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TRANSIENT REGIMES OF HOT CARRIERS IN p-TYPE SILICON
L. Reggiani, J.-C. Vaissière, J.-P. Nougier, D. Gasquet
To cite this version:
L. Reggiani, J.-C. Vaissière, J.-P. Nougier, D. Gasquet. TRANSIENT REGIMES OF HOT CAR- RIERS IN p-TYPE SILICON. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-357-C7-367.
�10.1051/jphyscol:1981744�. �jpa-00221681�
JOURNAL DE PHYSIQUE
Colloque C7, supplément au n°10, Tome 42, ootobve 1981 page C7-357
TRANSIENT REGIMES OF H O T CARRIERS IN P-TYPE SILICON
. . - * * *
L. Reggiam, J.C. Vaissiere, J.P. Nougier and D. Gasquet
Gruppo Nazionale di Stvuttura delta Materia, Istituto di Fisiea, Universitd di Modena, Via Campi 213/A, 41100 Modena, Italy
*Vniversite des Saienaes et Techniques du Languedoe, Centre d'Etudes d'Electro- nique des Solides, Laboratoire associe au C.N.R.S. LA 21 et Gveao Microondes, 34060 Uontpelliev Cedex, France
Résumé : On compare les régimes t r a n s i t o i r e s obtenus, sur Si-p à 300 K, par simulation de Monte Carlo et résolution i t é r a t i v e de l'équation de Boltzmann : l'accord est excellent- Le régime t r a n s i t o i r e : a) dépend assez peu des for- mes de bandes d'énergie, b) dépend beaucoup de l'énergie i n i t i a l e , une aug- mentation de c e l l e - c i peut f a i r e d i s p a r a î t r e l a survitesse, c) dépend peu de la forme de la d i s t r i b u t i o n i n i t i a l e . Dans tous les cas, l e régime t r a n s i t o i - re déterminé en u t i l i s a n t les équations dynamiques donne un accord très sa- t i s f a i s a n t avec les précédentes simulations.
Abstract : A comparison is made between transient regimes obtained, on p-Si at 300 K, by a Monte Carlo simulation and by an iterative solution of the Boltzmann equation : the agreement is excellent. The transient regime : a) depends relatively few on the energy bandshapes; b) Is strongly dependent on the initial energy, the increase of which may lead to a disappearance of the velocity overshoot; c) Slowly depends on the shape of the initial dis- tribution function. In every cases, the transient regime computed using the balance equations is in very satisfactory agreement with the previous simu- lations.
1. INTRODUCTION
For studying transient regimes in homogeneous semiconductors, three methods can be used, i) iterative technique, ii) Monte Carlo simulation, iii) balance equa- tions. within the frame of the classical Boltzmann equation, the first two methods are exact numerical solutions ( [_ 1 - 6]). However, no direct comparison of the transient regime obtained by both methods have been performed till now. In this paper, various transient regimes are compared for the case of holes in Si using both methods with the same band structure and the same scattering mechanisms in order to
check their reliability on a practical case of interest. Transient regimes corres- ponding to various initial conditions are studied. The influence of valence band particularities (such as nonparabolicity and warping) are investigated. Further- more, comparison of these methods with the balance equations is carried out with the aim to provide a reference for estimating differences between realistic and simplified models used in practical calculations.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981744
A f t e r a b r i e f r e c a l l o f t h e methods used ( s e c t i o n 2 ) , we s h a l l compare ( s e c t i o n 3) the v a r i o u s models, then study the i n f l u e n c e s o f the d i s p e r s i o n r e l a t i o n
~ ( i )
( s e c t i o n 4), of t h e i n i t i a l energy d i s t r i b u t i o n ( s e c t i o n 5),and o f t h e i n i t i a l d i s t r i b u t i o n f u n c t i o n ( s e c t i o nG),
before g i v i n g an example showing an undershoot v e l o c i t y ( s e c t i o n 8 ) .2. PIETHODS USED
The values of t h e t r a n s p o r t q u a n t i t i e s o f i n t e r e s t f o r a t r a n s p o r t dynamic response can be obtained from a Monte Carlo procedure by s i m u l a t i n g t h e behaviour of a number o f independent p a r t i c l e s and t a k i n g ensemble averages over t h e whole gas, a t given values of time t (7
] .
