- Email: [email protected]
Effective threshold energy for pair production in nonpolar semiconductors
Solid State Communications, Vol. 20, pp. 169—172, 1976.
Pergamon Press.
Printed in Great Britain
EFFECTWE THRESHOLD ENERGY FOR PAIR PRODUCTION IN NONPOLkR SEMICONDUCTORS* R. Chwang and C.R. Crowd Departments of Electrical Engineering and Materials ‘Science, University of Southern California, Los Angeles, Ca. 90007, U.S.A. (Received 5Apr11 1976 byA.G. Chynoweth) A streaming model (high field) analysis is given for the average energy for and the probability of electron—hole pair production in a semiconductor when a quadratically energy dependent impact ionization cross-section exists above a threshold energy and competes with a nonpolar optical phonon scattering mechanism. A power series expansion method and tabulated results are provided to 3)}dx. treat the resulting probability integrals of the form f(’x” exp {—03x +x 1. INTRODUCTION PAIR PRODUCTION by impact ionization at high electric field in a semiconductor often characterized 14 is The parameters are, the by a three parameter theory. ionization threshold energy, ET, the nonpolar optical phonon scatteringmean free path, X 0~,,and the average energy loss per optical phonon scattering, E0~.When the carrier energy exceeds the threshold energy for pair production, the carrier is subjected to both phonon scattering and impact ionization. From density of states considerations alone the mean free path for ionization above the threshold should5’6 be has quadratically dependent on derived a simple model thedescribe carrier energy. Keldysh to this energy dependence of the impact ionization cross-section. In terms of the ionization mean free path X, for carriers an energy E above the threshold energy this can be expressed as
x~
~P
\E~)
(x)
=
ET+(EJ)+(n)EOP
scattering above the threshold ET is then pT(E)
=
±I 1 + 1’ ex E /‘ 1 1 q~kxop x) + (a + bx2) exP( (ax + ~~x3)Jdx ~—
~{— J (c
—
dE’ -~-~
—
)ci~ (3)
where x a
E
(4) (5)
ET ql~ X~,
/
b
ET
P
(6)
Note that, the energy E is measured from the threshold
(2)
where (x) is the mean distance required for pair production, (n) is the average number of phonon colisions and (E1) is defined as the average additional energy required for impact ionization above the threshold ET. Thus,
=
and
into consideration, the ionization coefficient a at electric field ~ then has to be expressed as24
=
carriers that reach the threshold energy ET are streaming along the field. The total density function for carrier
(1)
where P is a dimensionless quantity which is usually much greater than unity. X~is naturally infinite for carrier energy less than ET. Ifwe take this finite energydependent ionization cross-section above the threshold
a
required for pair production. The purpose of this paper is to derive an analytic expression for the term (E 1> based on a streaming model and to provide a method of handling the particular probability integrals encountered in such a case. At very high electric field, we wifi assume that
energy ET. The individual density function that corresponds to optical phonon scattering, p 0~(ff)dE,is p0~(E)dE= aexp (_(~+~3))~ (7)
f_
}
and to impact ionization, p1(E)dE, 2 exp (axis + ~~x3) dx Pi(E)dE = bx
(8)
ET + (E 1> *
represents an effective threshold energy
Research supported by the United States Army Research Office, under Grant No. DAHCO4-75-9-0002.
The moments of these distributions are of particular interest in that they determine the average energy gained and distance travelled before scattering. Computing the moments of the above three probability functions wifi 169
170
PAIR PRODUCTION IN NONPOLAR SEMICONDUCTORS
f
require the following evaluation of the integral,
x~exp + ~x3)) ~ Through a change of variable, we can rewrite equation(9)
where
5
\(n+1)/3
b
3)}dx /
=
‘1/3 1
fo(13)
=
I ~
(
(3) ~.PJ
E~
\q~X
)
.
Jfexp{_~x+x3)}dx.
(12)
and fi (13)
3((3)—
13f,,,..2((3).
(13)
Equation (13) can be used to decompose all the higher order f~(j3)(ii ~ 2) into the following two basic functions 3) exp (—(13x + x3)}dx (14) fo(1
$
and 3 )}dX
=
— df0 d13(13)
x exp {—(13x + x
(15)
Equations (14) and (15) cannot be integrated in simple closed form. However, if we expand the term exp (—13x) in the integrand, we obtain the following power series representations, 3 ) dx — j3 Jx exp (—x3) ~ o exp (—x 0 +~ x~exp (—x3 )dx
$
—~ ~ x 133 ~
exp (—x3)dx
1 ~
3
and
no
r
-~
3n!
n=o
2
~“
+
(_13)3r~1
~-)}.
