10 J Breakdown voltage BV'cBo (V)
2.4 Conclusions
Epitaxially grown base-collectorjunctions with a lightly doped collector are widely used in the fabrication of power transistors. Breakdown voltages of epitaxial bipolar n+pn-n+ transistors are numerically computed based on the more accurate ionization rates of R. Van Overstraeten and H. de Man [22]. The optimal values of collector dopingdensity Ne and the epitaxial layer thickness W subject to a minimum collector resisianceare numerically computed. Empirical expressions for optimal parameters Ne and W are also established.
In the evaluation of optimal parameters, the peak electric field for bulk breakdown is not used. By using the empirical expressions for Ne and W, more accurate values of other parameters of a transistor can be obtained.
32
!..};.
CHAPTER 3
OPTIMAL VALUES OF COLLECTOR PARAMETERS OF mGH VOLTAGE TRANSISTORS WITH GRADED COLLECTOR
3.1 Introduction
Works on current mode second breakdown show that a substantial measure of protection against failure by avalanche injection can be built into a transistor by grading the impurity distribution of the collector [18). The improvement is achieved directly from a substantial increase in the collector current required to trigger avalanche injection.
The current that initiates avalanche injection depends upon the doping density at the collector-substrate interface and it increases with the increase of doping density at the interface. The collector doping is optimized to achieve the minimum collector width for a specmed blocking voltage.
A graded collector impurity profile allows the current density to increase while pre- venting the maximum electric field from reaching a value that permits carrier multipli- cation. Appropriate collector design can make the critical current density considerably higher than would be possible with a uniformly doped collector designed to support the same breakdown voltage.
For single graded collector, the initial doping density and impurity gradient can be obtained for breakdown voltages specmed either in open-base or in open-emitter condi- tion 'under the requirement of achieving minimum collector resistance at free spreading of the depletion layer into the collector region. But a signmcant increase in current den- sity requires a very thick collector layer, particularly in the case of high voltage power transistor. This has a detrimental effect on the performance of the transistor and for the base drive required to hard saturation, and introduces technical problems concern- ing epitaxial layer growth. These difficulties can be eased considerably by using double graded collectors. This chapter deals with the determination of optimal parameters of
single and double graded collector of high voltage transistors. The optimal values of graded collector parameters will be compared with those of a uniformly doped collector.
3.2 Breakdown voltage of graded junctions
The doping density of the base region of an epitaxial bipolar transistor is much '1 higher than that of the collector region and therefore, the base- collector junction can be considered as a one-sided p+n junction for determination of the bulk brea.kdown voltage of the junction. For a one-sided graded junction, the brea.kdown voltage can be determined by the collector doping at the collector-base interface and the collector impurity gradient. For the open-emitter transistor, the avalanche brea.kdowncondition can be expressed by eqn. (2.4). For convenience, the eqn. (2.4) is again written below,
l
w -1."
(a.-a. lei.,Id - 1Ctpe 0 z -
o
&r an open-base transistor, the brea.kdown condition is given by,
(3.1)
(3.2)
Eqns. (3.1) and (3.2) are used for determination of brea.kdown voltages BVCBO
and BVCEO respectively. However, to solve the ionization integral, the mathematical expressionfor field distribution within depletion region is required.
3.2.1 Single graded collector
The doping profile of a single 'graded collector is shown in Fig. 3.1. The impurity concentration in the collector region is,
Nc(z)
=
No+
aW(z)where,
No= collector doping at the base-collector interface a= collector impurity gradient
W
=
collector layer width34
(3.3)
:>,
.,jJ.-{
~ impurity
gradient
c
OJa.
"0 CJl
.
C.-{0..
NO
.g
x
I~.•. ~---w--- ..• 1 Collector
Base~ll~c~ C_o_l_l_e_c_t_o-r-e_p-i_~_a_X_l_'a-l-l_a_y_e_r __ -...Isubstrate
Fig. 3.1 Proposed doping profile of a single graded collector.
Solution of one-dimensional Poisson's equation leads to the expression for electric field within the collector as,
q( 1 2)
E
=
Ec - - Noz+
-azf 2 (3.4)
Integrating eqn. (3.4) and using boundary condition, E
=
0 at z=
W, it can beshown that,
VB
=
EcW _ qNow2 _
qaw3
2f 6f
Also, the depletion layer width at breakdown is,
(3.5)
W=--+
Noa
(N
-
o)2+-
2f Eca qa (3.6)
for wider epitaxial layers that are able to contain the full depletion width.
