# 2002Khan

```Second International Conference on Electrical and Computer Engineering
ICECE 2002, 26-28 December 2002, Dhaka, Bangladesh
Analytical Expressions of Collector Current Density and Base
Transit Time in an Exponentially Doped Base Bipolar Transistor
for All Levels of Injection
Md. Ziaur Rahman Khan and M.M. Shahidul Hassan
Department of Electrical &amp; Electronic Engineering, BUET, Dhaka –1000.
Abstract - Analytical expressions of collector current
density and base transit time are developed for an
exponentially doped base modern bipolar junction
transistor (BJT). The present model incorporates
dopant dependence of carrier mobility, bandgap
narrowing and finite velocity saturation effects in the
calculation of collector current density and base transit
time. This model is applicable for all levels of
injection before the onset of Kirk effect.
1. Introduction
Modeling of collector current density and base transit
time is essential for design of high-speed bipolar
transistor. The two integral relations for the current
flow through the base region of a bipolar transistor,
and for the transit time, were generalized to the case of
a hetero structure bipolar transistor with a nonuniform
energy gap in the base region by H. Kroemer [1] in
1985. J. H. Van den Biesen [2] in 1986 used a
regional analysis to study the transit time of the BJT
as a function of base emitter bias. He subdivided the
total transit time from emitter to collector contact into
five components. But no closed form solution in [2]
was obtained. J. S. Yuan [3] in 1994 studied the effect
of the base profile on the base transit time of the
bipolar transistor for all levels of injection. He
proposed equations for the minority carrier
distribution within the base for different types of base
doping. Using boundary conditions and the proposed
equation he numerically evaluated the base transit
time. So the equation forms for Jn and &ocirc; B in his work
are not concise to express Jn and &ocirc;B as the function
defined by some existing models. K. Suzuki [4] in
1993 proposed electron current density Jn and base
transit time (&ocirc; B) models of uniformly doped bipolar
transistor for high level of injection. He incorporated
the electron velocity saturation effect in the collectorbase depletion region. Later, P. Ma [5] had simplified
the equations, but they are not simple enough to give a
physical insight into device operation. P. Ma, L.
Zhang in [6] improved the work of [5] considering the
velocity saturation of the electron in the depletion
region of base-collector and the electrical field
dependence on the minority carrier mobility.
However, the method is based on iteration techniques.
M. M. Shahidul Hassan and A. H. Khandoker [7]
ISBN 984-32-0328-3
developed a mathematical expression for Jn and &ocirc; B for
uniform base doping density for all levels of injection.
But the model ignored the velocity saturation of
electron at base-collector depletion region. D.
Rosenfeld [8] derived an analytical formula of base
transit time through a dopant-graded base considering
the dependence of mobility on the doping level. But
the equation is derived ignoring the bandgap
narrowing and velocity saturation effect. M. M. L.
Jahan and A. F. M. Anwar [9] developed an analytical
expression of base transit time for exponentially doped
base. But the expression is applicable for low injection
level only. In our present model, analytical
expressions for the collector current density and base
transit time for an exponential doped base are obtained
considering dopant dependence of carrier mobility,
bandgap narrowing effects and velocity saturation at
the base-collector junction. The expressions are
applicable for all levels of injection.
2. Derivation of the model equations
The base width of a modern high speed bipolar
transistor is very thin. So the carrier recombination
within the base can safely be neglected [4]. In the
absence of recombination, current density within the
base becomes constant.
The base transit time for all levels of injection is
derived solving the current transport equation for
electron profile n(x) which is given by [5]
dn ( x)
− J n ( x) = q Dn ( x) dx + q &micro; n ( x )n ( x) E ( x ) (1)
where Dn (x) is the electron diffusion coefficient , E(x)
is the electric filed, Jn (x) is the current density and
&micro;n (x) is the electron mobility.
The electric field in the base is given by [3]
2
kT  1 dp
1 d n ie ( x) 
E (x ) =
− 2
(2)


q  p dx n ie ( x) dx 
where p is the hole concentration in the base and n ie is
the effective intrinsic carrier concentration and it is
given by [9]
2V g
Vt
2
2  N A( x) 

n ie ( x) = nio 
 Nr 
120
(3)
where n io is the intrinsic carrier concentration in the
base, NA (x) is the base doping profile, Vg and Nr are
constants.
The exponential base doping profile is [3]
ηx
N A ( x) = N A (0 ) exp( − )
Wb
(4)
where Wb is the base width and η is the slope of base
doping and NA (0) is the peak base doping.
