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FREQUENCY AND WAVEVECTOR DEPENDENT DIFFUSION COEFFICIENT OF ELECTRONS FROM
MONTE CARL0 CALCULATIONS
C. Jacoboni, L. Reggiani, R. Brunetti
To cite this version:
C. Jacoboni, L. Reggiani, R. Brunetti. FREQUENCY AND WAVEVECTOR DEPENDENT DIF-
FUSION COEFFICIENT OF ELECTRONS FROM MONTE CARL0 CALCULATIONS. Journal de
Physique Colloques, 1981, 42 (C7), pp.C7-123-C7-128. �10.1051/jphyscol:1981713�. �jpa-00221649�
FREQUENCY AND WAVEVECTOR DEPENDENT DIFFUSION COEFFICIENT OF ELECTRONS FROM MONTE CARLO CALCULATIONS
C. Jacoboni, L. Reggiani and R. Brunetti
Gruppo Nazionale di Struttuva delta Materia, Istituto di Fisiea dell' Vniver- eitd di Modena, Via Campi 213/A, 41100 Modena, Italy
Résumé.- Quand un gradient de concentration 'àn/àx de particules est présent dans un système physique et ses variations temporelles et/ou spatiales sont très rapides, une généralisation de la loi de Fick conduit à la définition d'un coefficient de diffusion D(q,co), qu'on doit écrire en fonction du vecteur d'onde q et de la pulsa- tion UJ de la composante de Fourier de <)n/3x. L'expression générale pour D(q,U5) est obtenue, et la méthode de Monte Carlo est appliquée à son calcul. Les valeurs numé- riques des résultats sont données dans le cas du Si.
Abstract.- If in a physical system a particle-concentration gradient " ? n / ^ x is present which varies rapidly in time and/or in space, a generalization of Fick's law leads to a definition of a diffusion constant D(q,to) which is a function of the wavevector q and of the frequency cx> of the Fourier component ofdn/Bx. A general expression of D(q,"J) is obtained, and a Monte Carlo procedure is presented which leads to its evaluation. Numerical results are presented for the case of Si.
1. Introduction. - In the last years a considerable amount of work has been devoted to the study of hot-carrier diffusion properties at small distances and/or short times, that is , in conditions such that Fick's law does not hold. This problem may become very important in connection with the modelling of small semiconductor devices or when alternating fields are applied, with frequency comparable with the inverse of some electronic relaxation time and/
or wavevector comparable with the inverse of the electronic mean free path.
The aim of this paper is to present, for such types of problems, an extension of the the- ory which leads to Fick equation by making use of the Fourier analysis, which allows us to define a frequency and wavevector dependent diffusion coefficient D(q,u;). For q and oj different from zero it describes the diffusion properties of the system in steady-state condi- tions when a particle gradient S>n(q,a>)/"2 x is present. Furthermore, it will be shown that D(q,u>) reduces to the known static value when both q and to tend to zero.
2. Theory. - Let us consider a system of particles subject to an external uniform static e l - ectric field E. A particle source of sinusoidal type:
S ( x , t _ ) =. C | ' / + Csrt(fK-vt) j
Q )is then introduced, where x is the direction of the drift produced by E, together with a t r a p - ping mechanism with a constant rate / . The source amplitude C and y are connected thro- ugh the condition of particle conservation over a wavelength X ;
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981713
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The p a r t i c l e s a r e t h u s t r a p p e d and r e d i s t r i b u t e d in s p a c e . If we make u s e of t h e more con- venient complex formalism, the F i c k ' s l a w f o r s u c h a system would be w r i t t e n i n the one-di- mensional c a s e a s :
where D i s t h e diffusion constant and v i s the steady-state d r i f t velocity.
The g e n e r a l solution of Eq.(3) i s : d
where n, i s a c o n s t a n t a v e r a g e t e r m , nl and 9 a r e the amplitude a n d the phase-shift of t h e harmonic d i s t u r b a n c e , r e s p e c t i v e l y .
