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Influence of carrier energy quantization on the gate-induced drain breakdown
I I, pp.1933-1936, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-l IO1195 $9.50 + 0.00
So/id-S/are Elecfronics Vol. 38. NO.
Pergamon
003%1101(95)00021-6
INFLUENCE OF CARRIER ENERGY QUANTIZATION THE GATE-INDUCED DRAIN BREAKDOWN BOGDAN Institute
of Microelectronics
MAJKUSIAK
and TOMASZ
and Optoelectronics, Warsaw University 00-662 Warsaw, Poland
ON
JANIK of Technology,
(Received 4 Ju1.v 1994; in reoised form 22 November
Koszykowa
75,
1994)
Abstract-The carrier
energy quantization in the p+-type drain surface region located under the gate electrode of an MOS transistor is investigated. The influence of this phenomena on the gatedrain voltage indispensable for tunnelling of an electron from the valence band to the conduction band is theoretically considered and evaluated. The calculation of the gate induced drain leakage current under nonuniform electric field with inclusion of the carrier energy quantization is carried out as well.
1. INTRODUCTION
According to the scaling rules, a decrease in the planar dimensions of MOS transistors in VLSI circuits must be accompanied with reduction of the gate oxide thickness. One of limitations of this trend is the band-to-band tunneling occurring in the gateto-drain overlap region-the effect known as the gate-induced drain leakage (GIDL)[I-51. The theory of the band-to-band tunneling[6] which has become an object of a great interest again[7-11] and all theoretical models of the GIDL current has been worked out for geometrically unlimited systems such as the p-n junction. Therefore, the common feature of theoretical works published on the GIDL effect is that the potential well at the semiconductor surface is considered classically, i.e. the continuous spectrum of energy is assumed. However, due to the high concentration of dopants in the drain region the potential well at the semiconductor surface is very narrow. Consequently, the carrier energy quantization is very remarkable which in turn must affect the GIDL current or the gate-induced drain breakdown. Consideration of the influence of carrier energy quantization on the gate-induced drain breakdown voltage and current is the aim of this work.
2. THEORY
Considerations will be performed for the p-channel MOS transistor with the (100) oriented silicon substrate. If the gate is on the zero potential and the drain on the large negative potential, as in the case of the enhancement MOS transistor in the OFF state, the p+-typ drain region located under the gate electrode is in the deep depletion state. The electron-hole pairs generated thermally in the surface region are separated by the electric field: electrons are swept towards the substrate and holes towards the
drain electrode. If the gate-induced band bending in the drain surface region is high enough, the bandto-band tunnelling is possible, which produces additional electron-hole pairs and constitutes the drain-bulk leakage current. Assuming for simplification that the potential barrier at the oxide side of the oxide-semiconductor interface is infinitive, the shape of the potential energy well can be approximated by a truncated parabola: V(x) = cc
x < 0
(1)
where x,, = (2c,q5:/qNA)‘/* is the depletion width (VA is the drain acceptor concentration) given by [ 121:
region and 4:
is a parameter of the parabola approximating the potential distribution (4: 2 4, - ks T/q for small and medium concentrations). In (3) L, is the Debye length for an intrinsic semiconductor, F(r+) is the modified Kingston function based on the Fermi-Dirac statistics, & is the Fermi potential, ur = q&/ks T is the normalized Fermi potential and 4, is the surface potential in the drain region (referring to the drain bulk region). Equation (2) leads to the following expression for an electric field in the depletion region:
where F, = 2$:/xd 1933
is the surface electric
field.
Bogdan Majkusiak and Tomasz Janik
1934
If the spectrum of allowed energy at the semiconductor surface was continuous, the following condition indispensable for the gate-induced drain breakdown should be fulfilled:
(5) which can be also written
in the following
I .
Ejtl’
49, =Eg +
I
2 4.
.
