30
TRANSIENT SIMULATION OF MOSFETS
M. Razaz and S. F. Quigley
Department of Electronic and Electrical Engineering,
University of Birmingham, UK.
SUMMARY
C o n s i d e r a b l e r e s e a r c h has been c a r r i e d out i n t h e p a s t on the d e v e l o p m e n t of f a i r l y a c c u r a t e s t e a d y s t a t e models f o r s m a l l s i z e MOS d e v i c e s . These models a r e , i n g e n e r a l , n o t c a p a b l e of d e a l i n g with the device s w i t c h i n g a n d AC c h a r a c t e r i s t i c s . This paper p r e s e n t s a t r a n s i e n t model which can s i m u l a t e a c c u r a t e l y both t r a n s i e n t and s t e a d y s t a t e behaviour- of micron and submicron MOS d e v i c e s i n two s p a c e d i m e n s i o n s . The model has been implemented i n a f l e x i b l e and u s e r - f r i e n d l y s i m u l a t o r which can e a s i l y h a n d l e b o t h p l a n a r and non-planar MOS s t r u c t r e s . t y p i c a l s i m u l a t i o n r e s u l t s f o r t h e t r a n s i e n t r e s p o n s e of s u c h d e v i c e s a r e p r e s e n t e d and d i s c u s s e d .
1 INTRODUCTION
MOS t e c h n o l o g y ovex t h e p a s t two decades has b e e n c h a r a c t e r i z e d by a s t e a d y r e d u c t i o n of t h e minimum d e v i c e f e a t u r e s i z e . Due t o the i n h e r e n t two- o r t h r e e - d i m e n s i o n a l geometry of such devices and c o m p l i c a t e d i n t e r a c t i n g i n t e r n a l mechanisms i n v o l v e d , e v e n s o p h i s t i c a t e d a n a l y t i c a l models, which r e q u i r e p r a c t i c a l c h a r a c t e r i z a t i o n of many p a r a m e t e r s , f a i l to p r e d i c t t h e d e v i c e b e h a v i o u r a c c u r a t e l y . Thus numerical s i m u l a t i o n becomes a n e c e s s i t y for a c c u r a t e p r e d i c t i o n of t h e d e v i c e s t a t i c and dynamic c h a r a c t e r i s t i c s .
I n t h e p a s t , c o n s i d e r a b l e e f f o r t has been p u t i n t o t h e development of r e l i a b l e and a c c u r a t e s t e a d y s t a t e n u m e r i c a l models of MOS d e v i c e s . The e x t e n t of t h i s work has b e e n i n d i c a t e d i n r e c e n t review papers [ 1 , 2 ] . However, s u c h models can o n l y r e v e a l t h e DC behaviour of a MDSFET.
T r a n s i e n t m o d e l l i n g i s n e c e s s a r y f o r the d e t e r m i n a t i o n of
AC and dynamic c h a r a c t e r i s t i c s and a l s o f o r i n v e s t i g a t i n g t r a n s i e n t breakdown mechanisms such as p a r a s i t i c b i p o l a r

31 action in MOSFETs and parasitic thyristor action in CMOS structures.
Comparitively little work has been done on transient modelling of MOS devices [3,4,5]. The model in [3] employs stream functions and those in [4,5] are based on the finite difference approach, and cannot deal effecitively with non-planar MOS devices which are used in today's VLSI technology. This paper presents an accurate two-dimensional transient model for a MOSFET. The potential and charge distributions within the device, and hence 'the contact currents, at any instant in time are determined by solving the time-dependent Poisson and continuity equations subject to appropriate initial and boundary conditions. A triangular-based finite element method is used for space discretization and the
Crank-Nicolson method for time domain solution. Care has been taken to use appropriate generation mechanisms as well as a physically meaningful mobility model that takes account of the electric field, doping density, temperature, and various scattering mechanisms. The transient model has been incorporated into a user-friendly simulator [6] which has its own preprocessor, mesh generator and postprocessor.
The latter performs various postprocessing tasks, such as current and field calculation, DC or AC analysis and graphical display of device behaviour in one-, two- or three-dimensional forms.
Transient simulation of an MOS device tends, in general, to be computationally expensive. Therefore much effort has been spent on making the model efficient for a given device, firstly by using justifiable approximations, secondly by using a flexible grid with combined coarse and refined meshes and finally by employing fast direct-solution algorithms with bandwidth reduction facility. The accuracy of the model has been tested against published MOSFET results that have been obtained by an entirely different steady state approach. The agreement has been found to be good. Typical computed results are presented for the transient response and DC threshold characteristic of two short-channel planar and recessed nMOS transistors.
2 THE TRANSIENT MODEL
To develop a transient model for a MOSFET, it is necessary to find the potential and charge distribution within the device at any instant of time. This information can then be used to derive instantaneous contact currents and hence the dynamic response to an input excitation. The potential y, for a given charge distribution, can be determined from the Poisson equation:

32 v.evv = g ( n - p - N ) (l) where N is the net doping profile and g the Si or Si02 permittivity, n and p are the electron and hole densities, which can be found from the continuity equations: q l t • - "
V
-
J P
q R ( 3
>
The individual and total current densities are given by
J n
= - q ( n p n
W - D n vn ) (4)
J = - q ( p/J p
W + D p V p ) (5)
J
T
= J n
+ J p " < H £ <6)
The symbols in (2)-(G) have their usual meanings. Since the current densities are directly dependent on mobilities and diffusion coefficients, it is important that they are modelled accurately. Mobilities and diffusion coefficients are affected by lattice impurity scattering, carrier heating due to electric fields, and surface roughness scattering due to the Si-Si0
2
interface. In the vicinity of the interface, the mobility of the carriers is also modified by quantum mechanical effects due to the extremely narrow potential wells. The models used for mobilities and diffusion coefficients of carriers in the bulk and at the interface are similar to those in [7,2],
The net recombination rate R is, in general, given by
R
=
R
H
+ R S
R H
+ R
SURF
+ R
AUG
( 7 ) where R
SRH
, R S U R F a n d R
AUG are the Schockley-Read-Hall, surface, and Auger recombinations and R n
represents the impact ionization.
The expression used for R is

33
2
"SRH T n
(p+p t
) + T p
(n+xi t
) *
; where r»j is the intrinsic concentration, r n
and x p
the electron and hole lifetimes, and nt and p t
the trap levels, which are assumed to be equal to nj . The doping dependence of -i n
and r p
are taken from [9], For the recombination at the MOS interface, r n
and T
P
in (8) are replaced by the inverse of the electron and hole surface recombination velocities.
RAUG and RJI are represented by the following expressions:
R
AUG
= ( c n n + C p
P
> < " P - »J > <
9
>
R
TT
= -q|J la - q|J la (10)
II
XM
n
1
n ^' p
1
p
v
' where c n
and c p
are the Auger capture coefficients [0] and an and apthe impact ionization coefficients [10],
To compute y, n and p distributions within the device, and hence to obtain a complete model of the device behaviour, equations (l)-(3), together with their auxiliary equations are solved at each timestep sequentially using efficient LU decomposition algorithms. The time scheme is handled by employing the Crank-Nicolson method [11]. An alternative method is to linearize the matrix equations and to solve them simulataneously using a full implicit time scheme. Although the latter approach has a faster convergence and its use may be mandatory for large drain biases when a steady state approach is employed, it requires at least three times more storage which can be problematic for computers with small core memory. A number of assumptions can be made to reduce the computational effort needed for modelling a MOSFET using the sequential approach. An obvious simplification is to ignore the solution of the majority carrier continuity equation by assuming that the quasi-Fermi level of the majority carriers is flat across the device. This simplification is acceptable for steady state calculations, provided that impact ionization is negligible. However, it is not so useful for modelling of transients, during which appreciable substrate current flows, and it cannot give an accurate model of avalanche breakdown. Solving the minority carrier continuity equation in one dimension along the Si-S.i02 interface, rather than two-dimensionally, can

34 also save computing time. The timestep of the
Crank-Nicolson method, and hence the overall solution time is directly affected by the local dielectric relaxation time. This restriction may become severe in a MOSFET, due to the heavily doped source and drain regions and the sudden increase of the channel charge density caused by a large applied gate bias. To alleviate this restriction and to speed up the solution time, the heavily doped regions of the source and drain diffusions may be modelled by
Dirichlet boundary conditions. This has the effect of partially reducing accuracy, but a considerable gain in speed justifies its use.
3 SIMULATION RESULTS AND DISCUSSION
The two structures modelled are shown in fig. la and lb. The first is a planar n-channel MOSFET of 0.8 p m
gate length, 0.1 pm oxide thickness and 0.2 pm junction depth.
The device length is 1 pm and the substrate doping is 10
1S cm-3 , The gate niaterial is aluminium, giving a flat-band voltage of -,88V. The second is a similar device but with its channel recessed by 0.1 pm. The doping profile of the planar device is shown in fig. 2. It corresponds to an annealed phosphorus ion implantation into a constantly doped substrate. The doping profile of the recessed device is similar. The recessed channel is normally introduced to minimize the threshold voltage shift that occurs in short-channel MOSFETs due to the penetration of the drain field into the channel region. The effect of the recess is to reduce the effective junction depth of the drain diffusion and thus decreasing its influence on the channel.
Vs Vg Vd Vs Vg Vd
( a ) P l a n a r d e v i c e ( b ) Recessed d e v i c e
F i g . 1 MOS s t r u c t u r e s modelled