For our case, 8192 p a r t i c l e s have been con- s i d e r e d and t h e space d i s t r i b u t i o n has been assumed t o be o f a D i r a c t y p e ; d i f f e - r e n t i n i t i a l energy d i s t r i b u t i o n s w i t h d i f f e r e n t mean values have been considered, as w i l l be discussed i n t h e f o l l o w i n g . As i n t h e s t a t i o n a r y case, t h e s e l f s c a t t e - r i n g mechanism has been i n c l u d e d i n t h e s i m u l a t i o n . O f course, t h e r e s u l t s have been found t o be independent from t h e choice o f t h e s e l f - s c a t t e r i n g parameterr ,
con- t r a r y t o what s t a t e d i n r e f s .[
8] [
9],
provided t h a tr
i s l a r g e enough.The i t e r a t i v e method used f o r s o l v i n g t h e Boltzmann equation was described i n r e f . L 1 0 1
,
and has been extended here t o t h e case o f a two bands semiconductor.The method o f balanced equations was r e c e n t l y shown [ll] t o be a good appro- x i m a t i o n deduced from t h e Boltzmann equation. I t s a p p l i c a t i o n proved t o be success- f u l c11
] [
1 2 1 i n n-Si, p-Ge and n-GaAs.3. COMPARISON BETWEEN THE MODELS
I n order t o compare t h e rlonte Carlo and i t e r a t i v e s o l u t i o n s on t h e same mi- croscopic model, and t o a v o i d two much time consuming computations, we used a simpli- f i e d band s t r u c t u r e model c o n s i s t i n g i n a s i n g l e s p h e r i c a l band w i t h an e f f e c t i v e mass which, t o account f o r nonparabolic e f f e c t s , has been taken t o be dependentupon the e l e c t r i c f i e l d s t r e n g t h through the s t a t i o n a r y value o f t h e h o l e mean energy.
Acoustic s c a t t e r i n g , which accounts f o r o v e r l a p e f f e c t s , and non-polar o p t i c a l scat- t e r i n g , have been considered. The values used i n c a l c u l a t i o n s are r e p o r t e d i n Table l, and F i g . 1 shows t h e dependence o f t h e e f f e c t i v e mass w i t h t h e e l e c t r i c f i e l d s t r e n g h . The r e l i a b i l i t y o f t h e microscopic model, which d i f f e r s from t h a t o f Ref. [ l 3 1 o n l y f o r warping e f f e c t s , i s checked w i t h t h e comparison o f t h e o r e t i c a l and experimental r e s u l t s o f t h e s t a t i o n a r y values o f t h e d r i f t v e l o c i t y and l o n g i t u d i n a l d i f f u s i o n c o e f f i c i e n t (14
] ,
as shown i n F i g . 2. The overlapp was taken i n t o account by s e t t i n g G ( C , ~ ' ) = 1/2, thus having t h e e f f e c t o f deviding by 2 t h e value o f E;given i n t a b l e 1.
The i n i t i a l d i s t r i b u t i o n f u n c t i o n was assumed t o be t h e thermal e q u i l i b r i u m Maxwell Boltzmann d i s t r i b u t i o n a t 300 K.
Q u a n t i t y U n i t One s p h e r i c a l parabol i c band
V a l u e
b o l i c and one non p a r a b o l i c
Table 1. Constants used i n t h e c a l c u l a t i o n s f o r t h e v a r i o u s band models used i n t h e present paper. The n o t a t i o n s a r e t h e usual ones and t h e q u a n t i t i e s a r e d e f i n e d , o r given i n t h e references l i s t e d . The non para- b o l i c band c o n s i s t s i n two a r a b o l i c p o r t i o n s o f masses m10 and mll, l i n k e d by a l i n e a r p o r t i o n f153
.
F i g . 1 : Equivalent e f f e c t i v e mass as a Fig. 2 : Comparison between the e x p e r i - f u n c t i o n o f the e l e c t r i c f i e l d f o r t h e mental r e s u l t s , of d r i f t v e l o c i t y and one band model, see Ref.
[
151.
d i f f u s i o n , a n d t h e t h e o r y u s i n g t h e oneband model (taken from
[
14]
).
Figures 3a and 3b show t h e t r a n s i e n t d r i f t v e l o c i t y and t h e t r a n s i e n t energy f o r a s e t o f e l e c t r i c f i e l d s suddenly a p p l i e d a t t = 0 ( E = 10, 20 and 50 kV cm-').
The values obtained w i t h the Monte Carlo ( c i r c l e s , t r i a n g l e s and s t a r s ) a n d the i t e - r a t i v e method ( d o t t e d curves) a r e found t o c o i n c i d e w i t h i n t h e s t a t i s t i c a l uncer- t a i n t y . T h i s evidence proves t h e complete re1 i a b i l i t y o f both methods t o solve t h e Boltzmann equation under t r a n s i e n t c o n d i t i o n s . I n p a r t i c u l a r , d r i f t v e l o c i t y i s found t o e x h i b i t overshoot f o r f i e l d s t r e n g t h s above about 20 kV/cm. No overshoot was found f o r t h e energy.
On f i g u r e 3 a r e a l s o p l o t t e d t h e t r a n s i e n t regimes computed u s i n g t h e balance equations : these r e s u l t s a r e i n agreement w i t h i n 10 % w i t h t h e "exact" ones, which i s much s a t i s f a c t o r y if one takes i n t o account t h e g r e a t s i m p l i c i t y and r a p i d i t y o f t h i s method. These r e s u l t s a r e s i m i l a r t o those p r e v i o u s l y obtained [ll] [ l 2 1 on p-Ge, n-Si and n-GaAs.
Fig. 3 : T r a n s i e n t d r i f t v e l o c i t i e s (a) and energies ( b ) i n p-Si a t 300 K, f o r v a r i o u s values o f suddenly a p p l i e d e l e c t r i c f i e l d s a t t = 0. Comparison between : i t e r a t i v e s o l u t i o n o f t h e Boltzmann equation ( d o t t e d curves), Monte Carlo s i m u l a t i o n ( c i r c l e s , t r i a n g l e s and s t a r s ) , and balance equa- t i o n s ( f u l l curves).
INFLUENCE OF THE BANDSHAPES ON THE TRANSIENT REGIMES :
During t h e t r a n s i e n t regimes, t h e "average p o s i t i o n " o f t h e c a r r i e r s i n t h e
/c/
space vary, so as t h e i r "average mass". Therefore, one c o u l d expect t h a t f o r various bandshapes t h e t r a n s i e n t motion i s n o t t h e same. T h i s i s shown on Fig. 4 where t h e t r a n s i e n t d r i f t v e l o c i t y i s p l o t t e d f o r an e l e c t r i c f i e l d s t e p o f E = 50 kV cm-', a p p l i e d a t t = 0, and f o r various d i s p e r s i o n laws ~ ( k ) . The f o l l o w i n g mo- dels have been used, t h e i r parameters are l i s t e d t a b l e 1 :( i ) One s p h e r i c a l p a r a b o l i c band described i n t h e previous s e c t i o n ( i i ) Two s p h e r i c a l p a r a b o l i c bands w i t h e f f e c t i v e masses mh and me
( i i i ) T w o s p h e r i c a l bands, a l i g h t h o l e band p a r a b o l i c and t h e heavy h o l e band non
p a r a b o l i c c o n s i s t i n g i n two p a r a b o l i c p o r t i o n s , o f e f f e c t i v e masses m10 and mll, l i n k e d by a l i n e a r p o r t i o n C15
] .
( i v ) One warped sphere band described i n
[
13] .
O f course o n l y t h e warped sphere model i s a b l e t o e x h i b i t anisotropy. As i s shown on F i g . 4, t h e d r i f t v e l o c i t y (and so i s t h e mean energy) i s higher along a
<loo> d i r e c t i o n than along a < I l l > d i r e c t i o n . T h i s r e f l e c t s t h e lower value o f t h e heavy h o l e e f f e c t i v e mass along t h i s d i r e c t i o n . I n p a r t i c u l a r , i t has t o be noted t h a t t h e a n i s o t r o p y e x h i b i t e d by t h e d r i f t v e l o c i t y reaches i t maximum v a l u e around t h e peak o f t h e overshoot where, f o r the f i e l d s t r e n g t h considered, v a r i a t i o n s o f about 30 % have been found.
Fig. 4 c l e a r l y shows t h a t d i f f e r e n t bandshapes g i v e d i f f e r e n t t r a n s i e n t d r i f t v e l o c i t i e s , even when t h e steady s t a t e v e l o c i t i e s a r e equal.
F i g u r e 4 : T r a n s i e n t v e l o c i t i e s f o r d i f f e r e n t bandshapes : warped spheres, E//<100> (A) and E//<111> ( o ) ; two s p h e r i c a l bands, one p a r a b o l i c and one non p a r a b o l i c ( 8 ) ; two s p h e r i c a l p a r a b o l i c bands (--) ; one s p h e r i c a l p a r a b o l i c band
(-1.
5. EFFECT OF THE INITIAL ENERGY
I n order t o i n v e s t i g a t e t h e e f f e c t o f t h e i n i t i a l energy on the t r a n s i e n t behaviour, an e l e c t r i c f i e l d step o f 50 kV/cm was a p p l i e d a t t = 0 t o p-Si a t 300 K w i t h the one s p h e r i c a l parabol i c band model discussed above, corresponding t o t h r e e d i f f e r e n t i n i t i a l c o n d i t i o n s described by t h e f o l l o w i n g d i s t r i b u t i o n f u n c t i o n s f o ( l ) ( 6 i s the D i r a c f u n c t i o n ) :
c o n d i t i o n s 1 : f o ( t ) .c 6(0) : a1 l the c a r r i e r s a r e i n i t i a l l y l o c a t e d a t
2
= 0,thus v ( 0 ) = 0 and
-
E ( 0 ) = 0.c o n d i t i o n s 2 : fo(%) a 6 ( k o ) where c ( k 0 ) = 0.75 eV : a l l t h e c a r r i e r s have t h e i n i t i a l energy 0.75 eV, b u t t h e i r v e l o c i t i e s a r e randomly d i s t r i b u - t e d : i ( 0 ) = 0 and E(0) = 0.75 e V . 1 1 7 1
c o n d i t i o n s 3 : f o ( x ) = thermal o q u i l i b r i u m d i s t r i b u t i o n a t 300 K : v(0) = 0 and - - ~ ( 0 ) = 0.039 eV.
O f course t h e c o n d i t i o n number 1 i s u n r e a l i s t i c ( t h e case number 2 could describe c a r r i e r s i n j e c t e d using an e l e c t r o n gunn), b u t these t h r e e cases are i n - t e r e s t i n g f o r t e s t i n g t h e r e l i a b i l i t y o f t h e methods s i n c e they r e p r e s e n t very extreme c o n d i t i o n s .
F i g u r e 5a and 5b show t h e r e s u l t s obtained on t h e t r a n s i e n t v e l o c i t y and e n e r gy u s i n g t h e t h r e e techniques o f s i m u l a t i o n (Monte Car1 o, i t e r a t i v e , balance equa- t i o n s ) . As can be seen on F i g . 5a, t h e overshoot i s maximum w i t h t h e c o n d i t i o n l, w h i l e i t disappears f o r t h e c o n d i t i o n 2. I n f a c t , i n t h e former case t h e c a r r i e r s s t a r t t o a c c e l e r a t e before t o d i s s i p a t e energy and, provided t h a t t h e f i e l d s t r e n g t h i s h i g h enough, as i t i s i n t h i s case, due t o t h e streaming e f f e c t o f t h e f i e l d they have t h e p o s s i b i l i t y t o reach v e l o c i t y values h i g h e r than t h e s t a t i o n a r y one. I n t h e case number 2 t h e c a r r i e r s simultaneously s t a r t t o a c c e l e r a t e and d i s s i p a t e energy through phonon emission; t h i s process i s q u i t e randomizing i n momentum and leads t o a smoth increase w i t h time o f t h e d r i f t v e l o c i t y , so t h a t overshoot e f f e c t s d i - sappear. Thus the i n f l u e n c e o f t h e i n i t i a l energy and d i s t r i b u t i o n i s q u i t e impor- t a n t on the t r a n s i e n t v e l o c i t y . The s l i g h t discrepancy appearing between t h e Monte Carlo and i t e r a t i v e s i m u l a t i o n s comes t o t h e f a c t t h a t , w i t h our i t e r a t i v e t e c h n i - que, d i s c o n t i n u i t i e s a r e forbidden, so t h a t we a r e unable t o simulate d e l t a func- t i o n s : these were then replaced by narrow Gaussian d i s t r i b u t i o n s , which tends t o lower t h e overshoot i n t h e case number 1. As concerning t h e t r a n s i e n t energies (see F i g . 