Bysubstituting Eq.(4) i n t o Eq.(2), we obtain:
and Eq .(3) becomes:
As f a r a s we d e a l with problems l i n e a r i n n , t h i s equation c a n be used a l s o a t frequen- c i e s and w a v e v e c t o r s s o high that F i c k ' s law does not hold
/1,2/ ;i n t h i s c a s e D , however, must be t a k e n a s a function of q a n d
5. T h u s from E q . 6 ) we obtain an e x p r e s s i o n f o r D(q,
L*)
, #
)in t e r m s of the r e l a t i v e amplitude n /no and of t h e phase-shift of t h e d i s t u r b a n c e :
1I ' M o - 1'y
"' g ' / n , i e + ' ( " - y % ) - q
,Using C h a m b e r s method we c a n obtain a n i n t e g r a l e x p r e s s i o n f o r n(x,t):
where P ( x ' , t l , x , t ) i s t h e probability t h a t , without t r a p p i n g , a p a r t i c l e i n x' a t t ' will be in x
a t t . T h e i n t e g r a l of t h e f i r s t t e r m in t h e b r a c k e t s in Eq.(7) i s equal t o n,, s i n c e going f r o m
the time -
@t o t a l l t h e p a r t i c l e s have been t r a p p e d and r e d i s t r i b u t e d by t h e s o u r c e .
P ( x ' , t V , x , t ) is a function of x ' , t ' , x , and t only through the d i f f e r e n c e s f
=x-x' and 8 =
t
-t ' , s o that E q s .(7) and ( 4 ) yield:
and Eq.(6) yields finally:
Eqs.(lO) and (9) provide an expression f o r D(q,@ ,%
)valid f o r any frequency, wavevec- t o r , and particle lifetime . When i s much l a r g e r than both U-2nd any microscopic ti- me, D(q,
w )- D(q, & ,
w ) i s obtained f o r steady-state conditions. P (
$ , p) must be d e r i - ved from the knowledge of the particle dynamics; a Monte C a r l o method f o r the determination of R will be shown l a t e r . F o r finite 'tid a t r a n s i e n t diffusivity is obtained, whose value dep- ends upon the dynamical initial conditions when p a r t i c l e s a r e c r e a t e d .
3. Comparison with previous definitions f o r D(q) and D(w).
- The problem of defining aq-dependent diffusion coefficient D(q) D(q,O) and a n w-dependent diffusion coefficient D ( w )
=
D(0, u ) h a s been analysed i n s e v e r a l p a p e r s /1,3-6/ . F o r what c o n c e r n s the limitu-, 0 Eq.(lO) gives:
where R becomes:
This r e s u l t i s in agreement with that of /1/ .
As r e g a r d s the expression of D ( w ) , l e t us consider the limit of o u r r e s u l t s a s q approa- c h e s z e r o . R c a n be expanded i n s e r i e s of q:
Expressions f o r R,, R , and R c a n be obtained from the expansion of Eq.(g), a f t e r some non
totally trivial calculations: 1 2
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where C( @
)i s the a u t o c o r r e l a t i o n function of velocity fluctuations:
4 > being the ensemble a v e r a g e .
Substitution of E q s . ( l 3 ) into Eq.(lO) y i e l d s , f o r vanishing small q:
J
0
F o r 5 +
W :I n agreement with the r e s u l t s of P r i c e /3/ and Schlup / 4 / , we have thus found that f o r a time dependent density gradient the diffusion c o n s t a n t , defined by Eq.(5), c a n be obtained a s the F o u r i e r t r a n s f o r m of t h e a u t o c o r r e l a t i o n function of velocity fluctuations. Eq.(14) con- t a i n s a l s o t h e e x p r e s s i o n f o r a t r a n s i e n t D(
M , , k d ) o b s e r v e d a t small timesZ d , with the ini- t i a l conditions f o r the p a r t i c l e v e l o c i t i e s given by t h e steady-state distribution. I n particu- l a r , f o r vanishing small
LL)we obtain f o r t h e t r a n s i e n t diffusivity:
4. Monte C a r l o p r o c e d u r e f o r the calculation of D(q &&).
-I n o r d e r to obtain D(q ,y$) from
a Monte C a r l o simulation, i t i s sufficient t o include among t h e s c a t t e r i n g mechanisms a t r a p -
ping mechanism with a constant i n v e r s e r a t e 3 4
/1/. A f t e r a trapping p r o c e s s , the parti-
c l e s t a r t s a g a i n with the velocity i t had when decayed; velocities a r e thus d i s t r i b u t e d a c c o r -
ding t o the stationary-state distribution. If t h e p a r t i c l e i s t r a p p e d a f t e r i t h a s c o v e r e d a
d i s t a n c e 1 i n a time 8 , the quantity e x p [-i ( q 5 - w e ) i s r e c o r d e d . Then R ( q & v i s obta-
ined a s the mean value of t h e s e q u a n t i t i e s .
ned a s Fourier transforms of the autocorrelation function, a s in Eq.(15).
Fig. 1 r e f e r s t o a low field value at T= 77 K . As
qi n c r e a s e s , D(q, cu) decreases becau- s e when ql
(1 being the electron mean f r e e path
)becomes l a r g e r than unity, electrons con- tributing to the current a t any point x a r e coming with free fligths from regions f a r from x.
Therefore an average over several wavelengths of c a r r i e r concentration lowers the diffusion current. This nonlocal behaviour of D(q) has been discussed in /1/ . Similar considerations hold when
LO'C ( 7being the electron mean momentum relaxation time
)i s l a r g e r than unity;
--, -
0 w
N V
5
C.. '
2 - 0
lL LA
0"
0
- a
FREQUENCY ( lo3 G H z ) F R E Q U E N C Y ( 1 0 3 ~ H Z )
Fig.1 - Low-field diffusion c o e f f i c i e n t a s a function of Fig.2 - High-field diffusion c o e f f i c i e n t as a function of frequency
w
a t t h e d i f f e r e n t values of q indicated by t h e frequency (A: a t t h e d i f f e r e n t values of q indicated by t h e numbers on t h e curves, i n u n i t s of 10~cm;l numbers on t h e curves, in u n i t s of cnil.
at very large frequencies D(q,U) decreases because electrons contribute to the diffusion current averaging over several periods of concentration in the past. The low-q curves show a maximum at intermediate frequencies related to the presence of active and passive regions of energy for intervalley scattericg, a s discussed i n / 5 , 6 , 8 / , which in turn can be seen in a negative part of the autocorrelation function C( 8
).In /5/, however, this phenomenon i s seen in GaAs but not in silicon. From the results presented there, this fact seems to be as- sociated to the lack of a negative part of C ( 8
),a result which is in disagreement with the findings of the present authors. This feature i s still more evident in the case of high fields, a s shown in Fig.
2.
The crossing of two curves in Fig. 2 at high frequencies can be due to their different Doppler shifts, however in the region of high frequencies the statistical uncertainties of the results become quite l a r g e . The crossing at low frequencies i s due to the dependence of D upon q discussed below.
Figs. 3 and 4 show the results obtained for D(q,W) a s a function of q at low frequencies.
The two figures r e f e r to cases of low and high fields, respectively. As a general trend, D
decreases at increasing q for the reason discussed above; however, at high fields, a maxim-
um i s present at intermediate values of q. The physical interpretation of this maximum i s the
same seen above f o r D(0,
d )a s a function of
w. In fact the values of q at which D(q, i s
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Fig.3 - Low-field and low-frequency diffusion coeffic- Fig.4 - High-field,low-frequency diffusion c o e f f i c i e n t a s a c i e n t as a function of wavevector a t t h e indicated function of wavevector a t the indicated temperatures.
temperatures.
