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Effect of avalanche-induced light emission on the multiplication factor in bipolar junction transistors
SoWSfare ElectronicsVol. 34, No. 11, pp. 1191-I I%, 1991 Printed in Great Britain.All rightsreserved
0038-I 101/91 $3.00 + 0.00 Copyright0 1991 PergamonPressplc
EFFECT OF AVALANCHE-INDUCED LIGHT EMISSION ON THE MULTIPLICATION FACTOR IN BIPOLAR JUNCTION TRANSISTORS SHENGLYANG
JANG
Department of Eleotrouic Engineering, National Taiwan Institute of Technology, Taipei, Taiwan 10772, Republic of China (Received 21 March 1991; in revised form 6 May 1991)
Abstract-In Si bipolar junction transistors, the base and oolleotor ourrents, including avalanche effects, are oBen modeled without acoounting for the avalanche-induced light emission. The light absorption and light-induced carrier injection into the avalanche region were therefore negleoted. In this paper, we formulate the multiplication factor so as to acoouut for the avalanche-induced carrier injection into the breakdown region and we discuss the mechanism neglected earlier.
1. INTRODUCTION
In n-p-n Si bipolar junction transistors, the base and collector currents, including avalanche effects, are often modeled as[l-51: Z, = M’Zf
and Ze=G-(M’-l)Zf,
(1)
where M’ is the base-collector
avalanche multiplication factor, Z, and I,, are the base and collector currents, including avalanche multiplication, Zf and Is are the primary collector and base currents. The meaning of M’ is the ratio of the current observed with avalanche multiplication present to the current that would exist under the same conditions if we
could somehow turn off the avalanche mechanism. The multiplication ratio 5 reads: { =(M’-l)=&B
=
generated component of current Z, without avalanche *
(2)
In Si semiconductors, we can observe avalancheinduced light emission phenomena in reverse-biased p-n junctions[6], in MOSFETs[q and in BJTs[8]. The photon emission mechanisms could be[6-8]: (1) phonon-assisted, avalanche-induced electron-hole recombination; (2) intraband transition by holes between the split-off and normal valance bands; and (3) bremsstrahlung, that is, the “breaking” radiation of hot electrons in the Coulomb field of a charged center. The avalanche-induced light emission and the light absorption will affect the avalanche process, and the multiplication factor may depend on the lightabsorption process. However, the above equations do not take the light emission and absorption into account.
The light absorption will change the current injected from the base-emitter (BE) junction into the breakdown junction, i.e. the base-collector (BC) junction. Therefore the multiplication factor M’ defined above could not exactly describe the physical mechanisms in the avalanche breakdown. Hence, we formulate the multiplication factor so as to account for avalanche-induced carrier injection into the breakdown region and discuss the mechanism neglected in the previous literature. 2. MULTIPLICATION FACTOR
In this section, to conform to our previous publication[8], we discuss the case for an n-p-n bipolar junction transistor (BIT) with a reverse-biased BE junction and EB voltage V, larger than the BE breakdown voltage BV,, . We describe the case for a BIT with regions of uniform impurity density. In an n-p-n transistor, when Vcb> BV,,, we can get the BC photovoltage[8] such as shown in Fig. 1. This proves that the photon absorption occurs in the BC junction. The photon absorption can change the carrier injection into the BE junction from the case without the breakdown in the BE junction. When the EB junction is reverse-biased, the carrier transport
v,, l volt 1 Fig. 1. Direct current open BC voltage vs
1191
EB voltsge I’, for the transistors in Ref. [8].
SHENG-LYANGJNUG
1192 dx
Emitter
Base
Collector
-lb I
I I
I
I I
I
IE
I n
I -)rI,(O)
-bli-w In I It-l+% I I
I
I
I
I
I
I I
I
I
I
I
l
I
I
I
I
I
I
I
I
P
n
-
G
IC
Fig. 2. Schematic of an n-p-n transistor.
