse-biased pn junction, there is negligible diffusion of
majority carriers across the junction. The reverse saturation
current is due to drift of minority carriers across the junction and
depends on the rate of minority-carrier generation close to the
junction (within ~one diffusion length of the depletion region).
⇒ We can increase this reverse current by increasing the rate of
minority-carrier generation, e.g. by
� optical excitation of carriers (e.g. photodiode)
� electrical injection of minority carriers into the vicinity of the
junction…
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PNP BJT Operation (Qualitative)
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A forward-biased “emitter” pn junction is used to inject minority carriers
into the vicinity of a reverse-biased “collector” pn junction.
� The collector current is controlled via the base-emitter junction
Emitter
�np+ p-Collector
��� ��
�� ���
��� �
����� ��hole
Base
“F” “R”
��
�
��
� � 0 � � 0
��� ������ ���
� ≡ ���To achieve high current gain:
• The injected minority carriers should not recombine within the quasi-neutral
base region
• The emitter junction current is comprised almost entirely of carriers injected
into the base (rather than carriers injected into the emitter)
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Base Current Components
(Active Mode of Operation)
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The base current consists of majority carriers supplied for
1. Recombination of injected minority carriers in the base
2. Injection of carriers into the emitter
3. Reverse saturation current in collector junction (Reduces |�|)4. Recombination in the base-emitter depletion region
Emitter
�np+ p-Collector
��� ��
�� ���
��� �
����� ��hole
Base
“F” “R”
��
�
��
� � 0 � � 0
��� ������ ���
� ≡ ���
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BJT Circuit Configurations
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���
��� � 010��20��30��
Output Characteristics for Common-Emitter Configuration
CE: Common EmitterCB: Common Base CC: Common Collector
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BJT Modes of Operation
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p n p n p n
21E C
B
21E C
B
Qn
E
C
B
B
E
Qp
VBE
VBC
1 : F
2 : R
1 : F
2 : F
1 : R
2 : F1 : R
2 : R
SaturationActive
ReversedCut-Off
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BJT Electrostatics
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Under normal operating conditions, the BJT may be viewed electrostatically as
two independent pn junctions
������� �
Emitter
np+ p-
CollectorBase
pnp
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BJT Electrostatics
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Under normal operating conditions, the BJT may be viewed electrostatically as
two independent pn junctions
������� �
Emitter
np+ p-
CollectorBase
pnp
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BJT Electrostatics
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������� �
Emitter
np+ p-
CollectorBasepnp
� �
!�!�
"
"
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BJT Electrostatics
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#
Emitter
np+ p-
CollectorBasepnp
� �""
$
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BJT Performance Parameters (PNP)
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Emitter
�np+ p-Collector
�hole
Base
“F” “R”
��
���
� � 0 � � 0��� ������ ��� % ≡ ������ � ��
!�!�
Emitter Efficiency:
&' ≡ ������Base Transport Factor:
&() ≡ %&'Common-Base d.c. Current Gain:
1 2345
5 ↘ relative to 1 � 2
1 ↘ relative to 2
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Collector Current (PNP)
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Emitter
�np+ p-Collector
�hole
Base
“F” “R”
��
���
� � 0 � � 0��� ������ ���
!�!�
The collector current is comprised of:2 Holes injected from emitter, which do
not recombine in the base 3 Reverse saturation current of collector
junction
where ��. is the collector current
which flows when �� / 0
1 2345
�� / &()�� � ��.
�� / &() �� � � � ��.�� / &()1 � &() � � ��.1 � &()/ �� � ���.
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Notation (PNP BJT) , Assumptions
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0
Emitter
np+ p-
CollectorBase
� �"0" 0′0′0′′0′′
Assumptions:
1) 1-D structure
2) " ≪ 34 and is constant
3) Like diode gen-rec in depletion region is negligible
4) Uniform doping + step junction
5) Low-level injection everywhere!
