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© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 1
Bipolar junction transistor - Basics Introduction Walter Brattain, John Bardeen, and William Shockley invented the bipolar junction transistor (BJT) in 1949, while working for Bell Telephone Laboratories. This revolutionary invention changed the world. The invention of the BJT followed the invention of the point-contact transistor by Walter Brattain and John Bardeen. The point-contact transistor has several problems that prevented it from becoming a viable device. BJT is a three-terminal device. BJT is used as amplifier and switch.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 2
Circuit diagram of pnp transistor consisting of two diodes
The two n-type regions merge to form a very thin base. EB junction: Forward bias CB junction: Reverse bias
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 3
Band diagram (PNP)
Junction bias? Major current flows?
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 4
Basic amplifier circuits Common-base configuration
E
CII
=α (1)
α = current amplification in common base circuit Typical values: α > 0.99 (for state-of-the-art transistor)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 5
Common-emitter configuration
β = amplification in common-emitter circuit
α−α
=⎟⎠⎞
⎜⎝⎛ −α
=−
==β−
111 1
CE
C
B
CII
III
(2)
β > 100 for state-of-the art transistors
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 6
Common-collector configuration
αβ
=α
=B
C
B
E /I
III
(3)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 7
Nature of bipolar transistor BJT is a current amplifier (not a voltage amplifier). BJT is current-controlled current source. BJT base current controls the emitter current and thereby the collector current.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 8
Qualitative discussion of pnp transistor
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 9
Basic ideas EB junction is asymmetric:
EnEp II >> (4) The emitter hole current is controlled by EB junction. The base width is small.
pB LW << (5) Most holes diffusing into the base will reach the collector if condition of Eq. (5) is met. Thus the base current controls collector current.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 10
Discussion of currents
EB junction currents (EB junction is forward biased) (1) Holes diffusing from the E into the B (2) Electrons diffusing from the B into the E
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 11
Base currents (3) Recombination of holes injected into base (4) Most holes reach C since LP >> WB BC junction currents (BC is reversely biased) (5) Electron minority carrier current from C to B (6) Hole minority carrier current from B to C We know that current (5) and (6) can be neglected for most practical purposes.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 12
Basic equations What is the fraction of the emitter hole current that reaches the collector?
EpC IBI = (6) B = Base transport factor B = Probability that a hole injected into B reaches C B ≤ 1
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 13
What fraction of total emitter current is emitter hole current?
( )EpEnEEp IIIE +γ=γ= (7)
EfficiencyEmitter =γ
EEp to of Ratio II=γ
1≤γ
Ep
En1
Ep
En
EpEn
Ep 11II
II
III
−≈⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=γ
−
(8)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 14
Current amplification α
γ===α BII
BII
E
Ep
E
C (9)
We will later calculate B and γ in two ways: 1. Approximate calculation
2. Exact calculation
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 15
Approximate hole distribution in base (PNP) Long base (WB >> Lp)
( ) Pn /n e Lxpxp −∆=δ (10)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 16
Short base (WB << Lp) Exponential function can be linearized
( )1eisit 0At /n BE
0−=∆= kTeV
n ppx (11)
( )0
CB0 n
/nBn 1eisit At pppWx kTeV −=−=∆= (12)
( ) 0isThat Bn ==Wxp (13)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 17
Can you identify the diffusion triangle in the figure? Equation for diffusion triangle:
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−∆=
B
nn 1
Wxpxp
(14)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 18
Note Diffusion Current:
xpDeJ
dd
pp −= (15)
( )xpei.J d/d. slopep ∝ Short base changes slope (i. e. dp / dx)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 19
Approximate calculation: Emitter efficiency (PNP) Recall the Shockley equation:
( )1e00 p
n
nn
P
P −⎟⎟⎠
⎞⎜⎜⎝
⎛+= kTeVn
LDp
LDAeI
(16)
… where first summand within first parenthesis is due to hole injection … where second summand within first parenthesis is due to electron
injection Emitter is “long”, and therefore the electron current from base into emitter is given by
( )10p
n
nEn −= kTeVen
LDAeI
(17)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 20
Base is “short”, and therefore the hole current from emitter into base is given by
( )B
pn
p
pEp 1
0 WL
epLD
AeI kTeV −= (18)
where last term, i. e. ( Lp / WB ), is correction due to increase in slope
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 21
One obtains the emitter efficiency using Eqs. (8), (17), and (18)
11
0
0
nB
P
pn
n
Ep
En
pWD
nLD
II
−=−=γ (19)
A2i
2ip0
using Nnpnn == (20)
D2i
2in0
and Nnnnp == (21) one obtains:
Anp
DBn1NLDNWD
−=γ (22)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 22
How can we attain high emitter efficiency? For a high value of γ: 1. WB must be very short 2. NA >> ND (23)
That is,
Emitter doping >> Base doping
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 23
Example: Problem: Assume a PNP transistor with the following parameters: Emitter doping: NA = 1 × 1018 cm–3 Base doping: ND = 1 × 1017 cm–3 Dp = Dn WB = 100 nm Ln = 1 µm Calculate emitter efficiency. Solution:
990100
111Anp
DBn .NLDNWD
=−=−=γ
The problem assumed reasonable parameters. For such reasonable parameters, we obtain a high current gain.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 24
Approximate calculation: Base transport factor (PNP)
Thought experiment: Let’s assume that the BC junction would not influence the hole distribution. Warning: Strictly speaking, this is incorrect assumption!