between the EB junction and the BC junction creates a negative BC voltage. With increasing reverse biased voltage V,, impact ionization occurs. The impact ionization induces light emission in the BE junction, and the light is absorbed in the whole device volume. The photocarrier diffusion in the quasi-neutral base and collector regions, and the photocarrier separation in the BC depletion region create a positive open BC voltage. The photovoltage effect may dominate the open BC voltage. Figure 2 shows the transistor schematic, where -X0, -X, , 0, A’, , Xc and X, are the emitter ohmic contact, the EB space-charge-region (SCR) edge on the emitter side, the EB SCR edge on the base side, 0
Z,(O) =
-IS
Fig. 2, in the infinitesimally thin layer of thickness dx, we can write the hole-current increment dl,(x) as: dZ,(x) = cr,Z,(x) dx + a,Z,(x)
I,(x) = Z,(x)+l,(x). Equation
-(a,-a,)dx”
dx’ 1
-&
1
1s
+I,(-&) (7)
J
[I 1 - 1 exp UO-Cap x’
Ia,&+ #Jlexp
-&
1
-an)& --Cap
-&
0
(6)
(5) leads to:
PO
IJ
Similarly, we can get:
(5)
where the a, and an values are defined as the number of electron-hole pairs generated by an electron and a hole per unit distance traveled due to impact ionization. U is the hole generation rate due to light absorption in the avalanche region. Here, we neglect the thermal emission of holes and electrons in deep levels. The total emitter current can be expressed as:
* U
[a,4 + qlllexp -‘G Iw
dx + qU,
-(a,-a,)dx”
dx’
-4
(8)
a,) dx’
-&
L
the BC SCR edge on the base side, the BC SCR edge on the collector side and the collector ohmic contact, respectively. We now formulate the multiplication factor in BJTs. Assuming no diffusion current, we let[9]:
Define the multiplication
factor M as:
4
h-z =Z,(--x,)+z,(o)’
(9)
Equation (7) yields:
14- uo)l =
(3)
GZ.+ I,(-&)+I, H
’
(104
where
and 1,(x ) = qAne,,
(4)
at position x. Here A is the emitter area, p and n are the carrier concentrations, up and v” are the average hole and electron carrier velocities. Refer now to
-(a, - a,) dx”
H =exp
[Se
0 _x -(a,-a,)dx’
]
dx’,
(lob)
1 uw
Avalanche-induced light emission in BJTs and Z*=~~~~(lnp[~~~~-(a~.,-a.)dr.ldr.. Equation
WV
1193
where L* = ,/D nbrnb,c 1 and c2 are constants. If the light creates low-level injection and the flat quasiFermi level approximation across the Bc[lO] is applicable, we can write the boundary conditions as:
(8) leads to: An(Xb) = g z, - I,(-X.)
= Hz”(o)+ YZS+ zs,
(lla)
{exp(qybe/kT)
- l},
(17)
lb
and
where
&r(O) = _“, Y=[~X~exp[~~X~-(a,,-a~)dx”]dx’.
(llb)
where n, is the intrinsic carrier density, Vbc is voltage, T is the electron temperature and k is BoltxmaM constant. N.,, is the concentration shallow acceptors in the base. From eqns (16-18) get:
Using eqns (1 la) and (1 lb), we get: H-G-1-Y and the multiplication H+(l-z-z) M-
factor
zpc-x,>
(18) ab
4
An(x)=
-
r,Gx,)+r,(o) + zp(-x,)+z”(o)
earl
BC the of we
exp(-ax)
Dd’J2- Lii2)
l-Y
The multiplication factor M depends on the selfconsistent electron current Z,(O), hole current I,(-&) and absorption current I, in the avalanche region. When Z,(O)<
Q[exp(Xb/L,,) - exp(-aXb)]
x
+ 0 + Q[exp(-aXb)
1 MC-1-Y
+$-exp(-XblL,)
1
-
eXp(-d%/L,d
exp(x/L
nb
) 1,
(l9)
ab
=l-J”~a~exp[~~~~(a~-a~)~~~]~”
(13)
The impact-ikixation al&s Z,,(O),ZP(-X,) and Zs, the currents Z,,(O),I,( -X,) and Z, in turn affect the avalanche process. Next, we calculate currents Z,,(O)and I,( -X,). The generation rate G(x) of electron-hole pairs at distance x from the BE depletion edge on the base side is: G(x) = @art exp(-ax), (14) where a is the absorption coefficient of the semiconductor, and 9 is the quantum efficiency for electron-hole creation. @ is the light intensity generated by the impact ionization in the BIT at x = 0. In the quasi-neutral base, we assume there is no electric field, carriers flow only by diffusion, the minority carrier continuity equation reads: D d2An An nb--z+9atjexp(-ax)=0, &2
(1%
where f,,b is the lifetime of electrons, An is the excess electron concentration, Dmbis the electron diffusion constant. The solution of eqn (15) is: An =cl
0 =
2
{exp(qVbc /kT) - 1)
and Q= D&Z2
@q -
Lg2)’
The electron current density diffusing from the base to the collector is:
d-b
+
Q[exp(Xb/L,,,) - exp(-aXb)J-0
+ - f
eXp(Xb,L,,)
exp( -X6/L,)
ab
+ 0 - Q[exp(-uXb)-exp(-X,IL,&]
exp(-x/L,,)+c2exp(x/L,,)
@a9 exp(-ax), - D,,b(a2- Lg2)
where
(16)
+ $-exp(-xb/L,,b) 8b
SHENO-LYANG JANG
1194
The electron current ionization region is:
density
entering
the impact
where Lp, = JDFfF, c3 and c4 are constants. boundary conditions can be written as:
gab% x-0
Ap(Xc) = $
Pm2 @nt. = +(~2-Lii2)+2L
&Xco)
Q [exp(Xb /Lnb) - exp( - al%)] - 0
Ap(x) = -
- $- exp(Xb/L,,)
+
@atl D,(a’-
(25)
L;‘)
exp(-ax)
2 sinh
- exp(-Xb/Lnb)]
x[{(R + T)exp(W/L,)-Texp(-aW)}
$abexp(-Xb/Lnb)
.