Notations:
5 � / 5�67� / 6�3 � / 3�8 � / 8�9�:� / 9�:
5� / 56; / 63� / 38� / 8< := / <:
5 � / 5�67� / 6�3 � / 3�8 � / 8�9�:> / 9�:
≫ �
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“Game Plan” for I-V Derivation
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0
Emitter
np+ p-
CollectorBase
� �"0" 0′0′0′′0′′
minority-carrier diffusion equation (different set of B.C. for each region)
currents
5�6� 9�: 56 <: 5�6� 9�:
minority-carrier diffusion currents at depletion region edges
0′0 "0′′ 0′′ 0′@� A0BBC
@��A0C@�
0′′@� A0BC
@��A"C@�
�� / �!�5 D∆<D0 FGH: � !�5� D∆9�D0′′ FGBBH:BB�� A0BBC���A0C
�� / �!�5 D∆<D0 FGHI � !�5� D∆9�D0′′ FGBH:B�� A0BC���A"C
9�A0′′C<A0C
9�A0′C
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Minority Carriers
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0
Emitter
np+ p-
CollectorBase
� �"0" 0′0′0′′0′′
Carriers:
5�6� 9�: 56 <: 5�6� 9�:
Steady-State solution is: DJ∆9�D0J / ∆9�5 8 3 / 5 8 ∆9� 0BB / �KLGBB M�⁄ � �JLOGBB M�⁄
∆< 0 / �KLG M=⁄ � �JLOG M=⁄∆9� 0BB / �KLGB M>⁄ � �JLOGB M>⁄
∆< 0 / <:ALP��= Q'⁄ � 1C∆< " / <:ALP�>= Q'⁄ � 1C∆9� 0′′ / 9�:ALP��= Q'⁄ � 1C
∆9� " / 9�:ALP�>= Q'⁄ � 1C
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Minority Carriers –Active
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0
Emitter
np+ p-
CollectorBase
� �"0" 0′0′0′′0′′
5�6� 9�: 56 <: 5�6� 9�:
0′0 "0′′ 0′′ 0′9�A0BBC
< 0
0′′<: < " 9�A0BC9�:
9�:
Active: forward reverse
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Minority Carriers –Saturation
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0
Emitter
np+ p-
CollectorBase
� �"0" 0′0′0′′0′′
5�6� 9�: 56 <: 5�6� 9�:
0′0 "0′′ 0′′ 0′9�A0BBC
< 0
0′′<:
< " 9�A0BC
9�:9�:
Saturation: forward forward
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Minority Carriers –Cut-off
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0
Emitter
np+ p-
CollectorBase
� �"0" 0′0′0′′0′′
5�6� 9�: 56 <: 5�6� 9�:
0′0 "0′′ 0′′ 0′9�A0BBC < 0
0′′<:
< " 9�A0BC9�:9�:
Cut-off: reverse reverse
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Ideal BJT!
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0
Emitter
np+
CollectorBase� " 0"0′′0′′�np+ p-
�hole
Base
“F” “R”
���
����� ������ ���
1 2345
" ≪ 3 ( 1 / 0)∆< 0 / �KLG M=⁄ � �JLOG M=⁄ ≅ �KB � �JB0
0 "
< 0
<: < "∆< 0 / <:ALP��= Q'⁄ � 1C∆< " / <:ALP�>= Q'⁄ � 1C
∆< 0 / ∆< 0 � ∆< 0 � ∆< "" 0Swhere
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Current Equations - IE
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0
Emitter
np+
CollectorBase� " 0"0′′0′′
0 "
< 0
<: < "0′′ 0′′9�A0BBC9�:
���A0′′C / �!�5� D∆9�D0′′ FGBBH:BB/ �!�5�3� 9�: LP��= Q'⁄ � 1���A0C / �!�5 D∆TD0 FGH:/ !�5" ∆< 0 � ∆< "
/ !�5" <: LP��= Q'⁄ � 1 � ALP�>= Q'⁄ � 1C
/ !� 5<:" �5�9�:3� LP��= Q'⁄ � 1 � !�5<:" ALP�>= Q'⁄ � 1C�� / ��� 0 � �� 0BB /
Dominant term
���A0′′C→���A0C→
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Current Equations – IC , IB
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p-
CollectorBase �" 0" 0′0′
"
0
< "
��� 0 / ��� "→
0′9�A0BC9�:
0′
���A"C →���A0′C
No recombination in Base
�� A0BC / �!�5�9�:3� ALP�>= Q'⁄ � 1C�� / ��� " � �� 0B
�!� 5�9�:3� � 5<:3 ALP�>= Q'⁄ � 1C/ !�5<:" LP��= Q'⁄ � 1
�np+ p-
�hole
Base
“F” “R”
���
����� ������ ���
1 2345
� / �� � �� / !�59�:3� LP��= Q'⁄ � 1 � !�5�9�:3� ALP�>= Q'⁄ � 1CV
W usually >
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Considering Recombination in Base
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0 "
< 0
<: < "���A0C→ ���A0C→
∆< 0 / �KLG M=⁄ � �JLOG M=⁄ ∆< 0 / <:ALP��= Q'⁄ � 1C∆< " / <:ALP�>= Q'⁄ � 1CS
∆< 0 / <:ALP��=Q' � 1C sinhIOGM=sinh IM= � <:ALP�>=Q' � 1C sinh GM=sinh IM=�� / !� 5�9�:3� � 5<:3 coth IM= LP��=Q' � 1 � !�5<:3 ALP�>=Q' � 1C 1sinh IM=�� / !�5<:3 ALP��=Q' � 1C 1sinh IM= � !� 5<:3 coth IM= � 5�9�:3� ALP�>=Q' � 1C� / �� � ��
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Quasi-Ideal BJT!