The collector resistaxice per unit area is,
(3.7) Unfortunately, eqns. ( 3.3), (3.4) and ( 3.7) do not lead themselves to convenient analytical expressions for the values of No, a and W corresponding to minimum resis-:.
tance. Instead; .the solution has been obtained using the computer iteration procedure •.
shown .in figure 3.2. Romberg integration is used to carry out the integration of eqns.
(3.1) and (3.2). Solving eqns. (3.5) and (3.6) together with satisfying ionization integral i.e., eqn. (3.1) or (3.2) as specified by the breakdown, a number of collector parameters are found. The combination of parameters which minimizes the collector resistance is the optimum combination.
The peak field at breakdown Ec, obtained from the iteration procedure is shown in Figure 3.3 as a function of open-emitter bulk breakdown voltage BVCBO' Ec decreases with an increase in BVCBO -due to the widening of the depletion layer.
No
No
start
rake specified
•••alue of BVc~o
InitializE' N",
Initialize Ec
Sol •••e eqns.3.5 &
3.6 siMultanE'ous- ly to get If & a
Sol •••e E'qn. 3.1
Sol •••e eqn. 3.7 for R
Fig. 3.2 Flow chart of the computer program for optimization of collector parameters.
l I
tI
,... 1
IE
Ii
u
;; I I
'--'
I0 I
W
-0
I
(l) I
;,;:: II
II
U I
.>:
I+J
u I
(l) ,
(l)
I
I
..:£.
0 t
(l) I
0... I
I
I103
Breakdown voltage Bv"cBo (V)
Fig. 3.3 Numerically computed peak electric field as a function
of open-emitter bulk breakdown voltage Bv"CBO'
The empirical relationship is given as,
Ec=6.72xlO 5BV:'CBO
-3
V/emfor BVCBO voltages in excess of 200 V.
Figure 3.4, 3.5 and 3.6 show the computed data plotted on appropriate axes. The curves allow values of No, a and W numerically computed for a single graded collector that is applied to BVCBO'
The empirical expressions for optimized No, W, a and R are given bY7
W
=
2.64 X 10-6 BVCBOi
23BV:' -~
a = 3.65x 10 CBO
cm
." :The peak electric field Ec for specified open-base bulk breakdown voltage BVCEO
" and current gain hFEO under optimized collector layer decreases with 'an increase in BVCEO again due to widening of the depletion layer, and with an increase in hFEO due to the reduced multiplication needed to achieve the base current IB
=
O~Fig. :3.7 shows the values for Ec obtained numerically. The empirical expression for Ec is given by,for BVCEO in excess of 100 V.
Figs. 3.8,3.9 and 3.10 show the numerically computed data on appropriate axes.
--
,..,
I
E
()'--"
Zo
1
a
15~J
1
a
1310
2 10 3Breakdown voltage BV'cBO
(V)Fig. 3.4 Numerically computed initial doping density as a
function of open-emitter bulk breakdown voltage Bv"CBO'
10 18
10 15
102 103
Breakdown voltage B'I cBo (V)
Fig. 3.5 Numerically computed impurity gradient as a function
of open-emitter bulk breakdown voltage BV.
CBO'c o
:;:::;
Q)
0..
o
Q)I
I I
,...,
In
C
o
L...
.~ 102
3
'-" I
3
~1
J J
~
I
10
102 103
Breakdown voltage B'!cBo (V)
Fig. 3.6 Numerically computed depletion width as a function
of open-emitter bulk breakdown voltage Sv*CBO'
()
W
'"0
Q)
t;:
1 I
I
I
Ii I
hFEo =30 h FEo =50 h FEo =80
102 103
Breakdown voltage BV'cEo
(V)I I
I
I
Fig. 3.7 Numerically computed peak electric field as a
function of open-base bulk breakdown voltage s\f cEO'
10
1501
.-
C0...""0
o
;.-- o
c
10 13
102 103
Breakdown voltage BV cEo
(V)Fig. 3.8 Numerically computed initial doping density as a
function of open-base bulk breakdown voltage BV'cEo,
44
h FEo =30 h FEo =50 h FEo =80
10
14~
,
'~
,
'~
,
'~
,
'~,
'~
,
'~
,
.'~"
'~
,
'~
,
'~
,
'~
,
'~
,
'~
,
'~
,
'~
,
102 103
Breakdown voltage BV'cEo
(V)I
"
-''''/Fig. 3.9 Numerically computed impurity gradient as a function
of open-base bulk breakdown voltage
8V'CEO'-. ~. --
.---
,.-....
