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Comments on “theoretical C(V) equation of an amorphous-crystalline heterojunction at low frequency”
0038-I lOlj88 $3.00 + 0.00 Copyright 0 1988 Pergamon Press plc
Solid-Stare Elecrronics Vol. 31, NO. 8, pp. 134%1350, 1988 Printed in Great Britain. All rights reserved
LETTERS
TO THE EDITOR
COMMENTS ON “THEORETICAL C(V) EQUATION OF AN AMORPHOUS-CRYSTALLINE HETEROJUNCTION AT LOW FREQUENCY”? (Received 15 September 1987; in revised form 5 March 1988)
P,(X) = NC,expK&, - EC,)/KU
It is my opinion that in this paper[l], the expressions for the free-carrier concentration in the heterojunction space-charge region are accurate only if a limiting condition is fulfilled. The definition, the frequency aspect, and the results of the model in Ref. [l] are discussed. In his paper, “Theoretical C(V) equation of an amorphous-crystalline heterojunction at low frequency,” Rubinelli[ l] derived a heterojunction spacecharge-region capacitance C by defining C as:
c=_%=de, dV
dv
x evk4 (x)/W P&)
= Nv, cxpK&, -
b)IKTl
x ew[-d (x)IKTl Mx)
= Ncr expK&
-
(4)
&dIK~l
x exp{q[4 (x)- hI/fW P&) = NV2 ewWv2 - &PYKU x exp{q[A- ~(x)IIW,
(1)
(3)
(5) (6)
where Nc and Nv are the effective density states, EC is the edge of the conduction band, E, is the edge of the valence band, 4 (x) is the electrostatic potential, Q,= ‘p,Wx; Q,= x2GWx, (2) and 4, is the barrier height of the junction. The s Xl II, author defines 4 (x,) as the reference potential withwhere V is the applied voltage, x, and x, are the out defining x,. I think the author intended to let space-charge-region edges at the amorphous and the 4(x1) = 0. In any case, eqns (3)-(4) will be correct crystalline side of the junction, respectively. In his only if E,-, and E,, are treated as the conduction band edge and the valence band edge in the quasi-neutral paper, the integral boundaries for Q, were mistakenly written as x2 and xc, and x, was undefined. Also, in emitter region (not in the space-charge region), reequation (l), it will be more precise if V is defined as spectively, provided the junction is an abrupt the separation of the electron quasi-Fermi potential junction[4]. The same concept applies to no and po. EFNIq and the hole quasi-Fermi potential EFP/q Equations (9-o-() are correct only if EC2 and Ev2 are [V = (FFN - EFP)/q]. The reason is that (EFN - EFP)/q treated as the conduction band edge and valence is the actual potential governing the free carriers, and band edge in the quasi-neutral base region, rethus the capacitance, in the space-charge region[2]. spectively. Note that because of the reference is Further, the applied voltage is different from chosen at x = x, , 4 (x) < 0 in the space-charge re(EFN - EFP)/q due to the ohmic drop in the quasigion. neutral regions and to the minority carriers at the The junction capacitance model[l], for both edges of the space-charge region[3]. For low injec- heterojunction and homojunction, has shown that tions, the applied voltage can be approximated the l/C* approaches zero, or C approaches infinity, at large voltages (Figs 34 of Ref.[l]). This contrasts same as (EFN - EFP)/q; for high injections, however, the applied voltage is greater than (EFN - EFP)/q[3]. with the predictions made by Refs[2, $61. A recently Another problem arises when the author intends to developed homojunction capacitance model[2] has find the free-carrier concentration in the space-charge suggested that for large voltages (voltages approach region, from which Q, and Q2, and thus C, can be the built-in voltage), the barrier height has nearly derived. The free electron concentrations nr, and nR vanished, and the conventional free-carrier-voltage and free hole concentrations pr, and prz, where sub- exponential relation does not hold. As a result, rigorous mathematical analysis is difficult at this bias script 1 represents the left-hand side (or amorphous side) and subscript 2 represents the right-hand side region. Consequently, a qualitative treatment was (or silicon side) of the junction, are expressed as presented in Ref.[2] to solve C at large voltages. Figure 1 illustrates the comparison of the capacitance (eqn (6) of Ref.[l]): model of Ref.[Z] with measured dependencies[$ with a method based on device simulation[6], and with the conventional depletion model(71. It should be noted tF. Rubinelli, Solid-State Electronics 30, 593 (1987). where
1349
1350
Letters
to the Editor
C
Si/Si
( 10m7F/cm2 )
Step Junction
ND,= 10’Bcm-3 NAP= lCI’6cm-a
10
I
0.1
I
0.0
0.2
,
I
I
0.6
Vc
I I
I
0.4
I
I
0.8
1.0
v(volts)
Fig. 1. Comparison of a recent homojunction space-charge-region capacitance model[2] with the conventional depletion model[7], with an experimental measurement[$ and with a method based on device simulation (MEDUSA)[6]. Note that the depletion model, the same as that of the Refll], predicts that C approaches infinity as V approaches the junction built-in voltage. In the figure, V, is the voltage at which the capacitance model of Ref.[Z] reaches its maximum.
