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Monte Carlo simulation of charge transport in semiconductor devices
Microelectronic Engineering 19 (1992) 275-282 Elsevier
275
Monte Carlo simulation of charge transport in semiconductor devices P. L u g l i Dipartimento di Ingegneria Elettronica, Universita' di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy
Abstract Recent applications of Monte Carlo device simulation are reviewed, focussing mainly on GaAs structures. The results show the great versatility of the method, and its potential as an extremely powerful tool for device modeling.
1. I N T R O D U C T I O N Due to the continuous technological improvements, and the constant push for miniaturization of devices, there is an increasing need for physical simulators, able to combine a realistic description of carder transport to an intrinsic speed which allows their implementation in CAD tools. In fact, as the active region of devices shrinks and heterostructures come more heavily into play, non local and quantum effects become extremely important. The physics involved in charge transport of such structures then authomatically prevents the use of conventional simulators, and calls for more advanced modeling tools. The Monte Carlo (MC) approach is rapidly closing the gap with respect to more traditional, and widely used, Drift-Diffusion simulators, posing itself as the most valuable alternative. Indeed, the MC technique has moved a long distance since its first introduction as a tool for the study of charge transport in semiconductors. The outstanding improvements in MC device simulations and the vaste range of applications they have reached is due to the availability of very fast workstations and mainframes, which guarantee considerable speed at affordable costs, allowing the implementation of more and more sophisticated physical models, and to the optimization of the numerical algorithms. For instance, it is possible nowaday to incorporate a full band structure description, or to account for quantum or many body effects [1-5]. In the present paper, we will concentrate on some recent results obtained for GaAs unipolar and bipolar devices, which illustrate quite effectively the strength and the capabilities of the MC simulation. It is impossible, in the limited space available, to account for the enormous amount of work originated in recent years in the field, and we have to refer the interested reader to the vaste literature existing on the subject, starting for example from the bibliography that can be found in Refs. [1,4,5]. 0167-9317/92/$05.00 © 1992 - Elsevier Scienc6 Publishers B.V.' All rights reserved.
276
P Lugli / Monte Carlo simulation of charge transport
2. THE PHYSICAL MODEL The transport model developed for GaAs is based on a three-valley description of the conduction band and on three valence bands. Non parabolic isotropic dispersions are used, with non parabolicity factors treated as fitting paramenters and adjusted as to reproduce a variety of experimental results, including those provided by time resolved spectroscopy. The carrier interaction with polar optical, acoustic, equivalent and nonequivalent intervalley, intraband and interband phonons is considered, within the usual golden rule scheme [1]. In the presence of heavily doped regions, carrier-carrier scattering (including plasmon contributions) are taken into account. Impurity scattering is treated using the Brooks-Herring approach. A list of all relevant parameters is presented in Table 1. The main emphasis of the paper is on very high field effects, where impact ionization phenomena become crucial. We have adopted the model proposed by Kane [6], where the energy dependence of the ionization process is given by P(E)= (P/T) (E-Eth) a, E being the total carrier energy. Kane model gives a softer threshold for the ionization with respect to the more standard Keldish formalism (Fig. 1) [7]. The free parameters have been determined through the fit of the bulk ionization coefficient and are given in Table 1. Bandto-band tunneling and its dependence on electric field has also been accounted for on the basis of Kane's theory [8]. The simulation of impact ionization phenomena poses considerable numerical problems since it requires the knowledge of the high energy tail of the carrier distribution function. Furthermore, in the presence of carrier multiplication the number of simulated particles would grow above the initially set value, diverging as breakdown is approached. We have therefore developed a special multiplication technique for both energy and real space, which is an extension of the original idea of Philips and Price [9]. Each particle is assigned a statistical weight which varies with its position in the device and its energy. With such approach, it is possible to account for regions with very different doping levels (as in bipolar transistors) and to obtain a reliable statistics of rare processes, keeping at the same time a constant number of particles. For the simulation of devices, the MC algorithm is selfconsistently coupled to a Poisson solver, as described extensively in Ref. [1]. 10TM "-
GaAs
~" ""~ ~) r-"' 1013 1012
300K
...f----
;~00")1011 l O ~o
1.00
/ / // ,
I
/,
1.50
d
2.00
Kane H ,
I
2.50
energy (eV)
,
t
3.00
eo,,
3.50
Figure l Comparison of the room temperature scattering rates for electron impact ionization calculated from Kane (solid line) and Keldish (dotted line) models.
