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Calculation of the collector signal delay time of HBT's based on a piecewise-linear velocity profile
Solid-Stare
Elecrronics Vol. 36, No. 5, pp. 693-696,
1993
0038-I 101193 $6.00 + 0.00 Copyright Q 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
CALCULATION OF THE COLLECTOR SIGNAL DELAY TIME OF HBT’S BASED ON A PIECEWISE-LINEAR VELOCITY PROFILE J. J. Ltou and A. ZAJAC Electrical
and Computer
Engineering Department, University Orlando, FL 32816, U.S.A.
of Central
Florida,
(Received 3 October 1992; in revised form 5 November 1992) present a systematic method for calculating the frequency-dependent collector signal delay time r;T and the cutoff frequency fr of the heterojunction bipolar transistor (HBT). The method is developed based on the assumption thatf, is limited by r i-r. Furthermore, a piecewise-liner drift velocity profile is employed to account for velocity overshoot in the baseecollcctor junction. Previous works employing the low-frequency approximation or the step-wise drift velocity profile, which assumes a constant velocity in the overshoot region, are also briefly reviewed and discussed. Abstract-We
1. INTRODUCTION
2. MODEL DEVELOPMENT
The collector signal delay time r&r in the heterojunction bipolar transistor (HBT) has become a point of interest because it is often the limiting factor for the cutoff frequency fr of a HBT[l], particularly for a self-aligned HBT’s in which protons are implanted in the extrinsic collector to reduce the basecollector junction capacitance[2]. r&r is conventionally estimated as zcr/2 and:
For the piecewise-linear profile, the overshoot drift velocity r+,(x) between 0
dx/O),
(2)
where up is the peak drift velocity occurs at x = 0 and v, is the average velocity between W, and WC [Fig. l(b)]. Note that v, can be larger or smaller than v, depending on the bias conditions[6]. A general 7& model can be developed using the delay time concept[4] as follows. The induced current Jind through the basecollector junction has a form of:
WC
7CT =
= up + (a, - v,)x/Wo,
(1)
s0 where ~~7 is the collector transit time, IV, is the thickness of the base-collector space-charge layer, and u(x) is the free-carrier drift velocity. If v(x) is assumed equal to the constant saturation velocity v, (z 10’ cm/s), then 7CT = WC iv, and 7& = W,/2v,. In addition to the constant saturation velocity, other more realistic velocity profiles such as a step velocity profile [Fig. l(a)][3,4] and a piecewiselinear velocity profile [Fig. l(b)][5] taking into account velocity overshoot have been considered. Laux and Lee[3] have used the step profile as an example of their general formula and calculated 7& as a function of frequency for such a profile. Ishibashi[4] also calculated t& using the step profile, but he employed the low-frequency assumption which results in a frequency-independent 7& model. The purpose of this work is to develop an analytical and explicit 7& model based on the piecewiselinear profile without employing the low-frequency assumption. Under most bias conditions, the piecewise-linear velocity profile [Fig. l(b)] resembles closely the velocity profile in the base-collector generated by Monte Carlo junction region simulations[6,7]. 693
Jind = JO +j(t),
(3)
where J, is the d.c. current and j(t) is the a.c. current. Assuming j(t) has a sinusoidal waveform with an angular frequency w yields[4]: j(t) = j,(sin wr&/wzcr)sin Jlnd can also be expressed
w(t - 5cr).
(4)
as [4]: WC
Jmd
=
M,
(l/W,)
+j’(tN
dx,
(5)
5 (I where j’(t) is expressed in term of the drift velocity in the base-collector space-charge layer: i.(i)=j~sinw[f-~~d~/v,e)], for 0 < x < W,, j’(t)=j,sinw
t [
0
W0 d.x/v,(x)
s
- (X - Wo)lva
for W, < x < WC,
(6)
1, (7)
where: a% dx/v,(x)
s0
= [Wo/(r~ -
~,)Ilwe(~,lv,), (8)