2.0 -
F
form:
c
1.5 -
where Eg is the semiconductor band gap, F,(E,) is the surface electric field in the case of the band bending voltage, VFBDis the 4, = E,lq, VGDis the gate-drain flat-band voltage of the gate-oxide-drain system and ti is the oxide thickness. However, a great band bending at the highly doped drain surface being in the deep depletion state means a deep and narrow potential well. Therefore, the electron energy in this well is quantized. The spectrum of energy levels E,k of electrons can be obtained from the following equation[ 121:
1.0 . 10 I6
“.“..I
10 ”
. . .‘..‘.I
. ‘...‘..I
10 I8
...-..
10 I9
ACCEPTOR CONCENTRATION
10 *O
[cm -‘I
for tunneling of electrons from the valence band to the conduction band to Fig. 2. The surface potential
indispensable
occur: $-tunneling to the bottom of the conduction assuming the continuous electron energy spectrum, tunneling to the E,k energy level.
band E,,-
nelling to occur must be greater than the sum of the semiconductor band gap and the first energy level E,, , i.e. the threshold condition for the gate-induced drain breakdown takes the following form:
xd
E +&I 4s’ g 4
or in the equivalent X In
1
+I+:
=(i _a)n
form
(7)
IF3
l/-
44: j
where m, is the electron effective mass for the direction perpendicular to the semiconductor surface. For the (100) oriented silicon substrate WI,, can take two values (k = 1 or 2): m,, = 0.916 m, and m,, = 0.190 mo. Figure 1 shows the first energy levels obtained from (7) for N,, = IO’*cm-‘, the surface potential C#I,= E,/q, and temperature T = 300 K. As can be seen in Fig. 1. due to the energy quantization the minimum semiconductor band bending indispensable for the band-to-band tun-
where F,(E, + E,,) is the surface electric field in the case of the band bending 4, = (E, + E,,)/q. Figure 2 shows the dependence of the minimum band bending indispensable for the band-to-band
----
quantum classical
E [eVl 0.3 E3l %1
u.2
E
12
El,
0.1
0’ 00
10 ”
m 0
I
2
3
4
5
6
xlnml
Fig. I. Quantization of the electron energy in the P+-type drain surface region located under the gate electrode for NA = 10’8cm-x and at the band bending 4, = E,/q.
. . ......I
. 10 ‘s
......I
.
10 I9
ACCEPTOR CONCENTRATION
....aJ
10 X0
[cm ‘1
Fig. 3. The gate-induced breakdown voltage Voo for the classical and quantum approaches to the electron energy spectrum in the drain surface region (3.2 3V for the gate-SiO, conduction band work-function was assumed).
Influence
of carrier
tunneling on the acceptor concentration in the semiconductor region. The curve marked as q& = Eg corresponds to the classical approach to the energy levels in the potential energy well, according to which the relation (5) must be fulfilled. For this case, the decrease in the minimum surface potential with dopant concentration results from the band gap narrowing[ 131 included in the calculations. The other curves illustrate the quantum approach, ie. they show surface potential which must be induced to allow tunneling from the valence band to the level E,, in the conduction band (it means that q& = Eg + E,,). For N, = lOi cmm3 the difference between the surface potentials obtained from eqns (8) and (5) equals E,,/q = 0.268 V. Figure 3 shows the gate-induced drain breakdown voltage resulted from (6) for the classical approach (the dashed line) and from (9) for the quantum approach (the solid lines) in dependence on the acceptor concentration and oxide thickness. The classical breakdown voltage correspond to band bending denoted as qqSs= E, in Fig. 2 whereas the quantum breakdown voltage corresponds to band bending denoted as q& = Eg + E,, in Fig. 2. In calculations 3.2 eV is assumed for the gate to SiOZ conduction band work-function which is appropriate for the aluminium gate or the n + type polysilicon gate. The difference between both the approaches for N, = lOI9 cm-’ equals 0.464 and 0.721 V for t, = 3 and 8 nm. respectively. As can be seen in Fig. 3, the breakdown voltage increases with the dopant concentration and decreases with the oxide thickness. However, conditions (5) and (6) for the classical approach or (8) and (9) for the quantum approach can be treated as conditions critical for the gate-induced drain breakdown only for higher drain doping levels. They do not assure the flow of a remarkable band-to-band tunnel current at lower dopant concentrations-the electric field must be large enough to assure the sufficiently high tunneling probability. According to the classical approach the band-toband tunnel current density Jara, i.e. the GIDL current I,,, per the channel width unity W and the gate-to-drain overlap length unity AL, can be expressed in the following way:
J BTB= x
IBTB
” \,I neTe[F(.u)] dr.