The electron mobility in the base is given by [3]
&micro;max − &micro;min
(5)
&micro;n ( x) = &micro;min +
γ
 N A( x) 
 N ref 


If this mobility equation is used directly in (1) the
differential equation becomes intractable. So with
reasonable accuracy (5) is simplified to the following
form
a
(6)
&micro;n ( x) =
18


N
A −
− C N γA ( x) 



N =1

where a, A, B and C are constants.
)
If the electric filed is substituted in (1) the current
density equation becomes [4]
2
n ie
J n = − q D n ( x)
n( x ) +
The total charge storage of the injected carrier n(x) in
the base per unit area can be written as
Wb
Qlb = q ∫ nl ( x) dx
(8)
0
Substituting the values of different parameters in (7)
and using the conditions for low injection i.e.
NA (x)&gt;&gt;n(x), the equation of current density for low
injection region becomes
(9)
where vs is the saturation velocity of electron, n lo is
the normalized carrier concentration at base-emitter
junction for low level of injection,
η
)
 4 nie2 (0)
 
n ( 0 ) = −0 .5 N A ( 0 ) + 0 .5 N A ( 0) 1+
exp  V BE  
2
 N A (0)
 V t  
and at x=Wb , n l (W b ) = J ln
q
Similarly evaluationg the value of carrier
concentration for low injection and using (8), the
stored charge in the base for low injection region
becomes
Qlb = W b L nlo
(10)
η
where
(
)


 A   1 
L = q N A ( 0 ) 1 − L1    eηp − 1 
p
p
   


(
)
(
)
 A
 1 
 
 1
+ q N A ( 0 )  L1 η + L1 f 1( N )
 e− ηNγ − 1 +   eηp − 1  
p
N
γ
p

 

 


∑(
18
f 1( N ) = B
N =1

1
N
− C ) 

 (γN + p ) 
(N
A
Nγ
(0) ) and
2q v s E
L1 =
Jno
Similarly substituting the values of different
parameters in (7) and using the conditions for high
injection i.e. NA (x)&lt;&lt;n(x), the equation of current
density high injection region becomes
J hn = q v s G n ho
(11)
where, n ho is the normalized carrier concentration at
base-emitter junction for high level of injection


N A ( 0 ) J no
 and
G =
 s

f
Jno
+
Fq
(
0
)
vs N A 

A

F =   f s − 1 


s

∑(
18
+B
2 akTη N A (0 )
Jno =
,
Wb
p=1-2s, s=Vg /Vt
)
(
d  n(x )[n(x ) + N A ( x )] (7)


2
(
x
)
dx
NA
n ie ( x )


 N ( 0 ) f p Jno 
A
E =
, f =e ,
 Jno − Dq v s N A ( 0) 
p
f
18 
1


Np 
Nγ
−γNp
p 

 N A (0 ) f
− 2 B ∑ ( −C ) 
− f 
0 

 (γN + p ) 

The boundary condition used to solve the equation are
at x=0,
( x)
J ln = q v s E n lo
(
2A
1−
p
vs
1+
∑(
D=
−C
N =1
(
)
1

N
γ
N A ( 0) )  (γN − s )  f − (γN − s ) − 1


and the equation of stored charge density for high
level of injection is
Qhb = W b M nho
η
and
121
(12)
0 .5
where

 A   1 
A 
M = q N A (0) 1 + M 1    (1 − e−ηs) − M 1 η 
s
s
s 
   


 1
+ q N A (0) M 1 f 2 ( N )
 Nγ

q vs G
Jno
f 2( N ) = B
∑
(− C N
γ
A
)
We have solved (7) for two cases, low and high level
of injection. But in the intermediate region of
injection, the equation is not analytically tractable.
However a general expression for Jn and Qbn is
possible to obtain by exploring their behaviors in low
and high injection region.
According to (9) and (11) current density for low
injection and high injection region are
J ln = q v s E n lo
J hn = q v s G n ho
is normalized carrier concentration for all
n( 0 )
N A ( 0)
levels of injection and is given by
Comparing (13) with (9) and (11), we can write
for
W =G
of injection becomes
W ( L + 2M no)
τ B = ηq b ( E + 2 G )
no
vs
E + 2G no
1 + 2 no
o
o
o
(14)
η
-3
8.6
8.4
8.2
8.0
7.8
0.6
0.7 0.8 0.9 1.0 1.1 1.2
Base Emitter voltage (V)
1.3
Fig. 1(a) Variation of base transit time on
base-emitter voltage
b
lb
18
N A ( 0 ) = 2 x 1 0 c m , η= 3
and W b =150nm
7.6
According to (10) and (12), stored charge density for
low injection and high injection region are
W L
Q =
n
(17)
The variations of base transit time with different
parameters are shown in Fig. 1(a) and 1(b). The Fig.1
(a) shows that the base transit time increases with the
increase of base-emitter voltage. With the increase in
base-emitter voltage, the minority carrier injection in
the base increases. This reduces the aiding electric
filed in the base for exponentially doping profile. So
base transit time increases with the increase in baseemitter voltage.