1
(19)
147 10
+...
(3n
— 2)
r
P~n+ 3) 2
/
=
3fl
258 11.. (3n
=
—1) ~
(-i-) (i-)
(20)
(21)
3fl
=
P(n + 1)
(22)
n!,
one can compute f0 (13) and f1 (13) fairly easily via equations (18) and (19). 3. RESULTS AND DISCUSSIONS Table 1 lists the calculated values off 0 (13) and ft (13) for a range of 13 from 0 to 3.0. These computations 13were used, performed less than in 35 double terms in precision. the expansion Withininthe equations range of (l6Tand (1 7) were required to achieve convergence beyond the 14th significant figure. From equation (11), if the parameters ET and X~,are known for a given material, the scale of 13 can be appropriately converted into values of P. The main quantity of interest,
rbE2 lbE3
(16) n!
(3n + 2)! P (n + 1))
—
+ (3n + 1)! r (n + 1) (13)~+2 ~ (n + + (3n + 2)!
/3)1/3
~÷l) 3
I + 3)
I (~13)~fl /
~
=
+
3f~(13)= (n — 2)f~_
=
j3)~’~“
(18)
Via integrated by pal relationship
$
~
/
9ts, we obtain the following recurrence
=
‘~
With the aid of the standard Gamma function identities8
We denote the definite integral of interest as
f0 (13)
In + 2
(11)
2. METHOD OF COMPUTATION
f
~ I(
+ (3~)’ ~ (n +
0~/
f~ (13)
(i3)
t~—~--—)
\2/3
3 } in the integrand of equation term exp {—x (10) The can be viewed as a modified error function,7 which is further damped by the exp (—j3x) term. Thus, mathematically speaking, the expression (10) is proportional to the nth moment of a damped modified error function distribution.
f~(fl)
~
x~exp (—(13x +x
1/3
a(~-)
(10)
1
=
Toterms ease the ~ n! we can further (17) group in foactual and f1computation in the following manner.
(9) in a more convenient form
(
~
Vol. 20, No.3
=
3]}dE exp {— [aif + (b/3)E3]}dE exp (— [aE+ (b/3)E
___ f
0(i3)—13f~(i3) 1 ‘l3fo(6)
(23)
Vol. 20, No.3
PAIR PRODUCTION IN NONPOLAR SEMICONDUCTORS
Table 1. fotfl) and f~(fl),13= 0(0.1)3.0
13
foO3)
f~ (13)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
0.89297 95 0.84946 05 0.80899 40 0.771 32 34 0.73621 67 0.70346 33 0.67287 23 0.64427 02 0.61749 92 0.59241 61 0.5688899 0.54680 17 0.5260426 0.50651 34 0.48812 33 0.4707892 0.45443 51 0.43899 10 0.42439 32 0.41058 27 0.39750 55 0.38511 20 0.37335 63 0.3621962 0.35159 29 0.34151 05 0.33191 56 0.32277 77 0.3140683 0.30576 10 0.29783 16
0.45 137 26 0.41947 88 0.39027 92 0.36351 82 0.33896 67 0.31641 90 0.29569 04 0.27661 48 0.2590425 0.2428388 0.22788 23 0.2140634 0.20128 31 0.18945 18 0.17848 86 0.1683201 0.1588798 0.1501073 0.1419480 0.13435 20 0.12727 41 0.12067 30 0.1145112 0.1087543 0.10337 13 0.09833 35 0.09361 47 0.08919 12 0.08504 11 0.0811442 0.07748 23
171
Figure 1 is a plot of these three normalized moments for a range of 13 from 0 to 3.0. From this plot, we observe that the quantity
~ F—
bJ
~
9j.oo
0.50
1~00
150
VALUE OF
$
2~00
2.50
3.00
Fig. 1. Plot of the three normalized moments
~
a: a: a: a:
Using the definition of (I from equation (11),
(E1) q&X0~=
~
fo(L3)13fi03) lI3fo(i3)
(24)
=
q~X0~
13ff ((3)
~
00
0~S0
100
1 50
VALUE OF
Similarly, we can deduce the other normalized first moments corresponding to the two density functions defined in equations (3) and (7). Thus 3fo(13) (25)