ZAIAC
J. J. Llou and A.
694
In the case of low-frequency limitation. I/o, can be assumed much larger than W,,/o, and ( W,. - Wo)/q. This leads to [4]:
(a)
x “c .z
I
7CT=.[WoIL’c+(WC-
0”
Wo)
x [w,/~,+(wc-W")~,l/WC1/2 (14)
5 G
_---------
“s
‘Z
a
I I I 0
The results of T;‘~ as a function of w for the step-like velocity profile and r& in the low-frequency limitation for the piecewise-linear velocity profile were given in Refs [3] and [5], respectively. We now solve the frequency-dependent T;., lion-. the general model given in (9)-( I I). Equation (9) can be written as:
c
Wc
WO
x
(b)
(sin y)’ = yF(w),
0
WC
WO
x
Fig. 1. (a) The step-like drift velocity profile consisting a constant velocity U, in the overshoot region and the saturation velocity v, in the saturation region. (b) The piecewiselinear drift velocity profile consisting a linearly dependent velocity in the overshoot region and an average U, velocity in the rest of the space-charge region. II, can be smaller or larger than U, depending on the bias conditions.
is the time needed for electrons to travel from x = 0 to x = w,. Putting (6H8) into (5) and equating the resulting equation with (4), we obtain: (sin WT&)~/WZ& = (l/W,)
where: C, = [Wo/(& -
rr;, sin WC, dx [S 0 WC sin WC, dx + s WQ
=[w,/(v,-v,)]log,(v,/~,)+(x
=
3. ILMJSTRATION
1.6
- WJQ.
AND DISCUSSION
Figure 3(a)+b) plots F(w) vs the frequency for two velocity profiles reported in [6] (the profiles for V,, = 1.53V and VBE= 1.55 V, respectively, in Fig. 6 of Ref. 6). It can be seen that F is very small when the frequency is low and F starts to oscillate as the frequency is increased beyond 10” rddjs. If we assume z&(wT), where wT = 27-11~is the cutoff radian frequency, is the limiting factor for ~1~. as often the case for HBT’s, then:
1.8
Note that t&- is a function of w. Equation (9) can be simplified for some cases. For example, for a step-like velocity [Fig. l(a)], Q(X) = u, (u, is the constant velocity velocity overshoot region), and (9) is reduced (sin
where )’ = ruz& and F(a)) equals the right-hand side of (9). Equation (IS) is a transcendental equation, and there can be many solutions (eigenvalues) exist. These eigenvalues J, (i = 0, 1. 2,. ), which are the solutions of Y, are shown in Fig. 2. The number of the eigenvalues increases as F(w), which is the slope of the linear lint, is decreased. For instance, if F(w) = 0. then there are infinite eigenvalues and J,, = .)I, = 0, y2 = 3.14, .vj = 6.28. and etc. Note that JJ,)= 0 is the trivial eigenvalue for any F(w).
(9)
~,mx,~K4i - qJx x Wo+qm,~,
c,
1,
(15)
(10) (11)
-
(siny)*
---
yF(w)
1.4 1.2
/
special profile in the to:
WT~~)*/WT;~~
(l/w W,){2u, sin(w W,/2v,)sin
oC,
+ 2u, sin[w ( WC - W,)/Zu,]sin WC,},
0
(12)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4,s
5.0
y2
Y = Qx= CT
where: c, = W,/2v,,
0.5 yo y1
and
C, = Wo/vc + (W, - W,)/2&.
(13)
Fig. 2. Schematic illustration of the relation given in eqn (9), where _v~,~‘,.Y~, are the eigenvalues of the equation.
695
Collector signal delay time Profile I: VI, = 6 x 107 cm/s 100
(a)
10’
’
F
(a)
1
WC= 0.5 tnn
10-l
---
Profile I Profile II
10-2 10-3 5. -
1o-4 10-T
c4
10-6 10-7
w. -
10-S
Profile
= 0.1 pm
I
’ 1111n’ ’ Ii-
100 b 103
10-g
105
10’ o
10-10 103
105
107 co
109
1011
---
10’0
I I I,,,, II
1011
1012 w
1013
101 *b)
Profile I Profile II
I I ,I11111
10-3
1011
1013
(rad/s)
1 1
109
(rad/s)
100
I , /,I,,,
1
---
Profile Profile
’ ’ ’ I1lll’
10’0
Fig. 3. (a) F(o) calculated for two different velocity profiles. (b) Detailed F(w) characteristics at high frequencies.