n &gt;&gt; 1
o
s
(16)
Using (14) and (16) the base transit time for all levels
8.8
So, the empirical expression for current density then
becomes,
n
L + 2M no
no
η 1 + 2 no
9.0
E + 2G n
J =qv 1+2 n
n
n o &gt;&gt; 1
Qbn = W b
n &lt;&lt; 1
A best fit is found to be,
W=
for
The empirical expression for stored charge density in
the base then becomes,
o
for
n o &lt;&lt; 1
4. Results and Discussions
For all levels of injection the expression for current
density can be written as
(13)
J n = q v s W no
W =E
for
A best fit is found to be,
L + 2M n o
H =
1 + 2 no

N
1

(0 ) 
 (γN − s ) 
3. General Formulation
no
H =L
H =M
N =1
where,
Comparing (15) with (10) and (13) we see that
and
18
(15)
η
Base transit time (ps)
M1 =

 −ηNγ
1
 (e
− 1) +  (1 − e −ηs )

s 

Qbn = W b H n o
lo
Q hb = W b M n ho
η
For all levels of injection the expression for stored
charge density can be written as
From the Fig 1 (b) we see that the base transit time
increases with peak base doping concentration. As the
doping concentration in the base increases, the
impurity scattering increases. This reduces the carrier
mobility and increases base transit time.
122
[5]
Base transit time (ps)
9
8
7
[6]
6
5
VBE=1.0V
4
[7]
VBE=0.7V
3
10
17
10
18
-3
Peak base doping concentration (cm )
Fig. 1(b) Dependence of base transit time on
peak base doping concentration
[8]
From (17) we see that the base transit time is
proportional to the base width and inversely
proportional to the slope of base doping (η). So base
transit time increases with base width and decreases
with slope of base doping (η).
[9]
5. Conclusion
An empirical expression for base transit time of
exponentially doped base BJT is developed which is
applicable for all levels of injection. This work
incorporates dopant dependent mobility, bandgap
narrowing and finite velocity saturation effects. The
results obtained by proposed expressions are
compared with the result available in the literature and
found in good agreement.
References
[1]
[2]
[3]
[4]
H. Kroemer, “Two integral relationship
pertaining to the electron transport
through a bipolar transistor with
nonuniform energy gap in the base
region,” Solid State electronics, vol. 28
No. 11, pp. 1101-1103, 1985.
J. H. Van Den Biesen, “A Simple
Analysis of transit times in bipolar
transistors,” Solid-State Electronics, vol.
29, No. 5, pp 529-534, 1986.
J. S. Yuan, “Effects of base profile on
the base transit time of the bipolar
transistor for all levels of injection,”
IEEE Trans. Electron Devices, vol. 41.
No. 2, pp 212-216, Feb 1994.
K. Suzuki, “Analytical base transit time
model of uniformly-doped base bipolar
transistors for high-injection regions,”
Solid-State Electronics, vol. 36, No.1, pp
109-110, 1993.
123
P. Ma, L. Zhang, Y. Wang, “Analytical
relation pertaining to collector current
density and base transit time in bipolar
junction
transistor,”
Solid-State
Electronics, vol. 39, No. 1, pp 173-175,
1996.
P. Ma, L. Zhang, Y. Wang, “Analytical
model of collector current density and
base transit time based on iteration
method,” Solid-State Electronics, vol.
39, No. 11, pp 1686-1686, 1996.
M. M. Shahidul Hassan, A. H.
Khandoker, “New expression for base
transit time in a bipolar junction
transistor for all levels of injection,”
Microelectronics Reliability, vol-41, pp
137-140, 2001.
D. Rosenfeld and Samuel A. Alterovitz,
“Carrier transit time through a base with
dopant dependent mobility,” IEEE
Trans. Electron devices, Vol. 41. No. 5,
pp 848-849, May 1994.
M. M. L. Jahan, A. F. M. Anwar, “An
analytical expression for base transit
time in an exponentially doped base
bipolar junction transistor,” Solid-State
Electronics, Vol. 39, No. 1, pp 133-136,
1996.
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