(214
x exp((Xc - x)/L,)
>I Using eqn (2la), we can write the electron current Z,(O) entering the impact ionization region as: Z”(0) = Z,(O)+Znp(O),
(21b)
+ { T[exp( - a W) - exp( - W/L,)] -R exp(- W/L,)}exp(x
- Xc/L,)],
(26)
where
where
w=xco
Ad%,
4n(0)=
= 0,
(24)
1
+
ab
+ 0 + Q[exp(-aXb)
l},
where ZVdc is the concentration of shallow dopants in the collector. From eqns (23-25), we get:
nb
X
{exp(qVbc/kT)-
The
-xc,
R = ${exp(qVbc/kT)-
1)
Lnbsinh and x${exp(qVbc/kT)-I}+?$
Aq@qa2
exp(-axc)’
X-XC
2L,,bsinh +[--{Q[eXP(xb/Ld
=
-(u2
4@w2 _ Lp2j
exP(-aXc)
-exP(-cXZ’)])
- aXb) - exp( - xb /L,b)]}]-
Here Z.. represents the current injected into the BE junction without avalanche, and Zw denotes the current injected to the impact ionizatton region owed to the photon generation processes. In the quasi-neutral collector region, the 1-D steady-state continuity equation for minority carriers reads: D
Lp,2)
J,,(Xc) = -qD,g
A&b
+ { Q h-4
_
The hole current density diffusing from the collector to the base is:
(a2- L$2)
+
D,(a2
ab
and Z,(O) =
@al
T= nb
d2& AP --q+@arZ pEdx2
-{T[exp(-aW)-exp(-R
W/L,)]
exp(-WIL,I)}I.
The generation current in the BC junction
(27) region
is: J,, = q@ exp( - ax,) {1 - exp[ -a (Xc - NJ)]). (28)
exp(-ax)=O,
(22)
where rF is the lifetime of holes, Ap is the excess hole concentration, DP is the hole diffusion constant. The solution of eqn (22) is: Ap = c3 exp(-x/L,)+c4
x[{(R + T)exp(W/L,)-Texp(-aW)}
exp(x/L,)
@w - Dpc(a2 - L;*)
exp(-ax),
(23)
The collector current density Jc can be expressed as Jc = Jdbc+ J,,(Xb)+J,(Xc).
(29)
getting Jc = 0, we get the open BC voltage Vbc. In the quasi-neutral emitter region, the 1-D steadystate continuity equation for minority carriers reads: D
d2Ap AP ---++‘arZexp[a(x+Xe)]=O, pe cl2 rP
(30)
Avalanche-induced light emission in BJTs where T, is the lifetime of holes, DP is the hole diffusion constant. @’ is the light intensity seen at x= - Xe. The solution of eqn (30) is: Ap=cSexp(-(x+&)/L,)
The hole current diffusing from the emitter to the base can be rewritten as: I,(-Xe)=A.ZP(-Xe)=Z,(-Xe)fZ,(-Xe),(36) where
+ c6 exp((x + Xe)/L,)
qA@‘qa2
(31)
Ap(-xe)
P x {Z[exp( - au) - exp( - U/LJ] -Z[exp(U/L,)-exp(-au)]}.
= -$
(32) de
and Ap( -Xeo)
= 0.
@‘al
exp[a (x + Xe)]
- Lp2)
I,,(-Xe)
1
=
qAD,
u
Lps sinh r 0
(33)
where Ndc is the concentration of shallow donors in the emitter. From eqns (31-33), we get: D,(a’
qAD,
4p(-Xe)=(a2-L;2)+2L
where LP = &, c5 and c6 are constants. The boundary conditions take the form:
Ap(x) = -
1195
(37) (38)
$exp(-U/L,).
de
Here ZPTrepresents the c&rent diffusing to the BE junction without avalanche and Z, denotes the current diffusing to the impact ionization region owed to the photon generation processes. Using eqns (13,21b, 36), under the BE breakdown condition, we arrive at the multiplication factor
+
H+ x
M=
exp( - U/L,)
f de
x exp( - (x + Xe)/L,) +
U/L,)
-$-exp(1
1-Y
*
(39)
As we can see, the multiplication factor M depends on Z8, I,( -X,) and Z,(O), which in turn result from the photon absorption process. In Si bipolar junction transistors, the base and collector currents, including avalanche effects, now can be modeled as:
- exp(- U/L,)]
+Z[exp(-aU)
(1 - ~)~Zpp(-~,)+Z,(-x,)I+Z* &,(-X)+Z,,(-~,)+Z,(0)+Znp(0)
+ Z[exp(U/L,)
de
Z, = MK(O)+Z,(-X,)1 and
-ew(---VI
evKx
+
XeY&l ,
U=Xeo-Xe
Z, = MZ,,,,
@'atj and
D,(a2 - Lp2)’
The hole current density diffusing from the emitter to
= -qDPg
x- -xe
q@‘tfa2 =(a2 - Lp’) + 2L
X
+
qDP sinh U ( L, >
de
Z[exp( - a U) - exp( - U/L,)]
-
-$exp(-
U/LJ
{ +Z[exp(U/L,)
- exp(-au)]
(41)
3. CONCLUSIONS
exp(- U/L,)
$ [{
P
zs=z,-I,.
This is equivalent to the statement in eqn (1). In the derivation of eqn (39), we assume the avalancheinduced light emission is monochromatic. We can extend the above theory to cover the whole emission spectrum.
the base is: J,(-Xe)
(40)
Using eqn (39), and neglecting the current due to the photon absorption process, i.e. neglecting ZB, I,(-&) and Z,(O), we obtain:
where
z=
z,=z,-I,.
(34)
II
. (35)
The multiplication factor is conventionally defined as the ratio of current observed with avalanche multiplication present to the current without avalanche multiplication. This has neglected the effects of the impact-ionization-induced light emission. The empirical expession for the multiplication factor is often approximated by Miller’s relationship, with exponent n between 1 and 7[1,9,11]. However,
SHENG-LYANG JANG
1196
Miller’s equation is not very accurate[9]. The difficulty in using an exact empirical formulation in describing the multiplication factor may be inherent in the definition of the multiplication factor. In this paper, the formulation of the avalanche multiplication factor M in Si bipolar transistors has been revised, taking into account the avalancheinduced light emission, light absorption and lightinduced current injection. This revision makes the multiplication factor really describing the ratio of the current created by impact ionization processes to the current injected into the avalanche region. This makes the interpretation of the multiplication factor meaningful. REFERENCES
1. S. M. Sze, Physics of Semiconductor Dwices. Wiley, New York (1981).
2. J. J. Liou and J. S. Yuan, IEEE Trans. Electron Devices ED-37, 2274 (1990). 3. R. W. Dutton, IEEE Trans. Electron Devices ED-22, 334 (1975). 4. K. Sakui et al.. IEEE Trans. Electron Devices ED-36. 1215 (1989). 5. P.-F. Lu and T.-C. Chen, IEEE Trans. Electron Dwices ED-36, 1182 (1989). 6. D. K. Gauta& W.S. Khokle and K. B. Garg, Solid-St. Electron. 31, 219 (1988). 7. A. Toriumi, M. Yoshimi, M. Iwase, Y. Akiyama and K. Taniguichi, IEEE Trans. Electron Dwices 34, 1501 (1987). 8. S.-L. Jang and K.-L. Chem, Hot-carrier-induced photovoltage in silicon bipolar junction transistors. To be published (1991). 9. R. M. Warner Jr and B. L. Gmng, Transistors: Fundamentals for the IC Engineer. Wiley, New York (1983). 10. S.-L. Jang, Solid-St. Electron. 34, 373 (1991). 11. C. D. Bulucea and D. C. Prisecam, IEEE Trans. Electron Devices ED-20, 692 (1973).