(f)
c
01-
.0102
'-' E
3:
1~
-C-+-'
""0 I
.-
~ ..jc
0
1
.-
-+-'Q)0.
Q) ~
0 I
I
10
10 2
Breakdown voltage
I I I I 103
BV eEo
(V)Fig. 3.10 Numerically computed depletion width as a function
of open-base bulk breakdown voltage
8V'CEO'The empirical expressions for No, W, a and R under optimized collector layer are given by,
Figs. 3.11 and 3.12 present the optimized doping density at the collector-substrate interface N(W) and width W as a function of BVCEO including that apply to the uniform layer for comparison.
The optimized parameters for collector region so far analyzed are for epitaxial layers that are able to contain the full depletion width subjected to minimum collector resistance. However, a further lowering of collector layer can be obtained considering fully depleted collector. Optimization of collector layer for high impurity doping with minimum collector resistance leads to a reach-through condition at breakdown.
The voltage supported at low current density under .1\ reach-through condition is,
q 2 qa .3
VB
=
Ec - -NOW2e 1 - -W.6e • (3.9)whereW1is the collector layer width for fully depleted condition. Under the constraint
of E1 ~8 x 104Vfern where E1 is the field at collector-substrate interface, the optimum t values of collector parameters are computed. The results are plotted in Figs. 3.13 to
3.16.
1-.
]
r I
t
1--- 10 15-i I
L-I") ~
01
t)E
~O)U
='-"
0 ~u--- -'
J
0)'-"
3=
..cZ
I .•..•
0) 1
.•..• u
00I
10
14~>,'--
'+-.•..•0)
.- .•..•
(fJc I
c.-
J
0) I
"'00)
.•..•
CJl~c .•..•
•- (fJ 0..0
O:J O(fJ
10 13J
i I 102Breakdown voltage
Graded Graded Uniform Uniform
I I I 1 I
103 .
BV'cEO
(V)I
Fig. 3.11 Numerically computed doping density at the interface as a function of open-base bulk breakdown voltage BV'CEQ.
48
I
I
I
I
II Iii I
10
3Bv"cEO
(V)I I 10 2
Breakdown voltage
10
1
~
I
,I
-\...,; I
~I
h
FEo=30 Graded
I
. - -- . h
FEo=30 Uniform
.--....
en c
0
I...
I
() 2 I
'E
10 -;'-/
~3=
..c ~
-+-'-0
1
.-
~-t
C I
0
I
.-
...J-+-'Q) I,
0-
I
Q) I
0
1
I
I
;
Fig. 3.12 Numerically computed depletion width as a function of open-base bulk breakdown voltage Bv"CEO'
49
,...
E
()"'- >
'--"
()
W -0
t;:Q) ()
'C-+-'
() Q) Q)
.::£.
105"1
0
Q) I
0...
102 103
Breakdown voltage BV cEo (V)
'.
Fig, 3.13 Numerically optimized peak electric field as a
function of open-base breakdown voltage BV
CEO'" 10
15-)1 - h
FEo=20
- h
FEo=50
~
--- h
FEo=80
..• 4
r--. j
,..,
I
E
~I ,
()
,~
...,
..JI',~
0
I ',~
Z -l
,
'..••~,
>, I ,~
+'
,
.- c
(J)I '
,~',~<l)
10
14-1 ,
'~""0
~
,
'~,
01 '~
c j ,
.-
'~0-
,
0 '~
',~
""0
1
',~0
,,
:;;
.- c
10 13
10
210
3Breakdown voltage BV cEo (V)
Fig. 3.14 Numerically optimized initial doping density as a
function of open-base breakdown voltage BV
CEO'10 .,
~
,
'~
,
'~ ,
'~ ,
'~ ,
'~
,
'~ ,
'~
,
'~ ,
'~
, '~ ,
'~
, '~ ,
'~
,
'~
,
'~ ,
'~ ,
'~ , '~ ,
• I
• I
• ; I
102 103
Breakdown voltage BV cEo (V)
Fig. 3.15 Numerically optimized impurity gradient as a function of open-base breakdown voltage BV
cEQ•52
'>-
10
102 103
Breakdown voltage BV cEo (V)
Fig. 3.16 Numerically optimized depletion width as a function
of open-base breakdown voltage BV
CEO•The empirical expressions for Ee, No, a, Wand R are given as,
3.2.2 Double graded collector
The previous analysis for single graded collector imposes limitations on substantial increase on doping density at the collector-substrate interface. For optimized collector, practically increase in doping density is not at all significant. However, the use of an epitaxial collector with two linearly graded regions (Fig. 3.17) allows a significant increase in doping density at the collector-substrate interface at the expense of increased "
manufacturing complexity.