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0 "
< 0
<: < "→ ���A0C→���A0C
→ 4
�J / ∆�8 / !�8 _ ∆< 0 D0I:
/ !�28" ∆< 0 � ∆< "/ !�"<:28 ALP��=Q' � 1C � ALP�>=Q' � 1C
Valid for all regions of operation!
Hence:
�� / ��`(abc�� / ��`(abc � �J� / �`(abc � �J�defgh / !�59�:3� LP��=Q' � 1 � !�5�9�:3� ALP�>=Q' � 1C
� / !� 59�:3� �"<:28 LP��=Q' � 1 � !� 5�9�:3� �"<:28 ALP�>=Q' � 1C~4 j 10V ~7.5 j 10V 4 5
![Page 27: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/27.jpg)
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Quasi-Ideal BJT!
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Hence:
�� ≅ ��`(abc�� ≅ ��`(abc� ≅ �`(abc � �J
� / !� 59�:3� �"<:28 LP��=Q' � 1 � !� 5�9�:3� �"<:28 ALP�>=Q' � 1C~4 j 10V ~7.5 j 10V 4 5
�� / !� 5<:" �"<:28 LP��=Q' � 1 � !� 5�9�:3� �"<:28 ALP�>=Q' � 1C~7.5 j 10V~7 j 10m
![Page 28: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/28.jpg)
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CB: Common Base
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Output Characteristic Input Characteristic
��
��
���
���� / 01n�2n�3n�
�0.4Sa
tura
tio
n Active
Cut-Off
��.
Ideal:
very little
dependence to
output voltage
����+
-
����
+
-
&' / 1
np+ p-
��.
→ &()/ ���� / % / ����� / 5<:"5<:" �5�9�:3�/ 11 � 5�5 6;67� "3�⇈
Non ideal: &' / ������ / 1cosh IM=≅ 11 � KJ IM= p → &()/ &'%
Leakage current of BC diode
Determined by 9�: as 9�: ≫ <: np+ p-
��.
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29
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CE: Common Emitter
30
B+
- ���+
-
Output Characteristic Input Characteristic
��
�
���
�� Saturation Active
Cut-Off0.2
�� / 0�� � 0
very little
dependence to
output voltage
n
p+
p-��� ��
� / 010��20��30��
���.
�� � !�5<:" LP��=Q'" ≪ 3_�� � 0� � 0 r → S � � !�5�9�:3� LP��=Q' → �() / ��� / 55� ∙ <:9�: ∙ 3�"67�6; ⇈⇊
![Page 31: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/31.jpg)
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ICEO and ICBO
31
np+ p-
��.
n
p+
p-
���.���. ≫ ��.
n
p+
p-
���.
���������
��
���. / ��� � ��� / ��. � �()��.
![Page 32: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/32.jpg)
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5.
Ebers-Moll Equations
32
0 "
u<∆<�
∆<�
0 "
u<∆<�
0 "
u<∆<�
Normal Inverted
��v ��v��w ��w ��w � 0��w � 0
∆T� / 0 Normal ��x / y∆<�; ��x / {∆<�∆T� / 0 Inverted ��| / �{∆<� ; ��| / �y∆<� Sy / !�;=M= coth IM={ / !�;=M= csch IM=}�� / ��x � ��| / y∆<� � {∆<��� / ��x � ��| / {∆<� � y∆<�
~��x / ��� LP��=Q' � 1 ; ∆<� / 0��| / ��� 1 � LP�>=Q' ; ∆<� / 0
~�� / ��� LP��=Q' � 1 �&|��|�� / &x�� � ��� LP�>=Q' � 1&x ≡ ��x ��v⁄&| ≡ ��| ��w⁄
![Page 33: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/33.jpg)
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3.
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5.