Since we are interested in finding 7cr when the cutoff frequency occurs, only y, or y, is of interest because they are the eigenvalues most likely to satisfy the relation specified in (16) (see Fig. 2). It is important to point out that as F(w) is increased, the values of yi and yr are increased and decreased, respectively, toward unity. As a result, either y, or y2 can be used in finding w,; the difference is that y, is always smaller than unity and y, is always greater than unity for all frequencies. We choose y, in the following calculations. Figure 4(a, b) shows yz(w) characteristics for Profiles I and II. According to (16) the frequency at which y, is closest to unity is the cutoff frequency wr, and T&(u,) can be calculated from y2(wr)/wr. Thus jr z 64 GHz and r&r -N 2.5 x lo-” s for Profile I and ff z 200 GHz and 7& x 1.3 x lo-l2 s for Profile II. These values are somewhat larger than those obtained from the Monte Carlo simulation reported in predicted fro 50 GHz and Ref. [6], which Jr x 140 GHz for Profiles I and II, respectively. The overestimation of the present model may be due to the assumption that the cutoff frequency is limited only by the collector signal delay time. Note that yr(w) calculated for Profiles I and II [Fig. 4(b)] are larger than unity, and the reason for this occurrence was given in the discussions following (16).
’ ’ ’ fflll’
10”
1013
(fad/s)
I II
’ ’ ‘11lll’
1012
1013
o (rad/s)
Fig. 4. (a) y2(o) calculated for Profiles I and II. (b) Detailed y*(o) characteristics at high frequencies. Clearly the validity of the present model relies on the assumption that 7& is the dominant factor in the overall delay time. Otherwise the relation given in (16) does not hold and the proposed method in determining the cutoff frequency would not work. To assess this, we use a first-order model[8] and calculate the four delay times (the emitter charging time TV, base transit time 7a7, collector charging time tc, and collector signal delay time t&r) vs the collector current density for a typical HBT considered in [6], as 10-7
2s
10-a
-
10-g
-----
2
lo-lo
.!j
10-11
z-3
2
S
=E
. . . . . . r BT z
C +’ CT
---
10-12 10-13 10-14 lo-15
M 1
10
Collector
100
1000
current
10,000
density
100,000
(A/cm’)
Fig. 5. The four delay times governing the cutoff frequency versus the collector current density.
J. J. LIOU and A.
696
shown in Fig. 5. It indicates that, for the collector current density range most HBT’s are designed to operate (between IO4 and lO”A/cm’), the collector signal delay time is the largest among the four delay times. The method developed thus should be reasonably accurate inside this current range.
ZAJAC
(3) From the results in (2), determine both the cutoff radian frequency wT and the collector signal t&(mT) subjected to the relation of delay T;,(W,)UT+ I. Acknowledgemenl-This DSRjUCF Development
work was supported in part by the Grant (account No. 16-22-923).
4. CONCLUSION
In summary, a model is developed based on the delay time approach that can be used to determine both the cutoff frequency and the collector signal delay time of HBT’s. The present model considers a piecewise-linear velocity profile, rather than a constant or step profile used in the previous works, to account for velocity overshoot in the collector depletion region. The procedure is briefly summarized as follows. (I) Select the drift velocity profile (the peak velocity in the overshoot region and the average velocity in the rest of the collector junction region) in the HBT. (2) Input the velocity profile into the model and calculate the non-trivial eigenvalue yz as a function of the frequency w.
REFERENCES
I. P. Asbeck, Technology
Proc. 1989 Bipolar Trunsistor Meeting, Minneapolis (1989).
Circuit
and
2. K. N. Nagata, 0. Nakajima, Y. Yamauchi, T. Nittono, H. Ito and T. Ishibashi, IEEE Trans. Electron Deuices ED-35, 2 (1988). 3. S. E. Laux and W. Lee, IEEE Electron Device Left. 11, 174 (1990). lEEE Tram. Electron Deuice.v 37, 2103 4. T. Ishibashi, (1990). 5. J. J. Liou and H. Shakouri, Solid-Sl. Elecrron. 35, 15 (1992). 6. R. Katoh and M. Kurata, IEEE Truns. Electron Devices 36, 2122 (1989). 7. P. 1. Rockett. IEEE Trans. Elecfron Deoicrs ED-35, 1573 (1988). 8. M. Shur. GaAs Devices and Circuits. Plenum Press. New York (1987).