= q
energy
quantization
1935
denotes the distance from the semiconductor surface where the electron potential energy equals to the first allowed electron energy level E,, Since, in accordance with (4). linear field distribution is assumed, (IO) takes the following form: AY”l ES J BTB= n,,,(F) N A 5 /TV,)
(13)
dF
where F(x,) and F(x,) are calculated by inserting an approximate ?I value into (4). The curves marked as Jere = 1 PA/pm’ in Fig. 3 represents the gatedrain voltage required to assure the band-to-band tunnel current density of the level J BTB-- I PA/pm’. Equation (13) with .x0= 0 for the classical approach (the dashed line) and x0 = x,, for the quantum one (the solid line) was used in calculations. The obtained voltages increase with decreasing the dopant concentration and control the gate-induced drain breakdown voltage levels below 4 x 10’8cm-3. As seen in Fig. 3, due to the limited number of final states for electrons tunneling from the semiconductor valence band to the quantized conduction band in the surface region the carrier energy quantization results in a decrease in the band-to-band tunnel current. Therefore, also for lower doping levels the quantum gate-induced drain breakdown voltage is greater than the classically obtained one. Figure 4 shows the drain leakage current density as a function of the drain dopant concentration for the gate-drain voltage Van equal to 3 V. It results from (13) using x,, = 0 for the classical approach (the dashed line) and .x” = x,, for the quantum one (the solid line). For higher doping levels the band bending is small and the condition (8) in the case of the quantum considerations and the condition (5) in the case of the classical considerations are not fulfilled for the mentioned V,, voltage. This results in a sharp
10-3,
I
(10)
s In (IO) n,,,(F) is the band-to-band tunneling generation rate for the nonuniform electric field[ I I]. X, is the plane, where the potential equals E,/q: .Y’=.X,(I
-JT)
.x0equals 0 for the classical description quantum one, where
.y,,
=.y,(l--/I +)
(II) and x,, for the
ACCEPTOR
CONCENTRATION
Fig. 4. The density of the gate-induced
(12)
[cm
‘1
drain leakage current as a function of the drain dopant concentration (3.2 eV for the gate-SO? conduction band work-function was assumed).
1936
Bogdan Majkusiak and Tomasz Janik
decrease in the current density for higher doping levels. It is noticeable that the difference between both the approaches grows together with the increase of the dopant concentration. This is caused by the increase in the electron energy level quantization for higher doping levels.
avalanche breakdown occurring in the gate-to-drain overlap region may be a reason of an additional drain leakage and may cover the GIDL effect. Acknowledgement-This
work was supported by the KBN, Poland. Grant No. 800349101.
3. CONCLUSIONS REFERENCES
As it follows from the analysis presented above the carrier energy quantization in the drain surface region located under the gate electrode may have a significant effect on the gate-induced drain breakdown voltage. A resulted limitation of a number of energy levels taking part in the band-to-band tunneling process results in a decrease in the tunnel current and in an increase in the breakdown voltage. The breakdown voltage shift may be of practical importance especially in the case of the p-channel MOS transistor with the n +-type polysilicon gate, when the breakdown voltage can be especially low due to the large negative gate-drain work-function difference. The theoretical considerations were based on the assumption of idealized planar geometry of the gate-oxideedrain system. The considered effect may be weakened in practice since the nonuniformities along the channel caused by nonuniformities of oxide thickness. local oxide charge density, surface roughness as well as an influence of fringing fields will cause some “spread” on the eigenlevels and will smear out the breakdown voltage. Also, other generation mechanisms such as trap-assisted tunneling[l4] and
I.
2. 3. 4. 5.
W.-S. Feng, T. Y. Chan and C. Hu, IEEE Electron Device Lett. EDL-7, 449 (1986). S. Banerjee, W. Richardson, J. Coleman and A. Chatterjee. IEEE Electron Device Lelt. EDL-8, 347 (1987). J. Chen. T. Y. Chan. 1. C. Chen, P. K. Ko and C. Hu, IEEE Electron Device Lett. EDL-8. 515 (1987). C. Chang and J. Lien, IEDM Tech. Dig. 714 (1987). T. Y. Chan. J. Chen, P.K. Ko and C. Hu, IEDM Tech.
Dig. 718 (1987). 6. E. 0. Kane. J. Phys. Chem. Solids 12, 18 I (1959). 7. P. K. Chakraborty and J. C. Biswas, Solid-St. Electron 28, 493 (1985). 8. P. K. Chakraborty and J. C. Biswas, So/id-St. Elecrron 32, 633 (1989). 9. M. Takayanagi, S. Iwabuchi, T. Kobori and T. Wada. IEDM Tech. Dig. 311 (1989). 10. M. Takayanagi and S. Iwabuchi, IEEE Trans. Electron Devices 38, 1425 (1991). 11. P. K. Chakraborty. IEEE Trans. Elecrron Devices 40,
214 (1993). 12. T. Janik and B. Majkusiak, J. appl. Phys. 75, 5186 (1994). 13. S. C. Jain and D. J. Roulston, Solid-St. Elecfron. 34, 453 (1991). 14. F. Hurks, H. L. Peek, J. W. Slotboom and R. A. Windgassen, IEEE Trans. Electron Devices 40, 2273 (1993).
So/id-S/are Elecfronics Vol. 38. NO.
Pergamon
003%1101(95)00021-6
INFLUENCE OF CARRIER ENERGY QUANTIZATION THE GATE-INDUCED DRAIN BREAKDOWN BOGDAN Institute
of Microelectronics
MAJKUSIAK
and TOMASZ
and Optoelectronics, Warsaw University 00-662 Warsaw, Poland
ON
JANIK of Technology,
(Received 4 Ju1.v 1994; in reoised form 22 November
Koszykowa
75,
1994)
Abstract-The carrier
energy quantization in the p+-type drain surface region located under the gate electrode of an MOS transistor is investigated. The influence of this phenomena on the gatedrain voltage indispensable for tunnelling of an electron from the valence band to the conduction band is theoretically considered and evaluated. The calculation of the gate induced drain leakage current under nonuniform electric field with inclusion of the carrier energy quantization is carried out as well.
1. INTRODUCTION
According to the scaling rules, a decrease in the planar dimensions of MOS transistors in VLSI circuits must be accompanied with reduction of the gate oxide thickness. One of limitations of this trend is the band-to-band tunneling occurring in the gateto-drain overlap region-the effect known as the gate-induced drain leakage (GIDL)[I-51. The theory of the band-to-band tunneling[6] which has become an object of a great interest again[7-11] and all theoretical models of the GIDL current has been worked out for geometrically unlimited systems such as the p-n junction. Therefore, the common feature of theoretical works published on the GIDL effect is that the potential well at the semiconductor surface is considered classically, i.e. the continuous spectrum of energy is assumed. However, due to the high concentration of dopants in the drain region the potential well at the semiconductor surface is very narrow. Consequently, the carrier energy quantization is very remarkable which in turn must affect the GIDL current or the gate-induced drain breakdown. Consideration of the influence of carrier energy quantization on the gate-induced drain breakdown voltage and current is the aim of this work.
2. THEORY
Considerations will be performed for the p-channel MOS transistor with the (100) oriented silicon substrate. If the gate is on the zero potential and the drain on the large negative potential, as in the case of the enhancement MOS transistor in the OFF state, the p+-typ drain region located under the gate electrode is in the deep depletion state. The electron-hole pairs generated thermally in the surface region are separated by the electric field: electrons are swept towards the substrate and holes towards the
drain electrode. If the gate-induced band bending in the drain surface region is high enough, the bandto-band tunnelling is possible, which produces additional electron-hole pairs and constitutes the drain-bulk leakage current. Assuming for simplification that the potential barrier at the oxide side of the oxide-semiconductor interface is infinitive, the shape of the potential energy well can be approximated by a truncated parabola: V(x) = cc
x < 0
(1)
where x,, = (2c,q5:/qNA)‘/* is the depletion width (VA is the drain acceptor concentration) given by [ 121:
region and 4:
is a parameter of the parabola approximating the potential distribution (4: 2 4, - ks T/q for small and medium concentrations). In (3) L, is the Debye length for an intrinsic semiconductor, F(r+) is the modified Kingston function based on the Fermi-Dirac statistics, & is the Fermi potential, ur = q&/ks T is the normalized Fermi potential and 4, is the surface potential in the drain region (referring to the drain bulk region). Equation (2) leads to the following expression for an electric field in the depletion region:
where F, = 2$:/xd 1933
is the surface electric
field.
Bogdan Majkusiak and Tomasz Janik
1934
If the spectrum of allowed energy at the semiconductor surface was continuous, the following condition indispensable for the gate-induced drain breakdown should be fulfilled:
(5) which can be also written
in the following
I .
Ejtl’
49, =Eg +
I
2 4.
.
2.0 -
F
form:
c
1.5 -
where Eg is the semiconductor band gap, F,(E,) is the surface electric field in the case of the band bending voltage, VFBDis the 4, = E,lq, VGDis the gate-drain flat-band voltage of the gate-oxide-drain system and ti is the oxide thickness. However, a great band bending at the highly doped drain surface being in the deep depletion state means a deep and narrow potential well. Therefore, the electron energy in this well is quantized. The spectrum of energy levels E,k of electrons can be obtained from the following equation[ 121:
1.0 . 10 I6
“.“..I
10 ”
. . .‘..‘.I
. ‘...‘..I
10 I8
...-..
10 I9
ACCEPTOR CONCENTRATION
10 *O
[cm -‘I
for tunneling of electrons from the valence band to the conduction band to Fig. 2. The surface potential
indispensable
occur: $-tunneling to the bottom of the conduction assuming the continuous electron energy spectrum, tunneling to the E,k energy level.
band E,,-
nelling to occur must be greater than the sum of the semiconductor band gap and the first energy level E,, , i.e. the threshold condition for the gate-induced drain breakdown takes the following form:
xd
E +&I 4s’ g 4
or in the equivalent X In
1
+I+:
=(i _a)n
form
(7)
IF3
l/-
44: j
where m, is the electron effective mass for the direction perpendicular to the semiconductor surface. For the (100) oriented silicon substrate WI,, can take two values (k = 1 or 2): m,, = 0.916 m, and m,, = 0.190 mo. Figure 1 shows the first energy levels obtained from (7) for N,, = IO’*cm-‘, the surface potential C#I,= E,/q, and temperature T = 300 K. As can be seen in Fig. 1. due to the energy quantization the minimum semiconductor band bending indispensable for the band-to-band tun-
where F,(E, + E,,) is the surface electric field in the case of the band bending 4, = (E, + E,,)/q. Figure 2 shows the dependence of the minimum band bending indispensable for the band-to-band
----
quantum classical
E [eVl 0.3 E3l %1
u.2
E
12
El,
0.1
0’ 00
10 ”
m 0
I
2
3
4
5
6
xlnml
Fig. I. Quantization of the electron energy in the P+-type drain surface region located under the gate electrode for NA = 10’8cm-x and at the band bending 4, = E,/q.