Ebers-Moll Equations
33
~�� / ��� LP��=Q' � 1 �&|��|�� / &x�� � ��� LP�>=Q' � 1~�� / &|��| � ��. LP��=Q' � 1 �� / &x�� � ��. LP�>=Q' � 1
��. LP��=Q' � 1E C
B
& �
��. LP�>=Q' � 1
&����
�� ��� ��
![Page 34: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/34.jpg)
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Charge Control Analysis
34
Normal Inverted
��v ��v ��w ��w��v ≡ �x8�v
�� / ��v � ��w / �x 18�v � 18�v � �|8�w
�� / �x8�v � �| 18�w � 18�w&x ≡ ��x ��v⁄ &| ≡ ��| ��w⁄
�x �|��w ≡ � �|8�w��v / ��v � �x8�v ��w / ��w � �|8�w
Generally:
/ 8�v8�v � 8�v / 8�w8�w � 8�w
Small-
signal
model
�� / �x 18�v � 18�v � �|8�w � D�xD��� / �x8�v � �| 18�w � 18�w � D�|D� � / �x8�v � �|8�w � D�xD� � D�|D�
/ 8�v8�v�x ≡ ��v�v�v / �x8�v� / �v � �w / �x8�v � �|8�w
![Page 35: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/35.jpg)
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Transient Response
35
�����
���
���M�6
������ �
"
��
�K �J
���
� / ��� �����
���8� ��
��� / ���M ���
�8
��(
<
Bn
p+
p-
� ��C
���� �M
��
�W ��0
Cut-off
"
<
0
active
�"
<
0
saturation
�
"
<
0
active
�"
<
0
saturation
�
![Page 36: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/36.jpg)
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3.
4.
5.
Base Transient Time
36
"0
<active �� / �!�5 �∆<�0 FGHI
∆< A0C / !�5 ∆<A0, �C"�
� / !�"∆<A0C2/ !�5 2�!�"J�� / �A"J/25C8� / "J25
/ �8�
![Page 37: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/37.jpg)
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5.
Turn ON
37
Turn-on: � / ��� / �D�D� / � � �8 → � � / �8 � �LO� �=⁄
I.C.: � 0 / 0 → � � / �8 1 � LO� �=⁄
→ �� � / � �8� / �88� 1 � LO� �=⁄ 0 � � � �K�� � / ��� / ���M
�� �� / ��� → �� / 8 ln 11 � ���� 8�8
![Page 38: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/38.jpg)
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5.
Turn OFF
38
Turn-off: D�D� / ��� � �8 → � � / ���8 � �LO� �=⁄I.C.: � 0 ���� . / � ∞ ���� .x / �8
→ � � / �8 1 � � LO� �=⁄ � �→ ~�� � / ��� �W � � � ���� � / � �8� / �88� 1 � � LO� �=⁄ � � � � ��
��( / �� � �W → ��( / 8 ln 1 � ����� 8�8 � �
→
![Page 39: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/39.jpg)
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3.
4.
5.
Non-Ideal BJT
39
1. Base width modulation (Early effect)
2. Punch-through
3. Avalanche Breakdown
4. Geometrical effects
5. Generation-Recombination in depletion regions
6. High-level injection
Deviations from Ideality:
![Page 40: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/40.jpg)
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4.
5.
Base Width Modulation
40
�� ≅ ��� / !�5<:" LP��= Q'⁄ � ↗⟶ " ↘⟶ �� ↗ ���
��� � 010��20��30��
�7 Early voltage
Saturation
�� / ��L�=� ��A1 � ���7 CActive:HW:
Show that 7 / !6"���
Emitter
np+
Base�" �
p-
Collector �“F” “R”
![Page 41: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/41.jpg)
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2.
3.
4.
5.
Punch Through
41
�����!� !�
Emitter
p+
Base�p-
Collector �“F” “R”
n
���
![Page 42: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/42.jpg)
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3.
4.
5.
Avalanche Breakdown
42
���
��
���.��.���.
��.
���. / 1�K/� ��. n~4
![Page 43: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/43.jpg)
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5.
Geometry Considerations
43
p-n-p Individual device n-p-n Integrated circuit BJT
� ��Current Crowding
![Page 44: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/44.jpg)
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2.
3.
4.
5.
G-R in Depl. Region , High-level injection
44
p-n-p Individual device
���
ln � , �)�() ∝ L�=���
∝ L�=�J��
∝ L�=�J��
High-level injection, Kirk Effect
��� In depletion region
�()
��
��� high injection
���
![Page 45: Session 11: Solid State Physics Bipolar Junction Transistoree.sharif.edu/~sarvari/25772/PSSD010.pdf · Bipolar Junction Transistor ... minority-carrier diffusion equation ... Ebers-Moll](https://reader031.vdocuments.net/reader031/viewer/2022021717/5b4aca7f7f8b9a2d2f8c77df/html5/thumbnails/45.jpg)
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5.
Small-Signal Model
45
Low Freq:B
E
C
������+
-��
E
C
������+
-�� ����� ��
��> / � � �))�¡¡B
�� / ��� / ���¢/!�� / � �� / � . ��
�� / ��7� / � :
1 � ��: �
��� / ���£1 � ��: � �; / KJ¤:8��� / ���¢/! 8� / ��. 8�
High Freq: