- Email: [email protected]
Substrate hot electron injection modelling based on lucky drift theory
Microelectronic Elsevier
Engineering
22 (1993) 26 l-264 261
Substrate Hot Electron Injection Modelling Based on Lucky Drift Theory X.-J. Yuan, J.S. Marsland, and W. Eccleston Department of Electrical Engineering and Electronics, University of Liverpool, P.O. Box 147 Liverpool, L69 3BX, U.K.
1.
INTRODUCTION
Substrate hot electron (SHE) injection (also called emission) from a silicon substrate into a gate oxide has been extensively investigated because of its important role in both device reliability of MOSFETs and device operation of EEPROMs[l-21. Ning’s Lucky Electron Model[ l] is the most widely accepted model of SHE injection. However, the two parameters A and X in Ning’s expression ( P = Aexp(-d/h) ) often need to be readjusted for different processing and bias conditions in order to gain good agreement with the SHE injection probability. Monte Carlo computer simulation has been also used to simulate the hot electron injection process[3]. However, Monte Carlo methods consume a large amount of CPU time. In this paper a new model, based on Lucky Drift Theory, for predicting the SHE injection probability is presented. 2. MODEL 2.1. Lucky Drift Theory
The Lucky Drift Theory was first proposed by Ridley[4] for considering impact ionization in semiconductors. Shockley’s Lucky Electron concept[5] was extended by separating the rates of momentum and energy relaxation for carriers interacting with optical phonons. Ridley assumed that electrons can gain the threshold energy for ionization either by lucky ballistic motion or by a combination of lucky ballistic and lucky drift motion. Using this assumption an analytic expression was derived for the carrier ionization coefficients which gave good agreement with Baraff’s numerical results[6] over four orders of magnitude. Lucky Drift Theory has been further improved by Burt[7] and Marsland[8] to include the “soft threshold energy” and “dead space” concepts. In Lucky Drift Theory the probability of electrons elastically moving over a distance z starting from zero energy in the electric field direction can be expressed by: z
P(z)
=
expq +/exp+xPk 0
(z-z’) A
B
)
dz'
L
where X and X, are the electron mean free paths for momentum and energy relaxing collisions, respectively. In equation (1) the first term is the probability of reaching z without momentum relaxation and the integral is the probability of reaching z without energy relaxation but after the first momentum relaxing collision. hE can be derived from 0167-9317/93/!$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved.
X.-J. Yum et al. / Hot electron injection modelling
262
momentum and energy conservation. I
= e&Y(z) A2(2n+l) E
2ik.I
where @) is electric field, hw is the phonon energy, and n is the Bose-Einstein number. The probability of electrons reaching z without ionizing is given by: P,(z)
= 1
P,(z)
=
0 < z < 1,
exp(-j dz’ ) h,(d)
(3)
z > 1,
1,
where lo=Ei/e&lz) is the minimum distance over which an electron must move to gain the ionization threshold energy E, and X,(z) is the mean free path for ionization given by[8]: &(z)
=
121,
(4)
2P(Z-1J2
here p is the Keldysh factor. Divergent electric fields can be taken into account by further development of equations (1) - (4). 2.2. SHE Injection Probability Many structures and methods have been utilized to study SHE injection such as using optically induced SHE injection in a NMOS transistor or using a n-channel MOSFET with injection from an underlying p-n junction[l]. For all structures the energy band diagram near the Si/SiQ interface for the injection condition is shown in Fig. 1. Here d is the minimum distance which electrons must traverse to gain the kinetic energy equal to the interface barrier. Consider electrons created with no kinetic energy at a distance d from the interface. If the electrons move by lucky ballistic or lucky drift over the distance d without impact ionization then they will have sufficient energy to surmount the barrier. The injection probability for electrons starting from d can be expressed using equations (1) and (3) as: (5)
P, = P(d)P,(d)
Considering all the electrons at distances greater than d which can obtain sufficient kinetic energy for injection over the interface barrier if they are lucky to escape energy relaxation and impact ionization whilst travelling downfield, the SHE injection probability is given by: W
P.1nj = P(z) I
P,(z)
P,(W-z)
dz
(6)
0
Z
d
where W is the Si depletion width. P,(W-z)
FIG. 1. Si-SiO, interface energy band diagram.
263
X.-J. Yuan et al. / Hot electron injection modelling
is the probability of electrons undergoing an energy relaxing collision or impact ionization at z. This accounts for the multistage process in which electrons travel from W to z, undergo an energy relaxing collision or impact ionization at z and are then still able to surmount the Si/Si02 barrier if they are lucky travelling from z to the interface. P&W-z) can be derived from the basic assumptions of the Lucky Drift Theory. 3. SIMULATION RESULTS When the depletion field distribution is given, SHE injection probability at a given bias condition can be computed from equation (6). The Lucky Drift Model has been applied to devices reported on by Hemink et a2 [2]. The device is a novel EEPROM structure shown in Fig.%. At the programming condition the injector and substrate are grounded. A sufficiently high voltage is applied to the gate. Increasing the voltage on source and drain will extend the Si/SiQ interface depletion layer in the direction of the injector. At a certain drain voltage punch-through occurs and the p-n junction between injector and substrate will be forward biased. The electrons in the injector will be emitted into the depletion layer and will move towards the Si/SiO, interface. Some of them will gain sufficient energy and will -The injection surmount the barrier. mechanism is very efficient and homogeneous “gate at properly biased conditions. The absolute injection probability of the device can be determined from the measured gate and injector currents, P,=I,/Ii. Fig.3 shows the depletion electric field distribution for four devices at an injector current density of 100 A/cm-*. The drain voltages for the injection conditions are 3.3 V, 4.0 V, 4.5 V, and 6.7 V for devices 1 to 4 respectively. FIG. 2. The novel EEPROM structure (after Hemink [2]). The depletion depths, W; between the injector and Si/SiO, interface are all 0.5 pm. The minimum distance d can be determined from 4. field000 kV/cm) Ning’s effective potential barrier equation incorporating the Schottky and tunnelling device 1 barrier lowering effects [1]: 3.0
W(d)
B
=
3.leV-
p&
- a&
O-- device 2
(7)
--* - device 3
A- device 4
2.0
where Z& is the gate oxide field and the two constants a and fi have the values of 1.Ox 10e5e(cm2V)“3 and 2.59 x lOa e(cmV)“* respectively. The simulated injection probabilities as a function of gate voltages using the Lucky Drift Model (6) with the above injection conditions are shown as lines in Fig.4. The symbols in the figure are the measured results
1.0
0.0 0
0.1
0.2
0.3
0.4
depthturn) FIG. 3.
Electric field distributions in depletion region during injection condition.
264
X.-J. Yuan et al. I Hot electron injection modelling
by Hemink et aZ[2]. In the l.OE-02 pW simulation the optical phonon 0 measured 1 energy was taken to be 0.055 eV _--simulated 1 and the electron threshold energy 0 measured 2 for ionization was taken to be --~ simulated 2 l.leV. The momentum mean free * measured 3 path, h=70.2 A, and the Keldysh ~ simulated 3 factor, p=O.O3, are from a least L, measured 4 squares fit to experimental data for ionization coefficients in silicon@]. simulated 4 Fig.4 shows that the simulated and 10 11 12 13 14 15 gate voltage O/1 measured SHE injection probabilities are in good agreement for devices 1, 2, and 4. The FIG. 4. Measured and simulated SHE injection probabilities. deviation between simulated and measured results for device 3 might be from the neglect of flat band voltage of MOS structure. 4. DISCUSSION AND CONCLUSIONS Hemink et aZ[2] has also simulated the above EEPROM devices with Lucky Electron Model[l] (pinj = Aexp(-d/X)) and an empirical expression (Pinj = A,exp[-B,(q&/kT$]). Both methods involved fitting parameters to measured SHE injection probabilities. The Lucky Electron Model with two fitting parameters was in good agreement for only 3 of the 4 devices. The other model used 3 nonphysical fitting parameters. The model presented here can simulate SHE injection probabilities with a divergent depletion field profile in MOS structures. No parameters need to be fitted to measured injection probabilities. The momentum mean free path X and the Keldysh factor p were taken from the original Lucky Drift Model for calculating Si ionization coefficients. This model is able to predict the gate current resulting from SHE injection in MOS devices. ACKNOWLEDGMENTS The authors would like to thank Dr. G.J. Hemink of University of Twente, Netherlands for providing the measured SHE injection probabilities. This work is supported by SERC and MOD. REFERENCES 273. VI T.H. Ning, Solid-St. Electron. u(l978) El G.J. Hemink et al., Proc. of the ESSDERC’91, pp. 65-68. 131J.Y. Tang et al., J. Appl. Phys. 54 (1983) 5145. 141B.K. Ridley, J. Phys. C: Solid-St. Phys. .& (1983) 3373. 151W. Shockley, Solid-St. Electron. 2 (1961) 35. @I G.A. Baraff, Phys. Rev. 128 (1962) 2507. [71 M.G. Burt, J. Phys. C: Solid-St. Phys. ls (1985) L477. 181J.S. Marsland, Solid-St Electron. 30 (1987) 125.