0038-I 101/91 $3.00 + 0.00 Copyright0 1991 PergamonPressplc
EFFECT OF AVALANCHE-INDUCED LIGHT EMISSION ON THE MULTIPLICATION FACTOR IN BIPOLAR JUNCTION TRANSISTORS SHENGLYANG
JANG
Department of Eleotrouic Engineering, National Taiwan Institute of Technology, Taipei, Taiwan 10772, Republic of China (Received 21 March 1991; in revised form 6 May 1991)
Abstract-In Si bipolar junction transistors, the base and oolleotor ourrents, including avalanche effects, are oBen modeled without acoounting for the avalanche-induced light emission. The light absorption and light-induced carrier injection into the avalanche region were therefore negleoted. In this paper, we formulate the multiplication factor so as to acoouut for the avalanche-induced carrier injection into the breakdown region and we discuss the mechanism neglected earlier.
1. INTRODUCTION
In n-p-n Si bipolar junction transistors, the base and collector currents, including avalanche effects, are often modeled as[l-51: Z, = M’Zf
and Ze=G-(M’-l)Zf,
(1)
where M’ is the base-collector
avalanche multiplication factor, Z, and I,, are the base and collector currents, including avalanche multiplication, Zf and Is are the primary collector and base currents. The meaning of M’ is the ratio of the current observed with avalanche multiplication present to the current that would exist under the same conditions if we
could somehow turn off the avalanche mechanism. The multiplication ratio 5 reads: { =(M’-l)=&B
=
generated component of current Z, without avalanche *
(2)
In Si semiconductors, we can observe avalancheinduced light emission phenomena in reverse-biased p-n junctions[6], in MOSFETs[q and in BJTs[8]. The photon emission mechanisms could be[6-8]: (1) phonon-assisted, avalanche-induced electron-hole recombination; (2) intraband transition by holes between the split-off and normal valance bands; and (3) bremsstrahlung, that is, the “breaking” radiation of hot electrons in the Coulomb field of a charged center. The avalanche-induced light emission and the light absorption will affect the avalanche process, and the multiplication factor may depend on the lightabsorption process. However, the above equations do not take the light emission and absorption into account.
The light absorption will change the current injected from the base-emitter (BE) junction into the breakdown junction, i.e. the base-collector (BC) junction. Therefore the multiplication factor M’ defined above could not exactly describe the physical mechanisms in the avalanche breakdown. Hence, we formulate the multiplication factor so as to account for avalanche-induced carrier injection into the breakdown region and discuss the mechanism neglected in the previous literature. 2. MULTIPLICATION FACTOR
In this section, to conform to our previous publication[8], we discuss the case for an n-p-n bipolar junction transistor (BIT) with a reverse-biased BE junction and EB voltage V, larger than the BE breakdown voltage BV,, . We describe the case for a BIT with regions of uniform impurity density. In an n-p-n transistor, when Vcb> BV,,, we can get the BC photovoltage[8] such as shown in Fig. 1. This proves that the photon absorption occurs in the BC junction. The photon absorption can change the carrier injection into the BE junction from the case without the breakdown in the BE junction. When the EB junction is reverse-biased, the carrier transport
v,, l volt 1 Fig. 1. Direct current open BC voltage vs
1191
EB voltsge I’, for the transistors in Ref. [8].
SHENG-LYANGJNUG
1192 dx
Emitter
Base
Collector
-lb I
I I
I
I I
I
IE
I n
I -)rI,(O)
-bli-w In I It-l+% I I
I
I
I
I
I
I I
I
I
I
I
l
I
I
I
I
I
I
I
I
P
n
-
G
IC
Fig. 2. Schematic of an n-p-n transistor.