The second portion of the collector can be designed to control the breakdown"
voltage and to increase the critical current density for the onset of avalanche injection.
This double grading allows a substantial increase in critical current density with a small increase of collector resistance. The second region also needs to be optimized to give minimum resistance.
At large current densities, the electric field at the base-collector metallurgical junc- tion becomes zero and the peak electric field occurs in the second graded region.
Base
I-
gradient
a2
gradient
C\
• Wl
---1- w
2---J
Collector epitaxial layer
Collector s\,lbstrate
x
Fig. 3.17 Proposed doping profile of a.double graded collector.
The peak electric field Ec and breakdown voltage BilcED can be expressed as ( Appendix A ),
also,
and the breakdown condition is given by
(3.10)
(3.11)
(3.12) 1
_laW'
-l"(an-ap)dz'd----_ Q'ne 0 z
l+hFED a
The optimized values ofa2 and W2can be determined to achieve maximum collector doping at W at minimum collector resistance. The values are shown in Figs. 3.18 and 3.19. The values of doping density at the collector-substrate interface N(W) and total depletion width W are shown in Figs. 3.20 and 3.21 with their counterparts of uniformly doped collector for comparison.
The empirical expressions for a2 and W2 are written as
6( ~ 11
W2
=
4.95x 10- 1+
hFED)" BVCED'lll' em10
1.
h
FEo=20 h
FEo=50 h
FEo=80
~
,
'~
,
'~
,
'~
,
'~
,
'~
,
'~
,
'~
,
'~
,
'~,
'~
,
'~
,
'~,
'~
,
'~
,
'~,
10 15
102 103
Breakdown voltage BV cEo (V)
Fig. 3.18 Numerically computed second impurity gradient as a
function of open-base breakdown voltage BV
CEO'"...
(J)
c e
.u 10
2E
c o
~
<l>
a.
<l>
o
10
Fig. 3.19
10
210
3Breakdown voltage BV cEo (V)
Numerically computed width of the second
portion of the collector as a function of
open-base breakdown voltage 8V
CEO•Double graded Double graded Uniform
Uniform
10
13102 103
Breakdown voltage BV cEo (V)
I
I
Fig. 3.20 Numerically computed doping density at the interface
as a function of open-base breakdown voltage BV
CEO'l
..Jh
FEo=50 Double graded I
~
- - h
FEo=80 Double graded
0 __ -
h
FEo=50 Uniform ,,9
,...,
rnI I
.---h
FEo=80 Uniform #
c l ,,9 ,
0
"
l....
,~
(J
#
,~.- ,~
-.S
10J J ,,9 # ,~ ,~ ,~
3:.
I ;'~1 ,,9 ,~
.c ...
I,~
-l-'
j# ,~
""0
J ,~
.-
~ !,,9 ,~
i
,~
c J # ,~
0
I ,~
:;:; !
,,9 ,~
Q)
1 ,~
0..
# ,~
Q)
,~
0
J ,~ ,~
,~
,~~
10 !I
-J,
10 2 10 3
Breakdown voltage BV
cEo(V)
Fig. 3.21 Numerically computed total depletion width as a
function of open-base breakdown voltage BV
CEO'3.3 Conclusions.