Engineering
22 (1993) 26 l-264 261
Substrate Hot Electron Injection Modelling Based on Lucky Drift Theory X.-J. Yuan, J.S. Marsland, and W. Eccleston Department of Electrical Engineering and Electronics, University of Liverpool, P.O. Box 147 Liverpool, L69 3BX, U.K.
1.
INTRODUCTION
Substrate hot electron (SHE) injection (also called emission) from a silicon substrate into a gate oxide has been extensively investigated because of its important role in both device reliability of MOSFETs and device operation of EEPROMs[l-21. Ning’s Lucky Electron Model[ l] is the most widely accepted model of SHE injection. However, the two parameters A and X in Ning’s expression ( P = Aexp(-d/h) ) often need to be readjusted for different processing and bias conditions in order to gain good agreement with the SHE injection probability. Monte Carlo computer simulation has been also used to simulate the hot electron injection process[3]. However, Monte Carlo methods consume a large amount of CPU time. In this paper a new model, based on Lucky Drift Theory, for predicting the SHE injection probability is presented. 2. MODEL 2.1. Lucky Drift Theory
The Lucky Drift Theory was first proposed by Ridley[4] for considering impact ionization in semiconductors. Shockley’s Lucky Electron concept[5] was extended by separating the rates of momentum and energy relaxation for carriers interacting with optical phonons. Ridley assumed that electrons can gain the threshold energy for ionization either by lucky ballistic motion or by a combination of lucky ballistic and lucky drift motion. Using this assumption an analytic expression was derived for the carrier ionization coefficients which gave good agreement with Baraff’s numerical results[6] over four orders of magnitude. Lucky Drift Theory has been further improved by Burt[7] and Marsland[8] to include the “soft threshold energy” and “dead space” concepts. In Lucky Drift Theory the probability of electrons elastically moving over a distance z starting from zero energy in the electric field direction can be expressed by: z
P(z)
=
expq +/exp+xPk 0
(z-z’) A
B
)
dz'
L
where X and X, are the electron mean free paths for momentum and energy relaxing collisions, respectively. In equation (1) the first term is the probability of reaching z without momentum relaxation and the integral is the probability of reaching z without energy relaxation but after the first momentum relaxing collision. hE can be derived from 0167-9317/93/!$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved.
X.-J. Yum et al. / Hot electron injection modelling
262
momentum and energy conservation. I
= e&Y(z) A2(2n+l) E
2ik.I
where @) is electric field, hw is the phonon energy, and n is the Bose-Einstein number. The probability of electrons reaching z without ionizing is given by: P,(z)
= 1
P,(z)
=
0 < z < 1,
exp(-j dz’ ) h,(d)
(3)
z > 1,
1,
where lo=Ei/e&lz) is the minimum distance over which an electron must move to gain the ionization threshold energy E, and X,(z) is the mean free path for ionization given by[8]: &(z)
=
121,
(4)
2P(Z-1J2
here p is the Keldysh factor. Divergent electric fields can be taken into account by further development of equations (1) - (4). 2.2. SHE Injection Probability Many structures and methods have been utilized to study SHE injection such as using optically induced SHE injection in a NMOS transistor or using a n-channel MOSFET with injection from an underlying p-n junction[l]. For all structures the energy band diagram near the Si/SiQ interface for the injection condition is shown in Fig. 1. Here d is the minimum distance which electrons must traverse to gain the kinetic energy equal to the interface barrier. Consider electrons created with no kinetic energy at a distance d from the interface. If the electrons move by lucky ballistic or lucky drift over the distance d without impact ionization then they will have sufficient energy to surmount the barrier. The injection probability for electrons starting from d can be expressed using equations (1) and (3) as: (5)
P, = P(d)P,(d)
Considering all the electrons at distances greater than d which can obtain sufficient kinetic energy for injection over the interface barrier if they are lucky to escape energy relaxation and impact ionization whilst travelling downfield, the SHE injection probability is given by: W
P.1nj = P(z) I
P,(z)
P,(W-z)
dz
(6)
0
Z
d
where W is the Si depletion width. P,(W-z)
FIG. 1. Si-SiO, interface energy band diagram.