Elecrronics Vol. 36, No. 5, pp. 693-696,
1993
0038-I 101193 $6.00 + 0.00 Copyright Q 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
CALCULATION OF THE COLLECTOR SIGNAL DELAY TIME OF HBT’S BASED ON A PIECEWISE-LINEAR VELOCITY PROFILE J. J. Ltou and A. ZAJAC Electrical
and Computer
Engineering Department, University Orlando, FL 32816, U.S.A.
of Central
Florida,
(Received 3 October 1992; in revised form 5 November 1992) present a systematic method for calculating the frequency-dependent collector signal delay time r;T and the cutoff frequency fr of the heterojunction bipolar transistor (HBT). The method is developed based on the assumption thatf, is limited by r i-r. Furthermore, a piecewise-liner drift velocity profile is employed to account for velocity overshoot in the baseecollcctor junction. Previous works employing the low-frequency approximation or the step-wise drift velocity profile, which assumes a constant velocity in the overshoot region, are also briefly reviewed and discussed. Abstract-We
1. INTRODUCTION
2. MODEL DEVELOPMENT
The collector signal delay time r&r in the heterojunction bipolar transistor (HBT) has become a point of interest because it is often the limiting factor for the cutoff frequency fr of a HBT[l], particularly for a self-aligned HBT’s in which protons are implanted in the extrinsic collector to reduce the basecollector junction capacitance[2]. r&r is conventionally estimated as zcr/2 and:
For the piecewise-linear profile, the overshoot drift velocity r+,(x) between 0
dx/O),
(2)
where up is the peak drift velocity occurs at x = 0 and v, is the average velocity between W, and WC [Fig. l(b)]. Note that v, can be larger or smaller than v, depending on the bias conditions[6]. A general 7& model can be developed using the delay time concept[4] as follows. The induced current Jind through the basecollector junction has a form of:
WC
7CT =
= up + (a, - v,)x/Wo,
(1)
s0 where ~~7 is the collector transit time, IV, is the thickness of the base-collector space-charge layer, and u(x) is the free-carrier drift velocity. If v(x) is assumed equal to the constant saturation velocity v, (z 10’ cm/s), then 7CT = WC iv, and 7& = W,/2v,. In addition to the constant saturation velocity, other more realistic velocity profiles such as a step velocity profile [Fig. l(a)][3,4] and a piecewiselinear velocity profile [Fig. l(b)][5] taking into account velocity overshoot have been considered. Laux and Lee[3] have used the step profile as an example of their general formula and calculated 7& as a function of frequency for such a profile. Ishibashi[4] also calculated t& using the step profile, but he employed the low-frequency assumption which results in a frequency-independent 7& model. The purpose of this work is to develop an analytical and explicit 7& model based on the piecewiselinear profile without employing the low-frequency assumption. Under most bias conditions, the piecewise-linear velocity profile [Fig. l(b)] resembles closely the velocity profile in the base-collector generated by Monte Carlo junction region simulations[6,7]. 693
Jind = JO +j(t),
(3)
where J, is the d.c. current and j(t) is the a.c. current. Assuming j(t) has a sinusoidal waveform with an angular frequency w yields[4]: j(t) = j,(sin wr&/wzcr)sin Jlnd can also be expressed
w(t - 5cr).
(4)
as [4]: WC
Jmd
=
M,
(l/W,)
+j’(tN
dx,
(5)
5 (I where j’(t) is expressed in term of the drift velocity in the base-collector space-charge layer: i.(i)=j~sinw[f-~~d~/v,e)], for 0 < x < W,, j’(t)=j,sinw
t [
0
W0 d.x/v,(x)
s
- (X - Wo)lva
for W, < x < WC,
(6)
1, (7)
where: a% dx/v,(x)
s0
= [Wo/(r~ -
~,)Ilwe(~,lv,), (8)