between the EB junction and the BC junction creates a negative BC voltage. With increasing reverse biased voltage V,, impact ionization occurs. The impact ionization induces light emission in the BE junction, and the light is absorbed in the whole device volume. The photocarrier diffusion in the quasi-neutral base and collector regions, and the photocarrier separation in the BC depletion region create a positive open BC voltage. The photovoltage effect may dominate the open BC voltage. Figure 2 shows the transistor schematic, where -X0, -X, , 0, A’, , Xc and X, are the emitter ohmic contact, the EB space-charge-region (SCR) edge on the emitter side, the EB SCR edge on the base side, 0
Z,(O) =
-IS
Fig. 2, in the infinitesimally thin layer of thickness dx, we can write the hole-current increment dl,(x) as: dZ,(x) = cr,Z,(x) dx + a,Z,(x)
I,(x) = Z,(x)+l,(x). Equation
-(a,-a,)dx”
dx’ 1
-&
1
1s
+I,(-&) (7)
J
[I 1 - 1 exp UO-Cap x’
Ia,&+ #Jlexp
-&
1
-an)& --Cap
-&
0
(6)
(5) leads to:
PO
IJ
Similarly, we can get:
(5)
where the a, and an values are defined as the number of electron-hole pairs generated by an electron and a hole per unit distance traveled due to impact ionization. U is the hole generation rate due to light absorption in the avalanche region. Here, we neglect the thermal emission of holes and electrons in deep levels. The total emitter current can be expressed as:
* U
[a,4 + qlllexp -‘G Iw
dx + qU,
-(a,-a,)dx”
dx’
-4
(8)
a,) dx’
-&
L
the BC SCR edge on the base side, the BC SCR edge on the collector side and the collector ohmic contact, respectively. We now formulate the multiplication factor in BJTs. Assuming no diffusion current, we let[9]:
Define the multiplication
factor M as:
4
h-z =Z,(--x,)+z,(o)’
(9)
Equation (7) yields:
14- uo)l =
(3)
GZ.+ I,(-&)+I, H
’
(104
where
and 1,(x ) = qAne,,
(4)
at position x. Here A is the emitter area, p and n are the carrier concentrations, up and v” are the average hole and electron carrier velocities. Refer now to
-(a, - a,) dx”
H =exp
[Se
0 _x -(a,-a,)dx’
]
dx’,
(lob)
1 uw
Avalanche-induced light emission in BJTs and Z*=~~~~(lnp[~~~~-(a~.,-a.)dr.ldr.. Equation
WV
1193
where L* = ,/D nbrnb,c 1 and c2 are constants. If the light creates low-level injection and the flat quasiFermi level approximation across the Bc[lO] is applicable, we can write the boundary conditions as:
(8) leads to: An(Xb) = g z, - I,(-X.)
= Hz”(o)+ YZS+ zs,
(lla)
{exp(qybe/kT)
- l},
(17)
lb
and
where
&r(O) = _“, Y=[~X~exp[~~X~-(a,,-a~)dx”]dx’.
(llb)
where n, is the intrinsic carrier density, Vbc is voltage, T is the electron temperature and k is BoltxmaM constant. N.,, is the concentration shallow acceptors in the base. From eqns (16-18) get:
Using eqns (1 la) and (1 lb), we get: H-G-1-Y and the multiplication H+(l-z-z) M-
factor
zpc-x,>
(18) ab
4
An(x)=
-
r,Gx,)+r,(o) + zp(-x,)+z”(o)
earl
BC the of we
exp(-ax)
Dd’J2- Lii2)
l-Y
The multiplication factor M depends on the selfconsistent electron current Z,(O), hole current I,(-&) and absorption current I, in the avalanche region. When Z,(O)<
Q[exp(Xb/L,,) - exp(-aXb)]
x
+ 0 + Q[exp(-aXb)
1 MC-1-Y
+$-exp(-XblL,)
1
-
eXp(-d%/L,d
exp(x/L
nb
) 1,
(l9)
ab
=l-J”~a~exp[~~~~(a~-a~)~~~]~”
(13)
The impact-ikixation al&s Z,,(O),ZP(-X,) and Zs, the currents Z,,(O),I,( -X,) and Z, in turn affect the avalanche process. Next, we calculate currents Z,,(O)and I,( -X,). The generation rate G(x) of electron-hole pairs at distance x from the BE depletion edge on the base side is: G(x) = @art exp(-ax), (14) where a is the absorption coefficient of the semiconductor, and 9 is the quantum efficiency for electron-hole creation. @ is the light intensity generated by the impact ionization in the BIT at x = 0. In the quasi-neutral base, we assume there is no electric field, carriers flow only by diffusion, the minority carrier continuity equation reads: D d2An An nb--z+9atjexp(-ax)=0, &2
(1%
where f,,b is the lifetime of electrons, An is the excess electron concentration, Dmbis the electron diffusion constant. The solution of eqn (15) is: An =cl
0 =
2
{exp(qVbc /kT) - 1)
and Q= D&Z2
@q -
Lg2)’
The electron current density diffusing from the base to the collector is:
d-b
+
Q[exp(Xb/L,,,) - exp(-aXb)J-0
+ - f
eXp(Xb,L,,)
exp( -X6/L,)
ab
+ 0 - Q[exp(-uXb)-exp(-X,IL,&]
exp(-x/L,,)+c2exp(x/L,,)
@a9 exp(-ax), - D,,b(a2- Lg2)
where
(16)
+ $-exp(-xb/L,,b) 8b
SHENO-LYANG JANG
1194
The electron current ionization region is:
density
entering
the impact
where Lp, = JDFfF, c3 and c4 are constants. boundary conditions can be written as:
gab% x-0
Ap(Xc) = $
Pm2 @nt. = +(~2-Lii2)+2L
&Xco)
Q [exp(Xb /Lnb) - exp( - al%)] - 0
Ap(x) = -
- $- exp(Xb/L,,)
+
@atl D,(a’-
(25)
L;‘)
exp(-ax)
2 sinh
- exp(-Xb/Lnb)]
x[{(R + T)exp(W/L,)-Texp(-aW)}
$abexp(-Xb/Lnb)
.