that, except for the depletion model, C approaches zero when V, which is the separation of the quasiFermi potentials, approaches the built-in potential. A recent heterojunction capacitance model[S] also predicts the same C-V characteristics. The last comment to the paper concerns the applicable frequency range of the capacitance model. According to the definition of C [eqns (I)-(2)], the model is a qusi-static model, which assumes that the free carriers in the space-charge region travel with infinite velocity and that the free-carrier densities their steady-state deretain, during transients, pendence. Therefore the author made a good point that the model is only applicable for low frequencies. For high frequencies, as demonstrated in Refs[9, lo], the actual (non-quasi-static) junction capacitance depends heavily on the frequency of the excitation. Another minor correction concerns eqn (13) in the paper[ 11: l/C = d(4, - 4o)idQz - dQo/dQ, 3 dQ,/dQ,
should be dq&/dQ,.
Electrical
Engineering
University
of Central
P. 0.
25000,
Box
Department
J. J. LIOU
Florida
Orlando
FL 32816. U.S.A.
REFERENCES I. F. A. Rubinelli, Solid-St. Elecfron. 6, 593 (1987). 2. J. J. Liou, F. A. Lindholm and J. S. Park, IEEE Trans. Electron Dev. ED-34, 1571 (1987). 3. A. H. Marshak and C. M. Van Vliet, Proc. lEEE 72, 148 (1984). Thermal Physics, 2nd edn. 4. C. Kittel and H. Kroemer, Freeman, San Francisco (1980). 5. B. C. Bouma and Roelofs, Solid-St. Electron. 21, 833 (1978). Solid-St. Electron. 28, 6. S. W. Lee and E. J. Prendergast, 767 (1985). 7. W. Shockley, Bell Sysl. rech. J. 28, 435 (1949). 8. J. J. Liou, F. A. Lindholm and D. C. Malocha, J. uppi. Phys. In press. 9. M. A. Green and J. Shewchun. Solid-St. Electron. 17, 941 (1974). 10. J. J. Liou, Solid-St. Electron. 31, 81 (1987)
Solid-Stare Elecrronics Vol. 31, NO. 8, pp. 134%1350, 1988 Printed in Great Britain. All rights reserved
LETTERS
TO THE EDITOR
COMMENTS ON “THEORETICAL C(V) EQUATION OF AN AMORPHOUS-CRYSTALLINE HETEROJUNCTION AT LOW FREQUENCY”? (Received 15 September 1987; in revised form 5 March 1988)
P,(X) = NC,expK&, - EC,)/KU
It is my opinion that in this paper[l], the expressions for the free-carrier concentration in the heterojunction space-charge region are accurate only if a limiting condition is fulfilled. The definition, the frequency aspect, and the results of the model in Ref. [l] are discussed. In his paper, “Theoretical C(V) equation of an amorphous-crystalline heterojunction at low frequency,” Rubinelli[ l] derived a heterojunction spacecharge-region capacitance C by defining C as:
c=_%=de, dV
dv
x evk4 (x)/W P&)
= Nv, cxpK&, -
b)IKTl
x ew[-d (x)IKTl Mx)
= Ncr expK&
-
(4)
&dIK~l
x exp{q[4 (x)- hI/fW P&) = NV2 ewWv2 - &PYKU x exp{q[A- ~(x)IIW,
(1)
(3)
(5) (6)
where Nc and Nv are the effective density states, EC is the edge of the conduction band, E, is the edge of the valence band, 4 (x) is the electrostatic potential, Q,= ‘p,Wx; Q,= x2GWx, (2) and 4, is the barrier height of the junction. The s Xl II, author defines 4 (x,) as the reference potential withwhere V is the applied voltage, x, and x, are the out defining x,. I think the author intended to let space-charge-region edges at the amorphous and the 4(x1) = 0. In any case, eqns (3)-(4) will be correct crystalline side of the junction, respectively. In his only if E,-, and E,, are treated as the conduction band edge and the valence band edge in the quasi-neutral paper, the integral boundaries for Q, were mistakenly written as x2 and xc, and x, was undefined. Also, in emitter region (not in the space-charge region), reequation (l), it will be more precise if V is defined as spectively, provided the junction is an abrupt the separation of the electron quasi-Fermi potential junction[4]. The same concept applies to no and po. EFNIq and the hole quasi-Fermi potential EFP/q Equations (9-o-() are correct only if EC2 and Ev2 are [V = (FFN - EFP)/q]. The reason is that (EFN - EFP)/q treated as the conduction band edge and valence is the actual potential governing the free carriers, and band edge in the quasi-neutral base region, rethus the capacitance, in the space-charge region[2]. spectively. Note that because of the reference is Further, the applied voltage is different from chosen at x = x, , 4 (x) < 0 in the space-charge re(EFN - EFP)/q due to the ohmic drop in the quasigion. neutral regions and to the minority carriers at the The junction capacitance model[l], for both edges of the space-charge region[3]. For low injec- heterojunction and homojunction, has shown that tions, the applied voltage can be approximated the l/C* approaches zero, or C approaches infinity, at large voltages (Figs 34 of Ref.