___________________________ ~9j.
(26)
where
2.00
2~00
3.00
~
Fig. 2. Plot of the probability P1(13) that the next scattering will be via pair production vs the value of (3 from 0.0 to 3.0. tion to the scattering probability impact for large (3. total However, at large values of due (3, thetocontribuionization is small. Thus the moments (E 8)/q~X0~and
172
PAIR PRODUCTION IN NONPOLAR SEMICONDUCTORS
Vol. 20, No.3
ihe factor P in equation (1) can only be approxicarriers after each phonon scattering. Since the streaming mated within an order of magnitude for a given material approximation is only valid at extremely high fields, one must also account for the scattering trajectories to e.g. P 102 for Si and Ge.5 If we assume an electric field at 4 x 1 05 V/cm and that the values of ET and ~, deduce the exact effect of an energy dependent ionizafor electrons in Si at room temperature are 1.1 eV and tion cross-section on the avalanche coefficient. A study 48 A respectively,3 then equation (11) specifies that the using this approach (i.e. a characterization as a Markov range of (3 goes from 0.5 to 1.0 for a range ofF from process) will be published later. However, the present 204 to 25. Thus within an uncertainty factor of 3 in the simple treatment provides a useful upper bound for the value of P, we can still deduce impact ionization paraeffective threshold energy, and provides order of magmeters with reasonable accuracy from Figs. 1 and 2. We nitude estimates for the effect of a quadratic energy conclude that for carriers reaching beyond the threshold, dependent impact ionization cross-section on the. 0.5 ± 0.1 of them will create a new electron—hole pair secondary carrier generation. This is true because during their first scattering event, and on the average, fortunately P is not of the order of unity. If this were the impact ionization event takes place at 0.6 ±0.2 of so, then one would need to consider a strong effect q~X 0~ above the threshold energy. associated with successive phonon scattering of carriers In general, in a more exact treatment, one can of energy greater than the threshold energy. follow the successive energy distribution of the primary
REFERENCES 1. 2.
SHOCKLEY W., Solid StateElectron. 2,35 (1961). BARAFFG.A.,Phys. Rev. 128, 2507 (1962).
3. 4.
OKUTO Y. & CROWELL C.R., Phys. Rev. B6, 3076 (1972). OKUTO Y. & CROWELL C.R., Phys. Rev. BlO, 4284 (1974).
5.
KELDYSH L.V.,Sov. Phys. JETP 10,509 (1960).
6.
KELDYSH L.V., Soy. Phys. JETP 21, 1135 (1965).
7. 8.
ABRAMOWITZ M.,J. Math. Phys. 30,162(1951). ABRAMOWITZ M. & SEGUN A., Handbook ofMathematical Functions, p. 25~.Dover, New York (1968).
Pergamon Press.
Printed in Great Britain
EFFECTWE THRESHOLD ENERGY FOR PAIR PRODUCTION IN NONPOLkR SEMICONDUCTORS* R. Chwang and C.R. Crowd Departments of Electrical Engineering and Materials ‘Science, University of Southern California, Los Angeles, Ca. 90007, U.S.A. (Received 5Apr11 1976 byA.G. Chynoweth) A streaming model (high field) analysis is given for the average energy for and the probability of electron—hole pair production in a semiconductor when a quadratically energy dependent impact ionization cross-section exists above a threshold energy and competes with a nonpolar optical phonon scattering mechanism. A power series expansion method and tabulated results are provided to 3)}dx. treat the resulting probability integrals of the form f(’x” exp {—03x +x 1. INTRODUCTION PAIR PRODUCTION by impact ionization at high electric field in a semiconductor often characterized 14 is The parameters are, the by a three parameter theory. ionization threshold energy, ET, the nonpolar optical phonon scatteringmean free path, X 0~,,and the average energy loss per optical phonon scattering, E0~.When the carrier energy exceeds the threshold energy for pair production, the carrier is subjected to both phonon scattering and impact ionization. From density of states considerations alone the mean free path for ionization above the threshold should5’6 be has quadratically dependent on derived a simple model thedescribe carrier energy. Keldysh to this energy dependence of the impact ionization cross-section. In terms of the ionization mean free path X, for carriers an energy E above the threshold energy this can be expressed as
x~
~P
\E~)
(x)
=
ET+(EJ)+(n)EOP
scattering above the threshold ET is then pT(E)
=
±I 1 + 1’ ex E /‘ 1 1 q~kxop x) + (a + bx2) exP( (ax + ~~x3)Jdx ~—
~{— J (c
—
dE’ -~-~
—
)ci~ (3)
where x a
E
(4) (5)
ET ql~ X~,
/
b
ET
P
(6)
Note that, the energy E is measured from the threshold
(2)
where (x) is the mean distance required for pair production, (n) is the average number of phonon colisions and (E1) is defined as the average additional energy required for impact ionization above the threshold ET. Thus,
=
and
into consideration, the ionization coefficient a at electric field ~ then has to be expressed as24
=
carriers that reach the threshold energy ET are streaming along the field. The total density function for carrier
(1)
where P is a dimensionless quantity which is usually much greater than unity. X~is naturally infinite for carrier energy less than ET. Ifwe take this finite energydependent ionization cross-section above the threshold
a
required for pair production. The purpose of this paper is to derive an analytic expression for the term (E 1> based on a streaming model and to provide a method of handling the particular probability integrals encountered in such a case. At very high electric field, we wifi assume that
energy ET. The individual density function that corresponds to optical phonon scattering, p 0~(ff)dE,is p0~(E)dE= aexp (_(~+~3))~ (7)
f_
}
and to impact ionization, p1(E)dE, 2 exp (axis + ~~x3) dx Pi(E)dE = bx
(8)
ET + (E 1> *
represents an effective threshold energy
Research supported by the United States Army Research Office, under Grant No. DAHCO4-75-9-0002.