(214
x exp((Xc - x)/L,)
>I Using eqn (2la), we can write the electron current Z,(O) entering the impact ionization region as: Z”(0) = Z,(O)+Znp(O),
(21b)
+ { T[exp( - a W) - exp( - W/L,)] -R exp(- W/L,)}exp(x
- Xc/L,)],
(26)
where
where
w=xco
Ad%,
4n(0)=
= 0,
(24)
1
+
ab
+ 0 + Q[exp(-aXb)
l},
where ZVdc is the concentration of shallow dopants in the collector. From eqns (23-25), we get:
nb
X
{exp(qVbc/kT)-
The
-xc,
R = ${exp(qVbc/kT)-
1)
Lnbsinh and x${exp(qVbc/kT)-I}+?$
Aq@qa2
exp(-axc)’
X-XC
2L,,bsinh +[--{Q[eXP(xb/Ld
=
-(u2
4@w2 _ Lp2j
exP(-aXc)
-exP(-cXZ’)])
- aXb) - exp( - xb /L,b)]}]-
Here Z.. represents the current injected into the BE junction without avalanche, and Zw denotes the current injected to the impact ionizatton region owed to the photon generation processes. In the quasi-neutral collector region, the 1-D steady-state continuity equation for minority carriers reads: D
Lp,2)
J,,(Xc) = -qD,g
A&b
+ { Q h-4
_
The hole current density diffusing from the collector to the base is:
(a2- L$2)
+
D,(a2
ab
and Z,(O) =
@al
T= nb
d2& AP --q+@arZ pEdx2
-{T[exp(-aW)-exp(-R
W/L,)]
exp(-WIL,I)}I.
The generation current in the BC junction
(27) region
is: J,, = q@ exp( - ax,) {1 - exp[ -a (Xc - NJ)]). (28)
exp(-ax)=O,
(22)
where rF is the lifetime of holes, Ap is the excess hole concentration, DP is the hole diffusion constant. The solution of eqn (22) is: Ap = c3 exp(-x/L,)+c4
x[{(R + T)exp(W/L,)-Texp(-aW)}
exp(x/L,)
@w - Dpc(a2 - L;*)
exp(-ax),
(23)
The collector current density Jc can be expressed as Jc = Jdbc+ J,,(Xb)+J,(Xc).
(29)
getting Jc = 0, we get the open BC voltage Vbc. In the quasi-neutral emitter region, the 1-D steadystate continuity equation for minority carriers reads: D
d2Ap AP ---++‘arZexp[a(x+Xe)]=O, pe cl2 rP
(30)
Avalanche-induced light emission in BJTs where T, is the lifetime of holes, DP is the hole diffusion constant. @’ is the light intensity seen at x= - Xe. The solution of eqn (30) is: Ap=cSexp(-(x+&)/L,)
The hole current diffusing from the emitter to the base can be rewritten as: I,(-Xe)=A.ZP(-Xe)=Z,(-Xe)fZ,(-Xe),(36) where
+ c6 exp((x + Xe)/L,)
qA@‘qa2
(31)
Ap(-xe)
P x {Z[exp( - au) - exp( - U/LJ] -Z[exp(U/L,)-exp(-au)]}.
= -$
(32) de
and Ap( -Xeo)
= 0.