ZAIAC
J. J. Llou and A.
694
In the case of low-frequency limitation. I/o, can be assumed much larger than W,,/o, and ( W,. - Wo)/q. This leads to [4]:
(a)
x “c .z
I
7CT=.[WoIL’c+(WC-
0”
Wo)
x [w,/~,+(wc-W")~,l/WC1/2 (14)
5 G
_---------
“s
‘Z
a
I I I 0
The results of T;‘~ as a function of w for the step-like velocity profile and r& in the low-frequency limitation for the piecewise-linear velocity profile were given in Refs [3] and [5], respectively. We now solve the frequency-dependent T;., lion-. the general model given in (9)-( I I). Equation (9) can be written as:
c
Wc
WO
x
(b)
(sin y)’ = yF(w),
0
WC
WO
x
Fig. 1. (a) The step-like drift velocity profile consisting a constant velocity U, in the overshoot region and the saturation velocity v, in the saturation region. (b) The piecewiselinear drift velocity profile consisting a linearly dependent velocity in the overshoot region and an average U, velocity in the rest of the space-charge region. II, can be smaller or larger than U, depending on the bias conditions.
is the time needed for electrons to travel from x = 0 to x = w,. Putting (6H8) into (5) and equating the resulting equation with (4), we obtain: (sin WT&)~/WZ& = (l/W,)
where: C, = [Wo/(& -
rr;, sin WC, dx [S 0 WC sin WC, dx + s WQ
=[w,/(v,-v,)]log,(v,/~,)+(x
=
3. ILMJSTRATION
1.6
- WJQ.
AND DISCUSSION
Figure 3(a)+b) plots F(w) vs the frequency for two velocity profiles reported in [6] (the profiles for V,, = 1.53V and VBE= 1.55 V, respectively, in Fig. 6 of Ref. 6). It can be seen that F is very small when the frequency is low and F starts to oscillate as the frequency is increased beyond 10” rddjs. If we assume z&(wT), where wT = 27-11~is the cutoff radian frequency, is the limiting factor for ~1~. as often the case for HBT’s, then:
1.8
Note that t&- is a function of w. Equation (9) can be simplified for some cases. For example, for a step-like velocity [Fig. l(a)], Q(X) = u, (u, is the constant velocity velocity overshoot region), and (9) is reduced (sin
where )’ = ruz& and F(a)) equals the right-hand side of (9). Equation (IS) is a transcendental equation, and there can be many solutions (eigenvalues) exist. These eigenvalues J, (i = 0, 1. 2,. ), which are the solutions of Y, are shown in Fig. 2. The number of the eigenvalues increases as F(w), which is the slope of the linear lint, is decreased. For instance, if F(w) = 0. then there are infinite eigenvalues and J,, = .)I, = 0, y2 = 3.14, .vj = 6.28. and etc. Note that JJ,)= 0 is the trivial eigenvalue for any F(w).
(9)
~,mx,~K4i - qJx x Wo+qm,~,
c,
1,
(15)
(10) (11)
-
(siny)*
---
yF(w)
1.4 1.2
/
special profile in the to:
WT~~)*/WT;~~
(l/w W,){2u, sin(w W,/2v,)sin
oC,
+ 2u, sin[w ( WC - W,)/Zu,]sin WC,},
0
(12)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4,s
5.0
y2
Y = Qx= CT
where: c, = W,/2v,,
0.5 yo y1
and
C, = Wo/vc + (W, - W,)/2&.
(13)
Fig. 2. Schematic illustration of the relation given in eqn (9), where _v~,~‘,.Y~, are the eigenvalues of the equation.
695
Collector signal delay time Profile I: VI, = 6 x 107 cm/s 100
(a)
10’
’
F
(a)
1
WC= 0.5 tnn
10-l
---
Profile I Profile II
10-2 10-3 5. -
1o-4 10-T
c4
10-6 10-7
w. -
10-S
Profile
= 0.1 pm
I
’ 1111n’ ’ Ii-
100 b 103
10-g
105
10’ o
10-10 103
105
107 co
109
1011
---
10’0
I I I,,,, II
1011
1012 w
1013
101 *b)
Profile I Profile II
I I ,I11111
10-3
1011
1013
(rad/s)
1 1
109
(rad/s)
100
I , /,I,,,
1
---
Profile Profile
’ ’ ’ I1lll’
10’0
Fig. 3. (a) F(o) calculated for two different velocity profiles. (b) Detailed F(w) characteristics at high frequencies.