5b), no overshoot i s observed. F i n a l l y i t can be remarked, on F i g . 5a, t h a t t h e slopes o f v ( t ) a t t h e o r i g i n a r e i d e n t i c a l : t h i s i s q u i t e normal, and eq. (16) of reference
Cl11
shows t h a t , w i t h o u t any approximation:when v ( t = 0 ) = 0 :( d ~ / d t ) ~ = ~ = eE/m and thus does n o t depend on t h e i n i t i a l d i s t r i b u t i o n provided t h a t v(t=O)=O. However, as can be seen from eq. (24) o f r e f .
[l11 - ,
( d ~ / d t ) ~ = ~ depends on c(t=O) even when v(t=O)=O, as i s o b v i o u s l y confirmed by F i g . 5b.F i n a l l y one should note t h a t t h e balance equations g i v e t r a n s i e n t curves i n q u i t e good agreement w i t h t h e exact methods.
Si-holes
Ez5O k ~ c r n - '
W
0.5
- o r Monte-Carlo
1 \,
.---iterative -balance equations4
Q25
F i g . 5 : T r a n s i e n t v e l o c i t i e s ( a ) and energies (b) computed u s i n g t h e t h r e e t e c h n i - ques o f simulation: (Monte Carlo
.
), i t e r a t i v e ( - . ..), and balance eqs.(-)), f o r various i n i t i a l c o n d i t i o n s :b:
f o ( k ) -6 (0) ;6
: fo(k) a 6 ( k o ) where€(ko) = 0.75 eV ;
@
: f o ( k ) = thermal e q u i l i b r i u m d i s t r i b u t i o n a t 300 K.6. INFLUENCE OF THE INITIAL DISTRIBUTION FUNCTION
The balance equations do n o t take i n t o account t h e d i s t r i b u t i o n f u n c t i o n , they describe o n l y the n o t i o n o f t h e average v e l o c i t y and o f t h e average energy. I n order t o g e t an i d e a o f t h e e r r o r performed by n e g l e c t i n g t h e shape o f t h e d i s t r i b u t i o n f u n c t i o n , we study t h e t r a n s i e n t v e l o c i t i e s ( F i g . 6a) and energies ( F i g . 6b) f o r a step o f e l e c t r i c f i e l d E = 50 kV/cm a p p l i e d a t t = 0, t h e c a r r i e r s being,at t=O, d i s t r i b u t e d according t o t h e two f o l l o w i n g laws g i v i n g both v(O) = 0 and
2 2
E(O) = 0.5 eV : 1) f o ( z ) a exp ( - -h k /2m kBTo) w i t h 3 kB To/? = 0.5 eV (thermal e q u i l i b r i u m a t To = 3870 K). T h i s c o n d i t i o n i s o f course u n r e a i i s t i c , b u t c o r r e s - ponds t o a l i m i t i n g case where a l l t h e energies i n the { x } space a r e occupied.
2 2 2 2
2) f o ( x ) a e x p
[-
A (k-kl) /2m kB T ~ ] w i t h kl such t h a t h k 1/2m = 0.5 eV and T1 = 40 K. This corresponds t o c a r r i e r s l o c a t e d i n a narrow r e g i o n o f+
5.2 meV around the energy 0.5 eV.The t r a n s i e n t responses were computed using t h e i t e r a t i v e method, f o r t h e s i n g l e s p h e r i c a l p a r a b o l i c band model. F i g . 6a and 6b show t h a t t h e discrepancy between the two t r a n s i e n t responses i s n o t very l a r g e . As a consequence, t h e des- c r i p t i o n given by t h e balance equations i s , i n t h i s case two, q u i t e s a t i s f a c t o r y as shown i n F i g . 6.