The moments of these distributions are of particular interest in that they determine the average energy gained and distance travelled before scattering. Computing the moments of the above three probability functions wifi 169
170
PAIR PRODUCTION IN NONPOLAR SEMICONDUCTORS
f
require the following evaluation of the integral,
x~exp + ~x3)) ~ Through a change of variable, we can rewrite equation(9)
where
5
\(n+1)/3
b
3)}dx /
=
‘1/3 1
fo(13)
=
I ~
(
(3) ~.PJ
E~
\q~X
)
.
Jfexp{_~x+x3)}dx.
(12)
and fi (13)
3((3)—
13f,,,..2((3).
(13)
Equation (13) can be used to decompose all the higher order f~(j3)(ii ~ 2) into the following two basic functions 3) exp (—(13x + x3)}dx (14) fo(1
$
and 3 )}dX
=
— df0 d13(13)
x exp {—(13x + x
(15)
Equations (14) and (15) cannot be integrated in simple closed form. However, if we expand the term exp (—13x) in the integrand, we obtain the following power series representations, 3 ) dx — j3 Jx exp (—x3) ~ o exp (—x 0 +~ x~exp (—x3 )dx
$
—~ ~ x 133 ~
exp (—x3)dx
1 ~
3
and
no
r
-~
3n!
n=o
2
~“
+
(_13)3r~1
~-)}.
1
(19)
147 10
+...
(3n
— 2)
r
P~n+ 3) 2
/
=
3fl
258 11.. (3n
=
—1) ~
(-i-) (i-)
(20)
(21)
3fl
=
P(n + 1)
(22)
n!,
one can compute f0 (13) and f1 (13) fairly easily via equations (18) and (19). 3. RESULTS AND DISCUSSIONS Table 1 lists the calculated values off 0 (13) and ft (13) for a range of 13 from 0 to 3.0. These computations 13were used, performed less than in 35 double terms in precision. the expansion Withininthe equations range of (l6Tand (1 7) were required to achieve convergence beyond the 14th significant figure. From equation (11), if the parameters ET and X~,are known for a given material, the scale of 13 can be appropriately converted into values of P. The main quantity of interest,
rbE2 lbE3
(16) n!
(3n + 2)! P (n + 1))
—
+ (3n + 1)! r (n + 1) (13)~+2 ~ (n + + (3n + 2)!
/3)1/3
~÷l) 3
I + 3)
I (~13)~fl /
~
=
+
3f~(13)= (n — 2)f~_
=
j3)~’~“
(18)
Via integrated by pal relationship
$
~
/
9ts, we obtain the following recurrence
=
‘~
With the aid of the standard Gamma function identities8
We denote the definite integral of interest as
f0 (13)
In + 2
(11)
2. METHOD OF COMPUTATION
f
~ I(
+ (3~)’ ~ (n +
0~/
f~ (13)
(i3)
t~—~--—)
\2/3
3 } in the integrand of equation term exp {—x (10) The can be viewed as a modified error function,7 which is further damped by the exp (—j3x) term. Thus, mathematically speaking, the expression (10) is proportional to the nth moment of a damped modified error function distribution.
f~(fl)
~
x~exp (—(13x +x
1/3
a(~-)
(10)
1
=
Toterms ease the ~ n! we can further (17) group in foactual and f1computation in the following manner.