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 25
In this case, the following hole distribution would be obtained:
B1 /currentionrecombinatBase WpQ ∆∝τ∝ (24)
p2 /currentCollector LpQ ∆∝τ∝ (25) (Note: In Eqs. 24 and 25, we use that WB << LP)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 26
It is
21
1
2
1
21
2
Ep
C 11 QQQQ
QQQ
IIB −≈⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
τ+ττ
==−
(26)
Using Eqs. (24), (25), and (26) one obtains:
p
B1LWB −=
(27)
End of thought experiment. Warning: This thought experiment is an oversimplification and the result (Eq. 27) must not be used.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 27
Exact hole distribution in the base (PNP) Hole concentration at the emitter side of base
( ) ( ) kTeVkTeV ppxpp BE0
BE0
e1e0 nnnE ≈−==∆=∆ (28) Hole concentration at the collector side of base
( ) ( )0
BC0 nnBn 1e ppWxpp kTeV
C −≈−==∆=∆ (29) … note that VBC is negative Eqs. (28) and (29) are the boundary conditions for the hole concentration in the base
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 28
There is no electric field in the neutral region of the base. Therefore, transport can be described by the diffusion equation
( ) ( )2
p
nn2
n
2
dd
Lxpxp
xδ
=δ (30)
General solution of this equation is given by
( ) pnpn ee 21nLxLx CCxp −+=δ (31)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 29
The constants C1 and C2 will be determined by using the boundary conditions
( ) E21n 0 pCCxp ∆=+==δ (32)
( ) C21BnpBpB ee pCCWxp LWLW ∆=+==δ −
(33)
Solving Eqs. (32) and (33) for C1 and C2 yields
PBPB
PB
eeeEC
1 LWLW
LWppC −
−
−
∆−∆=
(34)
PBPB
PB
eee CE
2 LWLW
LW ppC −−
∆−∆=
(35)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 30
Insert the constants C1 and C2 into Eq. (31)
For 0C ≈∆p , the hole concentration in the base is given by
( )PBPB
PnPBPnPB
eeeeee
En LWLW
LxLWLxLWpxp −
−−
−
−∆≈δ
(36)
This function has an exponentially decreasing part and an exponentially increasing part.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 31
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 32
Discussion of slopes
Recall that the slope ( ) ]d/d[ nn xxpδ determines the diffusion current. Slope is larger at xn = 0 as compared to xn = WB. The difference in slope is due to recombination in base. Approximation for exponential function: For WB << LP, we can expand the exponential function into a power series:
...xxx +++=!2!1
1e2
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 33
Inserting this approximation into Eq. (36) and neglecting all quadratic and higher terms in Eq. (36) yields
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−∆=δ
B
nEn 1
Wxpxp
(37)
This equation represents the diffusion triangle in the base. The strictly triangular shape is valid for negligible recombination in the base.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 34
Mathematics of exponential functions Exponential function = Natural decay function For the next section, we need some mathematical relations for exponential functions and they are summarized below:
...718.211lime =⎟⎠⎞
⎜⎝⎛ +=
∞→
n
n n
...xxxx ++++=!3!2!1
1e32
Give some examples of natural (i. e. exponential) decays!
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 35
0e:Function 0
xxyy −=
0
0
0dd:Slope
xy
xy
x−=
=
0000de:Integral 0 xyxy xx =−∞
∫
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 36
Mathematics of hyperbolic exponential functions For the next section, we need some mathematical relations for hyperbolic exponential functions and they are summarized below:
( )
( )xx
xx
x
x
−
−
+=
−=
eecosh:functioncosHyperbolic
eesinh:functionsinHyperbolic
21
21
(Note: Hyperbolic cos function is also called “chain function”. Why?)