Since we are interested in finding 7cr when the cutoff frequency occurs, only y, or y, is of interest because they are the eigenvalues most likely to satisfy the relation specified in (16) (see Fig. 2). It is important to point out that as F(w) is increased, the values of yi and yr are increased and decreased, respectively, toward unity. As a result, either y, or y2 can be used in finding w,; the difference is that y, is always smaller than unity and y, is always greater than unity for all frequencies. We choose y, in the following calculations. Figure 4(a, b) shows yz(w) characteristics for Profiles I and II. According to (16) the frequency at which y, is closest to unity is the cutoff frequency wr, and T&(u,) can be calculated from y2(wr)/wr. Thus jr z 64 GHz and r&r -N 2.5 x lo-” s for Profile I and ff z 200 GHz and 7& x 1.3 x lo-l2 s for Profile II. These values are somewhat larger than those obtained from the Monte Carlo simulation reported in predicted fro 50 GHz and Ref. [6], which Jr x 140 GHz for Profiles I and II, respectively. The overestimation of the present model may be due to the assumption that the cutoff frequency is limited only by the collector signal delay time. Note that yr(w) calculated for Profiles I and II [Fig. 4(b)] are larger than unity, and the reason for this occurrence was given in the discussions following (16).
’ ’ ’ fflll’
10”
1013
(fad/s)
I II
’ ’ ‘11lll’
1012
1013
o (rad/s)
Fig. 4. (a) y2(o) calculated for Profiles I and II. (b) Detailed y*(o) characteristics at high frequencies. Clearly the validity of the present model relies on the assumption that 7& is the dominant factor in the overall delay time. Otherwise the relation given in (16) does not hold and the proposed method in determining the cutoff frequency would not work. To assess this, we use a first-order model[8] and calculate the four delay times (the emitter charging time TV, base transit time 7a7, collector charging time tc, and collector signal delay time t&r) vs the collector current density for a typical HBT considered in [6], as 10-7
2s
10-a
-
10-g
-----
2
lo-lo
.!j
10-11
z-3
2
S
=E
. . . . . . r BT z
C +’ CT
---
10-12 10-13 10-14 lo-15
M 1
10
Collector
100
1000
current
10,000
density
100,000
(A/cm’)
Fig. 5. The four delay times governing the cutoff frequency versus the collector current density.
J. J. LIOU and A.
696
shown in Fig. 5. It indicates that, for the collector current density range most HBT’s are designed to operate (between IO4 and lO”A/cm’), the collector signal delay time is the largest among the four delay times. The method developed thus should be reasonably accurate inside this current range.
ZAJAC
(3) From the results in (2), determine both the cutoff radian frequency wT and the collector signal t&(mT) subjected to the relation of delay T;,(W,)UT+ I. Acknowledgemenl-This DSRjUCF Development
work was supported in part by the Grant (account No. 16-22-923).
4. CONCLUSION
In summary, a model is developed based on the delay time approach that can be used to determine both the cutoff frequency and the collector signal delay time of HBT’s. The present model considers a piecewise-linear velocity profile, rather than a constant or step profile used in the previous works, to account for velocity overshoot in the collector depletion region. The procedure is briefly summarized as follows. (I) Select the drift velocity profile (the peak velocity in the overshoot region and the average velocity in the rest of the collector junction region) in the HBT. (2) Input the velocity profile into the model and calculate the non-trivial eigenvalue yz as a function of the frequency w.
REFERENCES
I. P. Asbeck, Technology
Proc. 1989 Bipolar Trunsistor Meeting, Minneapolis (1989).
Circuit
and
2. K. N. Nagata, 0. Nakajima, Y. Yamauchi, T. Nittono, H. Ito and T. Ishibashi, IEEE Trans. Electron Deuices ED-35, 2 (1988). 3. S. E. Laux and W. Lee, IEEE Electron Device Left. 11, 174 (1990). lEEE Tram. Electron Deuice.v 37, 2103 4. T. Ishibashi, (1990). 5. J. J. Liou and H. Shakouri, Solid-Sl. Elecrron. 35, 15 (1992). 6. R. Katoh and M. Kurata, IEEE Truns. Electron Devices 36, 2122 (1989). 7. P. 1. Rockett. IEEE Trans. Elecfron Deoicrs ED-35, 1573 (1988). 8. M. Shur. GaAs Devices and Circuits. Plenum Press. New York (1987).