(9) in a more convenient form
(
~
Vol. 20, No.3
=
3]}dE exp {— [aif + (b/3)E3]}dE exp (— [aE+ (b/3)E
___ f
0(i3)—13f~(i3) 1 ‘l3fo(6)
(23)
Vol. 20, No.3
PAIR PRODUCTION IN NONPOLAR SEMICONDUCTORS
Table 1. fotfl) and f~(fl),13= 0(0.1)3.0
13
foO3)
f~ (13)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
0.89297 95 0.84946 05 0.80899 40 0.771 32 34 0.73621 67 0.70346 33 0.67287 23 0.64427 02 0.61749 92 0.59241 61 0.5688899 0.54680 17 0.5260426 0.50651 34 0.48812 33 0.4707892 0.45443 51 0.43899 10 0.42439 32 0.41058 27 0.39750 55 0.38511 20 0.37335 63 0.3621962 0.35159 29 0.34151 05 0.33191 56 0.32277 77 0.3140683 0.30576 10 0.29783 16
0.45 137 26 0.41947 88 0.39027 92 0.36351 82 0.33896 67 0.31641 90 0.29569 04 0.27661 48 0.2590425 0.2428388 0.22788 23 0.2140634 0.20128 31 0.18945 18 0.17848 86 0.1683201 0.1588798 0.1501073 0.1419480 0.13435 20 0.12727 41 0.12067 30 0.1145112 0.1087543 0.10337 13 0.09833 35 0.09361 47 0.08919 12 0.08504 11 0.0811442 0.07748 23
171
Figure 1 is a plot of these three normalized moments for a range of 13 from 0 to 3.0. From this plot, we observe that the quantity
~ F—
bJ
~
9j.oo
0.50
1~00
150
VALUE OF
$
2~00
2.50
3.00
Fig. 1. Plot of the three normalized moments
~
a: a: a: a:
Using the definition of (I from equation (11),
(E1) q&X0~=
~
fo(L3)13fi03) lI3fo(i3)
(24)
=
q~X0~
13ff ((3)
~
00
0~S0
100
1 50
VALUE OF
Similarly, we can deduce the other normalized first moments corresponding to the two density functions defined in equations (3) and (7). Thus 3fo(13) (25)
___________________________ ~9j.
(26)
where
2.00
2~00
3.00
~
Fig. 2. Plot of the probability P1(13) that the next scattering will be via pair production vs the value of (3 from 0.0 to 3.0. tion to the scattering probability impact for large (3. total However, at large values of due (3, thetocontribuionization is small. Thus the moments (E 8)/q~X0~and
172
PAIR PRODUCTION IN NONPOLAR SEMICONDUCTORS
Vol. 20, No.3
ihe factor P in equation (1) can only be approxicarriers after each phonon scattering. Since the streaming mated within an order of magnitude for a given material approximation is only valid at extremely high fields, one must also account for the scattering trajectories to e.g. P 102 for Si and Ge.5 If we assume an electric field at 4 x 1 05 V/cm and that the values of ET and ~, deduce the exact effect of an energy dependent ionizafor electrons in Si at room temperature are 1.1 eV and tion cross-section on the avalanche coefficient. A study 48 A respectively,3 then equation (11) specifies that the using this approach (i.e. a characterization as a Markov range of (3 goes from 0.5 to 1.0 for a range ofF from process) will be published later. However, the present 204 to 25. Thus within an uncertainty factor of 3 in the simple treatment provides a useful upper bound for the value of P, we can still deduce impact ionization paraeffective threshold energy, and provides order of magmeters with reasonable accuracy from Figs. 1 and 2. We nitude estimates for the effect of a quadratic energy conclude that for carriers reaching beyond the threshold, dependent impact ionization cross-section on the. 0.5 ± 0.1 of them will create a new electron—hole pair secondary carrier generation. This is true because during their first scattering event, and on the average, fortunately P is not of the order of unity. If this were the impact ionization event takes place at 0.6 ±0.2 of so, then one would need to consider a strong effect q~X 0~ above the threshold energy. associated with successive phonon scattering of carriers In general, in a more exact treatment, one can of energy greater than the threshold energy. follow the successive energy distribution of the primary
REFERENCES 1. 2.
SHOCKLEY W., Solid StateElectron. 2,35 (1961). BARAFFG.A.,Phys. Rev. 128, 2507 (1962).
3. 4.
OKUTO Y. & CROWELL C.R., Phys. Rev. B6, 3076 (1972). OKUTO Y. & CROWELL C.R., Phys. Rev. BlO, 4284 (1974).
5.
KELDYSH L.V.,Sov. Phys. JETP 10,509 (1960).
6.
KELDYSH L.V., Soy. Phys. JETP 21, 1135 (1965).
7. 8.
ABRAMOWITZ M.,J. Math. Phys. 30,162(1951). ABRAMOWITZ M. & SEGUN A., Handbook ofMathematical Functions, p. 25~.Dover, New York (1968).