xxx
xxx
sinhcoshcoth :functioncotHyperbolic
coshsinhtanh :functiontanHyperbolic
=
=
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 37
xx
xx
xx
xx
sinh1
ee2cosech
:functioncosecanHyperbolic
cosh1
ee2sech
:functionsecanHyperbolic
=−
=
=+
=
−
−
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 38
Exact E, B, and C currents We have calculated the hole distribution in the base and can now calculate the currents of the three terminals E, B, and C by using the equation:
( )nn
p dxp
xdDAeI δ−=
(38)
Emitter current Emitter current is obtained by using Eqs. (31), (34), (35), (38)
( ) ( )12p
pnpEp 0 CC
LD
AexII −=== (39)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 39
⎟⎟⎠
⎞⎜⎜⎝
⎛∆−∆=
P
BC
P
BE
p
pEp cosech coth
LWp
LWp
LD
AeI (40)
Collector current
( ) ( )PBPB ee 12p
pBnpC
LWLW CCLD
AeWxII −=== − (41)
⎟⎟⎠
⎞⎜⎜⎝
⎛∆−∆=
P
BC
P
BE
p
pC cothcosech
LWp
LWp
LD
AeI (42)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 40
Base current
CEpCEB IIIII −≈−= (43)
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡∆+∆=
p
BCE
p
pB 2
tanhL
WppLD
AeI (44)
Eqs. (40), (42), and (44) are generally valid, i.e. for any bias
configuration and bias condition of the transistor. The equations can be
simplified for a transistor under regular operating conditions, which are
VBE = forward bias
VCB = reverse bias
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 41
Appropriate E, B, and C currents VBE = forward bias ∆pE ≠ 0 VCB = reverse bias ∆pC = 0 From Eqs. (40), (42), and (44) it follows that
p
BE
p
pEp coth
LWp
LD
AeI ∆= (45)
Using )3/()/1(coth xxx +≈ , one obtains
⎟⎟⎠
⎞⎜⎜⎝
⎛+∆=
p
B
B
pE
p
pEp 3L
WWL
pLD
AeI (46)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 42
Furthermore
p
BE
p
pC cosech
LWp
LD
AeI ∆= (47)
Using )6/()/1(cosech xxx −≈ , one obtains
⎟⎟⎠
⎞⎜⎜⎝
⎛−∆=
p
B
B
pE
p
pC 6L
WWL
pLD
AeI (48)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 43
Finally
CEpCEB IIIII −≈−= (49)
⎟⎟⎠
⎞⎜⎜⎝
⎛+∆=
p
B
p
BE
p
p61
31
LW
LWp
LD
Ae (50)
It follows that
Ep
BEB2
p
pB 22
pWAepWL
DAeI ∆
τ=∆=
(51)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 44
Base transport factor Using Eqs. (45) and (47) we calculate
p
B
pB
pB
Ep
C sech)(coth)(cosech
LW
LWLW
IIB ===
(52)
Using 2)2/1(1sech xx −≈ , one obtains
2
p
B211 ⎟
⎟⎠
⎞⎜⎜⎝
⎛−=
LWB
(53)
We now have a good expression for B. Compare this to Eq. (27)! (Recall: Do not use Eq. 27)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 45
We now have γ (i. e. the emitter efficiency, see Eq. 22) and B (i. e. the
base transport factor, see Eq. 53).
Since α = γ B, we can calculate the current amplification of a transistor:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛−=γ=α 2
p
2B
Anp
DBn2
11L
WNLDNWDB
(54)
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 46
Example
Problem: Calculate the Base Transport Factor for WB = 0.1 µm and for the following diffusion lengths:
(1) Lp = 0.1 µm and (2) Lp = 1 µm.
Solution: Calculating the base transport factor using
( )2pB /)2/1(1 LWB −= yields (1) B = 0.5 and (2) B = 0.995
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 47
Summary of operation regimes Cutoff VBE is too low to provide significant injection
Example: Given is a transistor with
2310i
317BaseD,
2p
Bp
m100100,cm10
cm10,/scm10
m1.0,m1
µ×==
==
µ=µ=
−
−
An
ND
WL
Calculate IEp for VBE = 0.3 V!
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 48
A106.1
cm10ee
9
B
PE
P
PEp
38
D
2i
nE 0
−
−
×=∆≈
===∆
WLp
LDAeI
Nnpp kTeVkTeV
Calculate IEP for VBE = 0.7 V!
mA9.8cm106.5
Ep
314E
=×=∆ −
Ip
Forward active Forward biased EB junction ∆pE ≠ 0
Reverse biased CB junction ∆pC = 0
Diffusion triangle in base
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 49
Saturation CB junction is forward biased as well. Simultaneous transistor action in both directions, i. e. both diodes are forward biased. If | VBE | > | VCB |, one obtains the following hole distribution in the base:
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 50
It is useful to consider the following thought experiment: Consider a transistor with VEB = 0.7 V = const..
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 51
Curves are displaced by 0.7 V.
The I-V curves can be identified as a diode characteristic plus a current from emitter.
© E. F. Schubert, Rensselaer Polytechnic Institute, 2003 52
Bridge between device physics and electrical circuit
Emitter efficiency (γ)
Base transport factor (B)
Current amplification in common base configuration (α)
Current amplification in common emitter configuration (β)
Circuit parametersDevice physics Material parameters
Mobilities Lifetimes Diffusion constants Doping
concentrations Physical constants Material constants