-
Si
-
holes 300K~r~crn"
-
F i g u r e 6 : T r a n s i e n t d r i f t v e l o c i t i e s (a) and energies ( b ) f o r p-Si, 300 K, i n a step o f E = 50 kV/cm, f o r two i n i t i a l d i s t r i b u t i o n s v(0) = 0 and ~ ( 0 ) = 0.5 eV ;
(.
. .
. . ) a l l t h e c a r r i e r s have an energy equal t o 500 k5.2 meV, and t h e d i r e c t i o n s o f t h e k v e c t o r a r e randomly d i s t r i b u t e d , i t e r a t i v e method ;(--
- ) m a x w i l l i a n d i s t r i b u t i o n centered a t k = 0, w i t h average energy 0.5 eV, i t e r a t i v e method ;(-)balance equations.
7. EXAMPLE : UNDERSHOOT VELOCITY :
L e t us apply a t time t = 0, a constant e l e c t r i c f i e l d El : a f t e r an overshoot ( i f El> 20 kV cm- 1 ) , t h e v e l o c i t y reaches i t s s t a t i o n a r y value. I f then, a t time tl, the f i e l d i s suddenly lowered and takes t h e constant value E2, t h e v e l o c i t y may exhi- b i t an undershoot value before s t a b i l i z i n g a t t h e steady s t a t e corresponding t o t h e f i e l d E2. This phenomenon i s shown on f i g u r e 7a and 7b where E2 was taken equal t o l 0 kV cm-', s t a r t i n g a t time tl = 1.5 ps.
he
s i m u l a t i o n was performed u s i n g t h r e e valuer of El (20, 50 and 100 X V cm-') w i t h t h e balance equations ( r e s p e c t i v e l y dot- ted, dashed and f u l l curves o f f i g u r e s 7a and 7b). I t was checked a t 50 kV cm-' by comparing t h e r e s u l t obtained, w i t h t h e i t e r a t i v e s o l u t i o n o f the Boltzmann equa- t i o n f o r t h e one p a r a b o l i c band model, t h e system being a t t < 0 i n thermal e q u i l i - brium : t h e agreement between t h e i t e r a t i v e method and t h e balance equation i s e x c e l l e n t . The d r i f t v e l o c i t y , shown i n F i g . 7a, e x h i b i t s an overshoot a t t h e i n - crease o f e l e c t r i c f i e l d . A n undershoot appears a t t h e decrease o f e l e c t r i c f i e l d , t h e more pronounced as t h e i n t e r m e d i a t e f i e l d E, i s higher. F i g u r e 7b shows t h a t noI
overshoot n o r undershoot are observed on t h e average energy. For b o t h t h e energy and (even more) t h e v e l o c i t y , t h e t r a n s i e n t regime i s almost t w i c e longer f o r t h e de- crease of t h e e l e c t r i c f i e l d than f o r t h e increase. The v e l o c i t y undershoot had a l - ready been p r e d i c t e d on mu1 t i v a l le y semiconductors (n-Si and n GaAs
[:
12]
).
However i t i s shown here t o be p o s s i b l e even w i t h a s i n g l e band s t r u c t u r e .When the e l e c t r i c f i e l d i s suppressed i n s t e a d o f being lowered (E2=0), the v e l o c i t y remains p o s i t i v e , b u t t h e decay t i m e decreases w i t h i n c r e a s i n g El.
F i g u r e 7 : T r a n s i e n t v e l o c i t y (a) and energy ( b ) responses t o an e l e c t r i c f i e l d El a p p l i e d a t t=O and E2 a p p l i e d a t t = t l . 1.5 ps, E2 = 10 kV cm-l. The i t e - r a t i v e method
( a )
was a p p l i e d f o r E l = 50tk<cm-1, and t h e balance equations were used f o r E l = 20 kV cm-l ( .. .
.), E1 = 50 kV cm-l (---) and E 1 = 100 kV cm-l(-).8. CONCLUSION
T r a n s i e n t regime o f h o t holes i n S i has been s t u d i e d u s i n g t h r e e d i f f e r e n t methods. I t e r a t i v e techniq"e and Monte Carlo s i m u l a t i o n have been proved t o produ- ce t h e same r e s u l t s when t h e microscopic model i s kept t o be t h e same. Due t o t h e l a c k o f experimental data, t h e present i n v e s t i g a t i o n has been devoted t o analyze s y s t e m a t i c a l l y some p h y s i c a l s i t u a t i o n s i n order t o o f f e r p l a u s i b l e expectations on t r a n s i e n t regime e f f e c t s . So, overshoot o f t h e d r i f t v e l o c i t y has been found t o be m a g n i f i e d by lowering t h e i n i t i a l c a r r i e r energy and, by c o n s i d e r i n g warping, when t h e f i e l d i s o r i e n t e d along <loo> d i r e c t i o n .
The t r a n s i e n t regimes were found t o depend s t r o n g l y on t h e i n i t i a l average v e l o c i t i e s and energies : d i m i n i s h i n g the i n i t i a l energy enhances t h e v e l o c i t y over- shoot. However, t h e t r a n s i e n t regime depends very few on various i n i t i a l d i s t r i b u - t i o n s corresponding t o t h e same i n i t i a l average v e l o c i t y and energy.
I n every case, i n s p i t e o f t h e g r e a t v a r i e t y of c o n d i t i o n s studied, t h e ba- lance equations gave r e s u l t s i n q u i t e s a t i s f a c t o r y agreement w i t h i t e r a t i v e o r Monte Carlo methods. T h i s s t r o n g l y j u s t i f i e s t h e use o f t h e balance equations, which i s a very f a s t and cheap method.
O f course t h e steady s t a t e regime i s completely independent from t h e i n i t i a l c o n d i t i o n s . On t h e basis o f these new r e s u l t s and o f o u r experience w i t h t h e t h r e e methods discussed above, we have drawn a general s y n t e t i c overview on t h e advanta- ges and l i m i t a t i o n s o f each method, which i s r e p o r t e d i n t a b l e s 2 and 3.
I
Balance equations Simple and f a s t method.A desk computer i s suf- f i c i e n t .
I t e r a t i v e technique
l
Monte Carlo s i m u l a t i o n An exact s o l u t i o n o f the Boltzmann equation i sobtained
The microscopic dynamic i s q u i t e t r a n s p a r e n t s i n ce each s c a t t e r i n g pro
-
d i r e c t way cess can be a n a l i z e d i n f u l l d e t a i l .
D i f f e r e n t i n i t i a l c o n d i t i o n s and i n i t i a l d i s t r i b u . t i o n functions can be simulated i n a rather simplc way.
Time dependent e l e c t r i c f i e l d s are e a s i l y i n - c l uded
- - -
Time dependent e l e c t r i c f i e l d s can be i n c l u d e d R e a l i s t i c band s t r u c t u r e models can be f u l l y accounted
Table 2 : P o s s i b i l i t i e s o f t h e v a r i o u s methods Balance equations
No r i g o r o u s j u s t i f i c a - t i o n o f the v a l i d i t y o f t h e equations can be g i v e n
The s t a t i o n a r y values o f the d r i f t v e l o c i t y , o f t h e mean energy and o f t h e e f f e c t i v e mass should be known a p r i o r i r e models cannot be i n - c l uded
To f u l l y e x p l * o i t t h e p o t e n t i a l i t y o f t h e method, very f a s t computers are needed
I t e r a t i v e technique The microscopic dynamics o f t h e s c a t t e r i n g process i s somewhat hidden
Table 3 : L i m i t a t i o n s o f t h e .methods
Monte Carlo s i m u l a t i o n D e t a i l s o f t h e d i s t r i - b u t i o n f u n c t i o n are n o t e a s i l y obtained
ACKNOWLEDGEMENTS : P a r t o f the computer f a c i l i t i e s have been k i n d l y provided by t h e computer Center o f t h e Elodena U n i v e r s i t y .