[l]). This contrasts same as (EFN - EFP)/q; for high injections, however, the applied voltage is greater than (EFN - EFP)/q[3]. with the predictions made by Refs[2, $61. A recently Another problem arises when the author intends to developed homojunction capacitance model[2] has find the free-carrier concentration in the space-charge suggested that for large voltages (voltages approach region, from which Q, and Q2, and thus C, can be the built-in voltage), the barrier height has nearly derived. The free electron concentrations nr, and nR vanished, and the conventional free-carrier-voltage and free hole concentrations pr, and prz, where sub- exponential relation does not hold. As a result, rigorous mathematical analysis is difficult at this bias script 1 represents the left-hand side (or amorphous side) and subscript 2 represents the right-hand side region. Consequently, a qualitative treatment was (or silicon side) of the junction, are expressed as presented in Ref.[2] to solve C at large voltages. Figure 1 illustrates the comparison of the capacitance (eqn (6) of Ref.[l]): model of Ref.[Z] with measured dependencies[$ with a method based on device simulation[6], and with the conventional depletion model(71. It should be noted tF. Rubinelli, Solid-State Electronics 30, 593 (1987). where
1349
1350
Letters
to the Editor
C
Si/Si
( 10m7F/cm2 )
Step Junction
ND,= 10’Bcm-3 NAP= lCI’6cm-a
10
I
0.1
I
0.0
0.2
,
I
I
0.6
Vc
I I
I
0.4
I
I
0.8
1.0
v(volts)
Fig. 1. Comparison of a recent homojunction space-charge-region capacitance model[2] with the conventional depletion model[7], with an experimental measurement[$ and with a method based on device simulation (MEDUSA)[6]. Note that the depletion model, the same as that of the Refll], predicts that C approaches infinity as V approaches the junction built-in voltage. In the figure, V, is the voltage at which the capacitance model of Ref.[Z] reaches its maximum.
that, except for the depletion model, C approaches zero when V, which is the separation of the quasiFermi potentials, approaches the built-in potential. A recent heterojunction capacitance model[S] also predicts the same C-V characteristics. The last comment to the paper concerns the applicable frequency range of the capacitance model. According to the definition of C [eqns (I)-(2)], the model is a qusi-static model, which assumes that the free carriers in the space-charge region travel with infinite velocity and that the free-carrier densities their steady-state deretain, during transients, pendence. Therefore the author made a good point that the model is only applicable for low frequencies. For high frequencies, as demonstrated in Refs[9, lo], the actual (non-quasi-static) junction capacitance depends heavily on the frequency of the excitation. Another minor correction concerns eqn (13) in the paper[ 11: l/C = d(4, - 4o)idQz - dQo/dQ, 3 dQ,/dQ,
should be dq&/dQ,.
Electrical
Engineering
University
of Central
P. 0.
25000,
Box
Department
J. J. LIOU
Florida
Orlando
FL 32816. U.S.A.
REFERENCES I. F. A. Rubinelli, Solid-St. Elecfron. 6, 593 (1987). 2. J. J. Liou, F. A. Lindholm and J. S. Park, IEEE Trans. Electron Dev. ED-34, 1571 (1987). 3. A. H. Marshak and C. M. Van Vliet, Proc. lEEE 72, 148 (1984). Thermal Physics, 2nd edn. 4. C. Kittel and H. Kroemer, Freeman, San Francisco (1980). 5. B. C. Bouma and Roelofs, Solid-St. Electron. 21, 833 (1978). Solid-St. Electron. 28, 6. S. W. Lee and E. J. Prendergast, 767 (1985). 7. W. Shockley, Bell Sysl. rech. J. 28, 435 (1949). 8. J. J. Liou, F. A. Lindholm and D. C. Malocha, J. uppi. Phys. In press. 9. M. A. Green and J. Shewchun. Solid-St. Electron. 17, 941 (1974). 10. J. J. Liou, Solid-St. Electron. 31, 81 (1987)