An analysis is presented to show that a bipolar transistor can be made resistant to failure from avalanche injection by incorporating graded impurity profiles within the collector. Calculations have been performed for both single and double graded collector layers and a comparison has been made with those of uniformly doped collector. In de- signing a transistor, the optimal parameters are usually evaluated under certain specified constraints, such as open-emitter breakdown voltage, maximum current gain hFEO, etc.
subject to minimum collector resistance. Meeting the requirements of specified break- down voltage and the minimum collector resistance, the computed optimal values of the collector parameters show that a. significant improvement can not be achieved with a.
single graded collector. However, the increase in permissible current density by using a double graded collector offers an improved measure of protection against failure by avalanche injection.
CHAPTER 4
CONCLUSIONS
In this work, an analysis is presented to show that epitaxial bipolar power transis- tors can be made resistant to failure from avalanche injection (leading to current mode second breakdown) by incorporating an appropriately graded impurity profile within the colleelor. The improvement has been achieved directly from an increase in the critical current density required to trigger avalanche injection. Calculations for the op- timal values of the collector parameters have been performed for both single and double
,
graded collector layers and a comparison has been made with those of uniformly doped colleelor. The results have been presented in a form that allows the optimum collector parameters for minimum collector resistance to be deduced, subject to a given open-base breakdown voltage BVCEO and maximum current gain hFEO.
The results demonstrate that a well designed transistor with single graded collector does not give any significant increase in collector doping density. On the other hand, the results show that a transistor with double graded collector offers the prospect of achiev- ing a substantial measure of protection against device failure from avalanche injection.
In chapter 3, the numerically computed results for both double graded and uniform collector are plotted in Iligs. 3.20 and 3.21 for collector doping density and collector width respectively. For a transistor with double graded colledmof open-base break- down voltage BVCEO =200 volts and maximum current gain hFEO = 50, the numerical solution gives the doping density at the collector-substrate interface N(W)
=
1.1X 1016cm-3 and the total collector width W = 51.4 /lm. The optimal values of collector doping density and width of its counterparts, uniform collector have been found to be 2.76 x 1014 cm-3 and 23.5 Ijm respeelively. The above comparison clearly shows that a transistor with double graded collector will provide high current protection with a
penalty in term of collector width compared to that offered by an optimum uniform collector.
In this thesis, the critical current densit,y for a transistor with double graded col- lector at the onset of avalanche injection has not been evaluated either numerically or analytically. An analysis for determination of the current density at the onset of avalanche injection may be conducted in future.
63
APPENDIX A
Calculation of impurity gradient and depletion width for second portion of the double graded collector
At a collector current density higher than J(Wt}, the peak electric field occurs in the second region and the field at the base-collector junction decreases to zero. The peak electric field moves towards the collector-substrate interface with further increase of current and at a current density Jo ::::qv,(No
+
alWI+
a2W2), the peak electric field occurs at the interface. The substrate is assumed to be heavily doped and the electric field sharply decreases to zero at the substrate. The field distribution considered for the evaluation ofa2 and W2 is shown in Fig. A.I. It is here considered that the electric field outside the second portion of the collector will be zero at the onset of avalanche injection. In this work, it is also assumed that the second portion will sustain the total breakdown voltage BVCEO•The basic equation to be solved is,
"
,"
,
.
:' i
, I
t ,
",J
---EdE ::::p - n
+
N(z)q dz (A.I)
where n and p are, respectively, the electron and hole concentrations at any point x, q is the electron charge and N(x) is the doping density in the collector at x. For
WI
<
z<
(WI+
W2).At the onset of avalanche injection, generation is sma.ll and therefore, the hole concentration p can safely be neglected and Poisson's equation becomes,
64
.
Base
gradient
a2
gradient
~
W1 ..-
---"f-- ""2--+1
Collector epitaxial layer
x
Collector substrate
•
"j,'1•••
Fig. A.IElectric field distribution within the collector at a current density J(W1
+
W2).The current density at which the peak electric field occurs at the interface is given as Jo
=
qv.(No+
alW1+
a2W2) and n= ft,
and then the eJectron concentration at the interface is given by,Substituting n from eqn.(A.4) into eqn.(A.3), it can be shown that, dE qa2
-= -(W2-z)
dz E
Integrating eqn.(A.5) and putting the boundary condition E(W2)
=
Ec,Ec--- qa2
w.
222E And the sustainable voltage is,
(A.4)
(A.5)
(A.G)
(A.7)
I.' •
BmLIOGRAPHY
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