35
Fig. 2 Doping p r o f i l e used for planar device
R e c e s s e d
MOSFET
44-U-44 Jq-UXlXXM-t t J-U J U U l i i - l ^ l t L t l X u X L U x u !
- 2 2 B 10 14
Vg ( V ) X 1 0
- 1
Fig. 3 Threshold c h a r a c t e r i s t i c s of the planar and recessed nMOS devices shown in f i g . 1

36
The DC transfer characteristics of both structures are presented in fig. 3, which shows that both devices are normally-on and that the ' threshold voltages are approximately -0.6V for the planar and -0.4V for the recessed structure. The transconductance of the recessed device is lower than that of the planar device by approximately a factor of four. This is due to the smaller amount of charge under the gate in the recessed case.
Fig. 4 shows the transient response of the planar device to a step in gate bias from 0V to 0.25V at Vd
S
=0.5V.
Fig. 4a-4d present the source, gate, drain and substrate currents respectively. The conduction and displacement components of the source, drain and substrate currents are also presented separately in fig. 5. The contact currents in each case takes about 30 ps to reach a steady state, and the characteristic consists of two distinct phases. The first, which lasts about 5 ps, is characterized by a strongly enhanced substrate current, corresponding to a strong flow of electrons from the source into the channel, and a diminished drain current, corresponding to a reduced flow of electrons from the channel into the drain. This phase of the transient is the period during which an extra charge under the gate builds up. It co-incides with the peak of the flow of displacement current into the gate.
The second phase is characterized by an almost constant source current and a slow fall in the gate current, matched by a slow rise in the drain current. This phase corresponds to the redistribution of potential along the newly enhanced channel. The form of the substrate current characteristic is interesting, changing sign twice during the transient. This is due to the opposing flow of positive conduction current, associated with positive charge being forced out of the device by the increased gate field, and negative displacement current, due to the redistribution of electric fields within the device.
The corresponding results for the recessed-channel device are shown in fig. 6. The form of the transients is much the same, but the planar device switches almost three times as fast as the recessed-channel device. This difference in speed is caused by the penetration of the drain field into the channel region, which lowers the gate capicitance to a much greater extent for the planar structure.

37
^Uiii|Hnpiir|nirpin|Hii|nnpnipiH[innim|i
- 2 |
6 12 IB 2 4 t (ps)
(a) Source current
I
Utl UlHll lllllUilUlllMMblUtlUltlUllllllUlUllHlluilLlig
B 12 18 24 t (ps)
(b) Gate current
IB 24 t (pa)
(c) Drain current
*2niiiliiiilini'itiiliiiiliiii1iiiiliiiiliiiilimliinlini1iml**il
0 8 12 IB 24 t (ps)
(d) Substrate current
Fig. 4 Contact current t r a n s i e n t s of the planar nMDS device for a s t e p change in gate voltage from Vgs=0V to Vgs=0.25V

38
I I I 1,1 I I I I l-U.L1XI4-U-U.I-l I I I 1-1
6 IS 18 24 t (ps)
E
O
V.
<
E
H r r r r H T T ' I • I ' I ' I ' I ' I ' ;
Of
~ 2 |
- 4 |
- B |
-B§
-10f
- 1 2 |
-14§
-1B§
-IBS
- 2 0 |
- 2 2 |
X d i s p ] cond i
-24f
- 2 B |
-281
>xj-Lt-i.i-ti-i-i-u-u
-
Lt
-HJ ' ' ' i . ' i - '
8 12 18 24 t (ps)
(a) Source current components
(b) Drain c u r r e n t components f—S i | n - i i ' l ' i I ' l l ' " I ' l l i T » T r r i ' | T | - i ;
:
N.
:
• \ .
x cond : w
2
H
' I
n
- 2
" y^"^ "
- 4 i / d i s p i
- 8 A / :
- 8 f i 1 1 1 1 1 i.l i.l.i.l i.l. i J . i . l J lg.1 M i l l '
8 12 18 2 4 t (pa)
(c) Substrate current components
Fig. 5 Conduction and displacement components of contact current transients for the planar nMOS device

39
4
UUIHIIllimimTmitmmtttmnnmmtTtTtiT^
E
3 u
^ 2
(0
H
( a ) ilUmUm'mUm.
4 B t (pa)
Source c u r r e n t
4 B
X 1 0
1 t (ps)
( b ) Gate c u r r e n t
: hiiiliiuUmtnuluiiliuUtutUmlmiliuilniri
4 B t (ps)
( c ) Drain c u r r e n t qlullllllllltllllllfl
4 B
X 1 0
1 t (ps)
( d ) S u b s t r a t e c u r r e n t
F i g . 6 C o n t a c t c u r r e n t t r a n s i e n t s of t h e r e c e s s e d nM3S d e v i c e f o r a s t e p change i n g a t e v o l t a g e from Vgs=0V t o Vgs=0.25V

40
I 1 I I 1 I H p - ^ T - T I T - T - l - l - T r T T T
| 80
<
E n
5B
5B
54 • . . . I . . . . I . . . . I • • • • I • 1 I I I I I I 1 xio* t (pa)
(a) Source c u r r e n t
— g p i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i n I.I . . . i . . . .H
X 1 0
1 t (ps)
(b) Gate current
X 1 0
1
X 1 0
1
*
(
P
Q )
t (pa)
(c) Drain current (d) Substrate current
Fig. 7 Contact current transients of the planar nMOS device for a step change in drain voltage from Vds=0.25V to Vds=0.5V

4 1
X10"
B 9"""" 12
X 1 0
1 t (pa)
(a) Source c u r r e n t
„—JuttTUitMiiJiiuliiHlnuimilHUluil.
3 B 8 12
X I O
1 t (ps)
(b) Gate current
XIO - 1 uiniilu|Hiuim|nntnnpmmtij
( c ) uuliiiilimln
6 g x i o
1 t (pa)
Drain c u r r e n t
(d]
6 g xio
1 t (pa)
Substrate current
Fig. 8 Contact current transients of the recessed nMDS device for a step change in drain voltage from Vds=0.25V to Vds=0.5V

42
F i g . 7 p r e s e n t s t h e t r a n s i e n t r e s p o n s e of t h e r e c e s s e d d e v i c e a t V g s
= - l V t o a s t e p i n d r a i n b i a s from 0.25V t o
0.5V. The s o u r c e c u r r e n t remains a l m o s t c o n s t a n t t h r o u g h o u t . The d r a i n c u r r e n t s t a r t s a t a very h i g h v a l u e and t h e n g r a d u a l l y d e c r e a s e s over t h e next 4 0 p s . The h i g h c u r r e n t c o r r e s p o n d s t o t h e removal of e l e c t r o n s from t h e d r a i n end of t h e channel due t o t h e e f f e c t of i n c r e a s e d d r a i n e l e c t r i c f i e l d on t h e i n v e r s i o n l a y e r . The form of t h e g a t e d i s p l a c e m e n t c u r r e n t , caused by t h e r e d i s t r i b u t i o n of p o t e n t i a l w i t h i n t h e c h a n n e l , i s s i m i l a r b u t o p p o s i t e i n s i g n . The .corresponding r e s u l t s f o r t h e p l a n a r d e v i c e a r e g i v e n i n f i g . 8 . Once a g a i n , t h e y a r e s i m i l a r i n form t o t h o s e of f i g . 7 , t h e main d i f f e r e n c e b e i n g t h e s p e e d of s w i t c h i n g .
4 REFERENCES
1 . W. L. Engl e t a l .
"Device m o d e l l i n g "
P r o c IEEE, v o l . 7 1 , p . 1 0 , 1983
2 . S . S e l b e r h e r r
"Numerical modelling of MOS devices: methods and problems"
L e c t u r e n o t e s , NASECODE IV, p . 122, 1985
3 . M. S . Mock
"A time-dependent numerical model of the insulatedgate field-effect transistor"
Solid-State Electron., vol. 24, p. 959, 1981
4. S. Y. Oh et al.
"Transient analysis of MOS transistors"
IEEE Trans. Electron Dev., vol. ED-27, p. 1571, 1980
5 . K. Yamaguchi
"A time d e p e n d e n t and two-dimensional n u m e r i c a l model f o r MOSFET d e v i c e o p e r a t i o n "
S o l i d - S t a t e E l e c t r o n . , v o l . 2 4 , p . 959, 1981
6 . M. Razaz and S . F . Quigley
"A t w o - d i m e n s i o n a l f i n i t e element dynamic s i m u l a t o r f o r s e m i c o n d u c t o r d e v i c e s "
P r o c . NASECODE IV , p . 488 , 1985
7 . C. J a c o b i n i . e t a l .
"A r e v i e w of some c h a r g e t r a n s p o r t p r o p e r t i e s of s i l i c o n " , S o l i d - S t . E l e c t r o n . , v o l . 20 , p . 77 , 1977

43
8. J. Dziewior and W. Schmid
"Auger coefficients for highly doped and highly exerted silicon", Appl. Phys. Lett. , p. 346 , 1977
9. A. Nakagawa
"One dimensional device model of the npn bipolar transistor including heavy doping effects under Fermi statistics"
Solid-St. Electron. , vol. 26 , p. 287 , 1983
10. R. Van Overstraeten et al.
"Measurement of the ionization rates in diffused silicon p-n junctions"
Solid-St. Electron. , no. 13 , p. 583 , 1970
11. G. Strang and G. J. Fix
"An analysis of the finite element method"
Prentice Hall , NJ , 1973