@‘al
exp[a (x + Xe)]
- Lp2)
I,,(-Xe)
1
=
qAD,
u
Lps sinh r 0
(33)
where Ndc is the concentration of shallow donors in the emitter. From eqns (31-33), we get: D,(a’
qAD,
4p(-Xe)=(a2-L;2)+2L
where LP = &, c5 and c6 are constants. The boundary conditions take the form:
Ap(x) = -
1195
(37) (38)
$exp(-U/L,).
de
Here ZPTrepresents the c&rent diffusing to the BE junction without avalanche and Z, denotes the current diffusing to the impact ionization region owed to the photon generation processes. Using eqns (13,21b, 36), under the BE breakdown condition, we arrive at the multiplication factor
+
H+ x
M=
exp( - U/L,)
f de
x exp( - (x + Xe)/L,) +
U/L,)
-$-exp(1
1-Y
*
(39)
As we can see, the multiplication factor M depends on Z8, I,( -X,) and Z,(O), which in turn result from the photon absorption process. In Si bipolar junction transistors, the base and collector currents, including avalanche effects, now can be modeled as:
- exp(- U/L,)]
+Z[exp(-aU)
(1 - ~)~Zpp(-~,)+Z,(-x,)I+Z* &,(-X)+Z,,(-~,)+Z,(0)+Znp(0)
+ Z[exp(U/L,)
de
Z, = MK(O)+Z,(-X,)1 and
-ew(---VI
evKx
+
XeY&l ,
U=Xeo-Xe
Z, = MZ,,,,
@'atj and
D,(a2 - Lp2)’
The hole current density diffusing from the emitter to
= -qDPg
x- -xe
q@‘tfa2 =(a2 - Lp’) + 2L
X
+
qDP sinh U ( L, >
de
Z[exp( - a U) - exp( - U/L,)]
-
-$exp(-
U/LJ
{ +Z[exp(U/L,)
- exp(-au)]
(41)
3. CONCLUSIONS
exp(- U/L,)
$ [{
P
zs=z,-I,.
This is equivalent to the statement in eqn (1). In the derivation of eqn (39), we assume the avalancheinduced light emission is monochromatic. We can extend the above theory to cover the whole emission spectrum.
the base is: J,(-Xe)
(40)
Using eqn (39), and neglecting the current due to the photon absorption process, i.e. neglecting ZB, I,(-&) and Z,(O), we obtain:
where
z=
z,=z,-I,.
(34)
II
. (35)
The multiplication factor is conventionally defined as the ratio of current observed with avalanche multiplication present to the current without avalanche multiplication. This has neglected the effects of the impact-ionization-induced light emission. The empirical expession for the multiplication factor is often approximated by Miller’s relationship, with exponent n between 1 and 7[1,9,11]. However,
SHENG-LYANG JANG
1196
Miller’s equation is not very accurate[9]. The difficulty in using an exact empirical formulation in describing the multiplication factor may be inherent in the definition of the multiplication factor. In this paper, the formulation of the avalanche multiplication factor M in Si bipolar transistors has been revised, taking into account the avalancheinduced light emission, light absorption and lightinduced current injection. This revision makes the multiplication factor really describing the ratio of the current created by impact ionization processes to the current injected into the avalanche region. This makes the interpretation of the multiplication factor meaningful. REFERENCES
1. S. M. Sze, Physics of Semiconductor Dwices. Wiley, New York (1981).
2. J. J. Liou and J. S. Yuan, IEEE Trans. Electron Devices ED-37, 2274 (1990). 3. R. W. Dutton, IEEE Trans. Electron Devices ED-22, 334 (1975). 4. K. Sakui et al.. IEEE Trans. Electron Devices ED-36. 1215 (1989). 5. P.-F. Lu and T.-C. Chen, IEEE Trans. Electron Dwices ED-36, 1182 (1989). 6. D. K. Gauta& W.S. Khokle and K. B. Garg, Solid-St. Electron. 31, 219 (1988). 7. A. Toriumi, M. Yoshimi, M. Iwase, Y. Akiyama and K. Taniguichi, IEEE Trans. Electron Dwices 34, 1501 (1987). 8. S.-L. Jang and K.-L. Chem, Hot-carrier-induced photovoltage in silicon bipolar junction transistors. To be published (1991). 9. R. M. Warner Jr and B. L. Gmng, Transistors: Fundamentals for the IC Engineer. Wiley, New York (1983). 10. S.-L. Jang, Solid-St. Electron. 34, 373 (1991). 11. C. D. Bulucea and D. C. Prisecam, IEEE Trans. Electron Devices ED-20, 692 (1973).