P. Lugli / Monte Carlo simulation of charge transport [ Electron parameters
(300K)
Acoustic deformation potential (eV) Effective mass ( m ' / m o ) a (nonparabolicity)(eV -1) Band-edge energy (eV) Intervalley deformation potential (eV/crn) from r (000) from L (111) from X (100) Intervalley phonon energy (meV) from I" (000) from L (111) from X (100) Number of equivalent valleys
Hole parameters (300K) Acoustic deformation potential (eV) Effective mass (m*/mo) Band-edge energy (eV) a (nonparabolicity) (eV -1) Interband deformation potential (eV/em) from H.H. form L.H. fromS.O. Interband phonon energy (rneV) from H.H. from L.H. from S.O. Parameter Value Density (g/cm 3) 5.36 Sound velocity (cm/s) 5.22.105 e~ 10.92 e0 12.90 Optical phonon energy (meV) 35.36 Electron ionization coefficients P / r (s -1) 6.22. 1012 a 3.2 Eth (eV) 1.439 Hole ionization coefficients PIT (.$--1) 3.11.1011 a 6.35 Eth (eV)
1.439
277
r (000)
L (111)
X (100)
7.0 0.063 0.610 1.439
7.0 0.222 0.244 1.739
7.0 0.58 0.061 1.961
0.0 7.0- 108 1.0- 109
7.0.108 1.0. 10 9 1.0. 10 9
1.0.109 1.0.10 9 1.0.10 9
0.0 27.8 29.3 1
27.8 27.8 29.3 4
29.3 29.3 29.3 3
H.H. 7.0 0.45 0.0 0.55
L.H. 7.0 0.085 0.0 4.0
S.O. 7.0 0.154 0.35 0.3
9.0.10 s 7.0.10 s 5.0.108
7.0- l0 s 9.0.108 5.0.10 s
5.0. l0 s 5.0. l0 s 9.0. l0 s
27.8 27.8 29.3
27.8 27.8 29.3
29.9 29.3 29.3
Table 1 Parameters used in the MC simulation for electrons (upper table) and holes (middle table). Other parameters, including the impact ionization ones, are reported in the lower table.
278
P. Lugli / Monte Carlo simulation of charge transport
3. HIGH FIELD TRANSPORT
We present now the results of the MC simulation of electron and hole transport in homogeneous high electric fields based on the model previoulsy described. All results are presented for 300 K and 500 K. The drift velocity versus electric field characteristics (Fig. 2) show the well known negative differential mobility region for electrons, with peak field around 4 kV/cm. The hole velocity also shows a slight decresase, after reaching a maximum around 7 kV/cm, in correspondence to a sizable transfer from the heavy-hole to the light-hole and split-off bands. Our results are in good agreement with other MC calculations that use full band structure models [10] and with available experimental results. 108
......
I
........
I
........
I
n-GaAs E
10 7
......
I
'
'
' A ' I
........
- p-Ga s T =
lo
[
. . . . .
300~
>= ~,~/X"
T-- 500 K
z~"
T - 500 K
10 5
1061 . . . . . . J . . . . . . . . 103
, ........ 104
, ........ 105
I 106
[ ...... I ........ 103
F ( V/cm )
t 104
. . . . . . . .
I
10 6
10 s
F ( V/cm )
Figure 2 Drift velocity versus electric field curve for electrons and holes at 300 K (solid lines) and 500 K (dashed lines) > (1) v
(1) t(D
.....1 ........ I ........ I ........ I
lO
-
. n-GaAs.
E
(1)
10 0
v
......P ........i ........i ...... _ P- -Ga#~ ~-
03 0) ¢(1)
10 -1
.=_
10 "1
c-
t(1)
>
102
...... I ........ L ........ I ........ I ~ 10 0 101 10 2 10 ~
F ( kV/cm )
((D ~ E
1 0. 2
...... I
........
10 0
I
101
........
I
........
10 2
I 10 3
F ( kV/cm )
Figure 3 Electron and hole mean energy as a function of the electric field at 300 K (solid lines) and 500 K (dashed lines)
275
Monte Carlo simulation of charge transport in semiconductor devices P. L u g l i Dipartimento di Ingegneria Elettronica, Universita' di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy
Abstract Recent applications of Monte Carlo device simulation are reviewed, focussing mainly on GaAs structures. The results show the great versatility of the method, and its potential as an extremely powerful tool for device modeling.
1. I N T R O D U C T I O N Due to the continuous technological improvements, and the constant push for miniaturization of devices, there is an increasing need for physical simulators, able to combine a realistic description of carder transport to an intrinsic speed which allows their implementation in CAD tools. In fact, as the active region of devices shrinks and heterostructures come more heavily into play, non local and quantum effects become extremely important. The physics involved in charge transport of such structures then authomatically prevents the use of conventional simulators, and calls for more advanced modeling tools. The Monte Carlo (MC) approach is rapidly closing the gap with respect to more traditional, and widely used, Drift-Diffusion simulators, posing itself as the most valuable alternative. Indeed, the MC technique has moved a long distance since its first introduction as a tool for the study of charge transport in semiconductors. The outstanding improvements in MC device simulations and the vaste range of applications they have reached is due to the availability of very fast workstations and mainframes, which guarantee considerable speed at affordable costs, allowing the implementation of more and more sophisticated physical models, and to the optimization of the numerical algorithms. For instance, it is possible nowaday to incorporate a full band structure description, or to account for quantum or many body effects [1-5]. In the present paper, we will concentrate on some recent results obtained for GaAs unipolar and bipolar devices, which illustrate quite effectively the strength and the capabilities of the MC simulation. It is impossible, in the limited space available, to account for the enormous amount of work originated in recent years in the field, and we have to refer the interested reader to the vaste literature existing on the subject, starting for example from the bibliography that can be found in Refs. [1,4,5]. 0167-9317/92/$05.00 © 1992 - Elsevier Scienc6 Publishers B.V.' All rights reserved.
276
P Lugli / Monte Carlo simulation of charge transport
2. THE PHYSICAL MODEL The transport model developed for GaAs is based on a three-valley description of the conduction band and on three valence bands. Non parabolic isotropic dispersions are used, with non parabolicity factors treated as fitting paramenters and adjusted as to reproduce a variety of experimental results, including those provided by time resolved spectroscopy. The carrier interaction with polar optical, acoustic, equivalent and nonequivalent intervalley, intraband and interband phonons is considered, within the usual golden rule scheme [1]. In the presence of heavily doped regions, carrier-carrier scattering (including plasmon contributions) are taken into account. Impurity scattering is treated using the Brooks-Herring approach. A list of all relevant parameters is presented in Table 1. The main emphasis of the paper is on very high field effects, where impact ionization phenomena become crucial. We have adopted the model proposed by Kane [6], where the energy dependence of the ionization process is given by P(E)= (P/T) (E-Eth) a, E being the total carrier energy. Kane model gives a softer threshold for the ionization with respect to the more standard Keldish formalism (Fig. 1) [7]. The free parameters have been determined through the fit of the bulk ionization coefficient and are given in Table 1. Bandto-band tunneling and its dependence on electric field has also been accounted for on the basis of Kane's theory [8]. The simulation of impact ionization phenomena poses considerable numerical problems since it requires the knowledge of the high energy tail of the carrier distribution function. Furthermore, in the presence of carrier multiplication the number of simulated particles would grow above the initially set value, diverging as breakdown is approached. We have therefore developed a special multiplication technique for both energy and real space, which is an extension of the original idea of Philips and Price [9]. Each particle is assigned a statistical weight which varies with its position in the device and its energy. With such approach, it is possible to account for regions with very different doping levels (as in bipolar transistors) and to obtain a reliable statistics of rare processes, keeping at the same time a constant number of particles. For the simulation of devices, the MC algorithm is selfconsistently coupled to a Poisson solver, as described extensively in Ref. [1]. 10TM "-
GaAs
~" ""~ ~) r-"' 1013 1012
300K
...f----
;~00")1011 l O ~o
1.00
/ / // ,
I
/,
1.50
d
2.00
Kane H ,
I
2.50
energy (eV)
,
t
3.00
eo,,
3.50
Figure l Comparison of the room temperature scattering rates for electron impact ionization calculated from Kane (solid line) and Keldish (dotted line) models.
P. Lugli / Monte Carlo simulation of charge transport [ Electron parameters
(300K)
Acoustic deformation potential (eV) Effective mass ( m ' / m o ) a (nonparabolicity)(eV -1) Band-edge energy (eV) Intervalley deformation potential (eV/crn) from r (000) from L (111) from X (100) Intervalley phonon energy (meV) from I" (000) from L (111) from X (100) Number of equivalent valleys
Hole parameters (300K) Acoustic deformation potential (eV) Effective mass (m*/mo) Band-edge energy (eV) a (nonparabolicity) (eV -1) Interband deformation potential (eV/em) from H.H. form L.H. fromS.O. Interband phonon energy (rneV) from H.H. from L.H. from S.O. Parameter Value Density (g/cm 3) 5.36 Sound velocity (cm/s) 5.22.105 e~ 10.92 e0 12.90 Optical phonon energy (meV) 35.36 Electron ionization coefficients P / r (s -1) 6.22. 1012 a 3.2 Eth (eV) 1.439 Hole ionization coefficients PIT (.$--1) 3.11.1011 a 6.35 Eth (eV)
1.439
277
r (000)
L (111)
X (100)
7.0 0.063 0.610 1.439
7.0 0.222 0.244 1.739
7.0 0.58 0.061 1.961
0.0 7.0- 108 1.0- 109
7.0.108 1.0. 10 9 1.0. 10 9
1.0.109 1.0.10 9 1.0.10 9
0.0 27.8 29.3 1
27.8 27.8 29.3 4
29.3 29.3 29.3 3
H.H. 7.0 0.45 0.0 0.55
L.H. 7.0 0.085 0.0 4.0
S.O. 7.0 0.154 0.35 0.3
9.0.10 s 7.0.10 s 5.0.108
7.0- l0 s 9.0.108 5.0.10 s
5.0. l0 s 5.0. l0 s 9.0. l0 s
27.8 27.8 29.3
27.8 27.8 29.3
29.9 29.3 29.3
Table 1 Parameters used in the MC simulation for electrons (upper table) and holes (middle table). Other parameters, including the impact ionization ones, are reported in the lower table.
278
P. Lugli / Monte Carlo simulation of charge transport
3. HIGH FIELD TRANSPORT
We present now the results of the MC simulation of electron and hole transport in homogeneous high electric fields based on the model previoulsy described. All results are presented for 300 K and 500 K. The drift velocity versus electric field characteristics (Fig. 2) show the well known negative differential mobility region for electrons, with peak field around 4 kV/cm. The hole velocity also shows a slight decresase, after reaching a maximum around 7 kV/cm, in correspondence to a sizable transfer from the heavy-hole to the light-hole and split-off bands. Our results are in good agreement with other MC calculations that use full band structure models [10] and with available experimental results. 108
......
I
........
I
........
I
n-GaAs E
10 7
......
I
'
'
' A ' I
........
- p-Ga s T =
lo
[
. . . . .
300~
>= ~,~/X"
T-- 500 K
z~"
T - 500 K
10 5
1061 . . . . . . J . . . . . . . . 103
, ........ 104
, ........ 105
I 106
[ ...... I ........ 103
F ( V/cm )
t 104
. . . . . . . .
I
10 6
10 s
F ( V/cm )
Figure 2 Drift velocity versus electric field curve for electrons and holes at 300 K (solid lines) and 500 K (dashed lines) > (1) v
(1) t(D
.....1 ........ I ........ I ........ I
lO
-
. n-GaAs.
E
(1)
10 0
v
......P ........i ........i ...... _ P- -Ga#~ ~-
03 0) ¢(1)
10 -1
.=_
10 "1
c-
t(1)
>
102
...... I ........ L ........ I ........ I ~ 10 0 101 10 2 10 ~
F ( kV/cm )
((D ~ E
1 0. 2
...... I
........
10 0
I
101
........
I
........
10 2
I 10 3
F ( kV/cm )
Figure 3 Electron and hole mean energy as a function of the electric field at 300 K (solid lines) and 500 K (dashed lines)