263
X.-J. Yuan et al. / Hot electron injection modelling
is the probability of electrons undergoing an energy relaxing collision or impact ionization at z. This accounts for the multistage process in which electrons travel from W to z, undergo an energy relaxing collision or impact ionization at z and are then still able to surmount the Si/Si02 barrier if they are lucky travelling from z to the interface. P&W-z) can be derived from the basic assumptions of the Lucky Drift Theory. 3. SIMULATION RESULTS When the depletion field distribution is given, SHE injection probability at a given bias condition can be computed from equation (6). The Lucky Drift Model has been applied to devices reported on by Hemink et a2 [2]. The device is a novel EEPROM structure shown in Fig.%. At the programming condition the injector and substrate are grounded. A sufficiently high voltage is applied to the gate. Increasing the voltage on source and drain will extend the Si/SiQ interface depletion layer in the direction of the injector. At a certain drain voltage punch-through occurs and the p-n junction between injector and substrate will be forward biased. The electrons in the injector will be emitted into the depletion layer and will move towards the Si/SiO, interface. Some of them will gain sufficient energy and will -The injection surmount the barrier. mechanism is very efficient and homogeneous “gate at properly biased conditions. The absolute injection probability of the device can be determined from the measured gate and injector currents, P,=I,/Ii. Fig.3 shows the depletion electric field distribution for four devices at an injector current density of 100 A/cm-*. The drain voltages for the injection conditions are 3.3 V, 4.0 V, 4.5 V, and 6.7 V for devices 1 to 4 respectively. FIG. 2. The novel EEPROM structure (after Hemink [2]). The depletion depths, W; between the injector and Si/SiO, interface are all 0.5 pm. The minimum distance d can be determined from 4. field000 kV/cm) Ning’s effective potential barrier equation incorporating the Schottky and tunnelling device 1 barrier lowering effects [1]: 3.0
W(d)
B
=
3.leV-
p&
- a&
O-- device 2
(7)
--* - device 3
A- device 4
2.0
where Z& is the gate oxide field and the two constants a and fi have the values of 1.Ox 10e5e(cm2V)“3 and 2.59 x lOa e(cmV)“* respectively. The simulated injection probabilities as a function of gate voltages using the Lucky Drift Model (6) with the above injection conditions are shown as lines in Fig.4. The symbols in the figure are the measured results
1.0
0.0 0
0.1
0.2
0.3
0.4
depthturn) FIG. 3.
Electric field distributions in depletion region during injection condition.
264
X.-J. Yuan et al. I Hot electron injection modelling
by Hemink et aZ[2]. In the l.OE-02 pW simulation the optical phonon 0 measured 1 energy was taken to be 0.055 eV _--simulated 1 and the electron threshold energy 0 measured 2 for ionization was taken to be --~ simulated 2 l.leV. The momentum mean free * measured 3 path, h=70.2 A, and the Keldysh ~ simulated 3 factor, p=O.O3, are from a least L, measured 4 squares fit to experimental data for ionization coefficients in silicon@]. simulated 4 Fig.4 shows that the simulated and 10 11 12 13 14 15 gate voltage O/1 measured SHE injection probabilities are in good agreement for devices 1, 2, and 4. The FIG. 4. Measured and simulated SHE injection probabilities. deviation between simulated and measured results for device 3 might be from the neglect of flat band voltage of MOS structure. 4. DISCUSSION AND CONCLUSIONS Hemink et aZ[2] has also simulated the above EEPROM devices with Lucky Electron Model[l] (pinj = Aexp(-d/X)) and an empirical expression (Pinj = A,exp[-B,(q&/kT$]). Both methods involved fitting parameters to measured SHE injection probabilities. The Lucky Electron Model with two fitting parameters was in good agreement for only 3 of the 4 devices. The other model used 3 nonphysical fitting parameters. The model presented here can simulate SHE injection probabilities with a divergent depletion field profile in MOS structures. No parameters need to be fitted to measured injection probabilities. The momentum mean free path X and the Keldysh factor p were taken from the original Lucky Drift Model for calculating Si ionization coefficients. This model is able to predict the gate current resulting from SHE injection in MOS devices. ACKNOWLEDGMENTS The authors would like to thank Dr. G.J. Hemink of University of Twente, Netherlands for providing the measured SHE injection probabilities. This work is supported by SERC and MOD. REFERENCES 273. VI T.H. Ning, Solid-St. Electron. u(l978) El G.J. Hemink et al., Proc. of the ESSDERC’91, pp. 65-68. 131J.Y. Tang et al., J. Appl. Phys. 54 (1983) 5145. 141B.K. Ridley, J. Phys. C: Solid-St. Phys. .& (1983) 3373. 151W. Shockley, Solid-St. Electron. 2 (1961) 35. @I G.A. Baraff, Phys. Rev. 128 (1962) 2507. [71 M.G. Burt, J. Phys. C: Solid-St. Phys. ls (1985) L477. 181J.S. Marsland, Solid-St Electron. 30 (1987) 125.