. . ......I
. 10 ‘s
......I
.
10 I9
ACCEPTOR CONCENTRATION
....aJ
10 X0
[cm ‘1
Fig. 3. The gate-induced breakdown voltage Voo for the classical and quantum approaches to the electron energy spectrum in the drain surface region (3.2 3V for the gate-SiO, conduction band work-function was assumed).
Influence
of carrier
tunneling on the acceptor concentration in the semiconductor region. The curve marked as q& = Eg corresponds to the classical approach to the energy levels in the potential energy well, according to which the relation (5) must be fulfilled. For this case, the decrease in the minimum surface potential with dopant concentration results from the band gap narrowing[ 131 included in the calculations. The other curves illustrate the quantum approach, ie. they show surface potential which must be induced to allow tunneling from the valence band to the level E,, in the conduction band (it means that q& = Eg + E,,). For N, = lOi cmm3 the difference between the surface potentials obtained from eqns (8) and (5) equals E,,/q = 0.268 V. Figure 3 shows the gate-induced drain breakdown voltage resulted from (6) for the classical approach (the dashed line) and from (9) for the quantum approach (the solid lines) in dependence on the acceptor concentration and oxide thickness. The classical breakdown voltage correspond to band bending denoted as qqSs= E, in Fig. 2 whereas the quantum breakdown voltage corresponds to band bending denoted as q& = Eg + E,, in Fig. 2. In calculations 3.2 eV is assumed for the gate to SiOZ conduction band work-function which is appropriate for the aluminium gate or the n + type polysilicon gate. The difference between both the approaches for N, = lOI9 cm-’ equals 0.464 and 0.721 V for t, = 3 and 8 nm. respectively. As can be seen in Fig. 3, the breakdown voltage increases with the dopant concentration and decreases with the oxide thickness. However, conditions (5) and (6) for the classical approach or (8) and (9) for the quantum approach can be treated as conditions critical for the gate-induced drain breakdown only for higher drain doping levels. They do not assure the flow of a remarkable band-to-band tunnel current at lower dopant concentrations-the electric field must be large enough to assure the sufficiently high tunneling probability. According to the classical approach the band-toband tunnel current density Jara, i.e. the GIDL current I,,, per the channel width unity W and the gate-to-drain overlap length unity AL, can be expressed in the following way:
J BTB= x
IBTB
” \,I neTe[F(.u)] dr.
= q
energy
quantization
1935
denotes the distance from the semiconductor surface where the electron potential energy equals to the first allowed electron energy level E,, Since, in accordance with (4). linear field distribution is assumed, (IO) takes the following form: AY”l ES J BTB= n,,,(F) N A 5 /TV,)
(13)
dF
where F(x,) and F(x,) are calculated by inserting an approximate ?I value into (4). The curves marked as Jere = 1 PA/pm’ in Fig. 3 represents the gatedrain voltage required to assure the band-to-band tunnel current density of the level J BTB-- I PA/pm’. Equation (13) with .x0= 0 for the classical approach (the dashed line) and x0 = x,, for the quantum one (the solid line) was used in calculations. The obtained voltages increase with decreasing the dopant concentration and control the gate-induced drain breakdown voltage levels below 4 x 10’8cm-3. As seen in Fig. 3, due to the limited number of final states for electrons tunneling from the semiconductor valence band to the quantized conduction band in the surface region the carrier energy quantization results in a decrease in the band-to-band tunnel current. Therefore, also for lower doping levels the quantum gate-induced drain breakdown voltage is greater than the classically obtained one. Figure 4 shows the drain leakage current density as a function of the drain dopant concentration for the gate-drain voltage Van equal to 3 V. It results from (13) using x,, = 0 for the classical approach (the dashed line) and .x” = x,, for the quantum one (the solid line). For higher doping levels the band bending is small and the condition (8) in the case of the quantum considerations and the condition (5) in the case of the classical considerations are not fulfilled for the mentioned V,, voltage. This results in a sharp
10-3,
I
(10)
s In (IO) n,,,(F) is the band-to-band tunneling generation rate for the nonuniform electric field[ I I]. X, is the plane, where the potential equals E,/q: .Y’=.X,(I
-JT)
.x0equals 0 for the classical description quantum one, where
.y,,
=.y,(l--/I +)
(II) and x,, for the
ACCEPTOR
CONCENTRATION
Fig. 4. The density of the gate-induced
(12)
[cm
‘1
drain leakage current as a function of the drain dopant concentration (3.2 eV for the gate-SO? conduction band work-function was assumed).
1936
Bogdan Majkusiak and Tomasz Janik
decrease in the current density for higher doping levels. It is noticeable that the difference between both the approaches grows together with the increase of the dopant concentration. This is caused by the increase in the electron energy level quantization for higher doping levels.
avalanche breakdown occurring in the gate-to-drain overlap region may be a reason of an additional drain leakage and may cover the GIDL effect. Acknowledgement-This
work was supported by the KBN, Poland. Grant No. 800349101.
3. CONCLUSIONS REFERENCES
As it follows from the analysis presented above the carrier energy quantization in the drain surface region located under the gate electrode may have a significant effect on the gate-induced drain breakdown voltage. A resulted limitation of a number of energy levels taking part in the band-to-band tunneling process results in a decrease in the tunnel current and in an increase in the breakdown voltage. The breakdown voltage shift may be of practical importance especially in the case of the p-channel MOS transistor with the n +-type polysilicon gate, when the breakdown voltage can be especially low due to the large negative gate-drain work-function difference. The theoretical considerations were based on the assumption of idealized planar geometry of the gate-oxideedrain system. The considered effect may be weakened in practice since the nonuniformities along the channel caused by nonuniformities of oxide thickness. local oxide charge density, surface roughness as well as an influence of fringing fields will cause some “spread” on the eigenlevels and will smear out the breakdown voltage. Also, other generation mechanisms such as trap-assisted tunneling[l4] and
I.
2. 3. 4. 5.
W.-S. Feng, T. Y. Chan and C. Hu, IEEE Electron Device Lett. EDL-7, 449 (1986). S. Banerjee, W. Richardson, J. Coleman and A. Chatterjee. IEEE Electron Device Lelt. EDL-8, 347 (1987). J. Chen. T. Y. Chan. 1. C. Chen, P. K. Ko and C. Hu, IEEE Electron Device Lett. EDL-8. 515 (1987). C. Chang and J. Lien, IEDM Tech. Dig. 714 (1987). T. Y. Chan. J. Chen, P.K. Ko and C. Hu, IEDM Tech.
Dig. 718 (1987). 6. E. 0. Kane. J. Phys. Chem. Solids 12, 18 I (1959). 7. P. K. Chakraborty and J. C. Biswas, Solid-St. Electron 28, 493 (1985). 8. P. K. Chakraborty and J. C. Biswas, So/id-St. Elecrron 32, 633 (1989). 9. M. Takayanagi, S. Iwabuchi, T. Kobori and T. Wada. IEDM Tech. Dig. 311 (1989). 10. M. Takayanagi and S. Iwabuchi, IEEE Trans. Electron Devices 38, 1425 (1991). 11. P. K. Chakraborty. IEEE Trans. Elecrron Devices 40,
214 (1993). 12. T. Janik and B. Majkusiak, J. appl. Phys. 75, 5186 (1994). 13. S. C. Jain and D. J. Roulston, Solid-St. Elecfron. 34, 453 (1991). 14. F. Hurks, H. L. Peek, J. W. Slotboom and R. A. Windgassen, IEEE Trans. Electron Devices 40, 2273 (1993).