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BALLISTIC TRANSPORT IN SEMICONDUCTORS : A DISPLACED MAXWELLIAN FORMULATION
E. Rosencher
To cite this version:
E. Rosencher. BALLISTIC TRANSPORT IN SEMICONDUCTORS : A DISPLACED MAXWELLIAN FORMULATION. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-351-C7-356.
�10.1051/jphyscol:1981743�. �jpa-00221680�
Colloque C7, supplément au n°10, Tome 42, ootobve 1981 page C7-351
B A L L I S T I C TRANSPORT IN SEMICONDUCTORS : A DISPLACED MAXWELLIAN FORMULATION
E. Rosencher
Centre Norbevt Segard, C.N.E.T., 38240 Meylan, Franee
Résumé. - Le formalisme de la Maxwellienne Déplacée, décrivant le transport d'électrons chauds dans les semiconducteurs, est étendu aux régimes non stationnaires où les temps de transit des porteurs dans les composants sont comparables aux différents temps de relaxation. Cette approche est appliquée au Silicium et à l'Arseniure de Gallium pour différentes températures afin de décrire la réponse d'un gaz d'électrons à une discontinuité spatiale ou temporelle du champ électrique. Les résultats des calculs sont analysés en termes de libre parcours balistique du gaz d'électrons dans le semicondeur.
Abstract. - The Displaced Maxwellian formulation for hot carrier transport in semiconductors is extended to non stationary conditions, where transit time of carriers in devices has the same order of magnitude as the different relaxation times. It has been applied to Si and GaAs at various lattice temperatures in order to describe the response of the electron gas to step changes in the electric field magnitude with respect to the time and space variables. The results of calculations are analyzed in terms of ballistic mean free path of the electron gas in the semiconductor.
Introduction. - At low temperature, the transit time of carriers in semiconductor devices may become comparable to the time between collisions. Therefore, the probability for the carriers to be ballistic all along their motion in device channels is no longer negligible [ 1 ]. This means that their velocity is a direct result of the potential drop, before any scattering occurs. Mainly three kinds of approach have been used in order to investigate this phenomenon : i) Monte-Carlo simulations, which are very computer time consuming ( 2 ) , ii) phenomenological formulation t 3 \ where the different relaxation times and effective masses are choosen in order to fit Monte-Carlo simulations and iii) the vacuum diode approach C.l ) , which completly neglects scattering. In this paper, we use an analytical approach based on a Displaced Maxwellian formulation ( 4 ) . It leads, starting from first principles to analytical expressions for the different relaxation times and for the transport equations. Therefore, parametric studies of non stationary transport in different semiconductors are by no mean a computationnal problem.
These studies, as well as the clear physical meaning of these transport equations, allow one to point out the physical parameters which govern ballistic transport.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981743
C7-352 JOURNAL DE PHYSIQUE
I - THE MODEL. - a) General frame. Let us briefly recall the basis of our calculations ( 4 l. Boltzmann's equation, in the relaxation time and effective mass approximation, reads :
where ~-'(l$l) considered as a probability of scattering, is derived from first principles for each type of interactions (5, 6 ) . Any observable quantity Q may then be predicted by multiplying ( 1) by the Q ([$l) function and integrating over velocity space, leading to general conservation equations ( 7 ) . Setting Q equal to
+
unity, m*v and m*?, one finds respectively the continuity equation, the momentum 2
and energy balance equations. Since Boltzmann's equation cannot be solved analytically, one must introduce a simplifying ansatz. The simplest way of describing the perturbed distribution function f is to assume Displaced Maxwellian statistics, with an electron temperature T, and a displacement in velocity space vd
( 4 l. The conservation equations can then be easily evaluated, assuming an electric
field E in the X direction :
a a I * 5 E - €
(n.~) + - ax (nvd. (Tmv + - k T ) ) =nqEvd - n . ( 4 )
d 2 e
where T (E)
is the mean carrier energy and.
v-space
lead to analytical expressions of macroscopic relaxation times TQ. These equations allow a fully analytical description for stationary and non stationary transport.
For multivalley semiconductors, these equations are written in each valley with coupling continuity equations :
where i and j refer to the different valleys, a; represents the number of
equivallent valleys i, and Tij and Tji are evaluated through equations ( 6 ) .
Equations (2) - (7) have been solved in order to describe the time response of an
electron gas submitted to a uniform electric field applied at t = 0. Figure 1 shows
typical results for Si and GaAs at 300 K. The parameters used in calculations have
1,
between our results and the ones obtained by this author, using Monte-Carlo simulations, unambiguous. The two fo&ulations lead to very similar results. Let us note that, in our case, the C. P. U. time on an IRIS 80 was within few seconds.
TIME
C I O - ' ~ ~ )
Fig. 1. - Time response of electron gas after electric field switching in Si and GaAs. Solid lines : present Displaced llaxwellian approach, dotted lines : Monte-Carlo simulations obtained by RUCH ( 2 l.
b) Spatial discontinuity treatment.- An important problem for the understanding of non stationary transport tin devices is the evolution versus distance of the electron gas when passing through a step like field discontinuity. However, equations (2) - (4) are known to be highly unstable. It is by no mean a computing problem, as stated earlier 7 1, but rather an unstability associated to the Displaced Maxwellian approximation. While some authors 7 l -studying Gunn effect- added a heat conduction term to equation ( 4 ) in order to obtain convergence, we used another stabilisation proceduce. It is usual, in plasma physics, to introduce in similar equations, a closing procedure : the cold plasma approximation [ 8 ] , which assumes that :
where P = nkTe is the carrier pressure. With this assumption, the conservation equations read :
a a
- n + - ( n v ) = O at ax d
These equations are similar to the phenomenological balance equations introduced by
SCHUR ( 3 l. Using this approximation, we have solved equations (9) - (11) in order
to study the steady state behaviour of an electron gas submitted to a step like
field discontinuity in high purity Si at 30 K and 300 K. Figure 2 ~ h o w s the
electron gas drift velocity as a function of distance from the discontinuity. The
C7-354 JOURNAL DE PHYSIQUE
results obtained are again in fair agreement with the ones obtained by RUCH using Monte-Carlo simulations 2 1.
I1 - BALLISTIC MEAN FREE PATH IN DEVICES. - In Fig. 2, it appears that, for a high applied field, the difference in non stationary regime distances between 30 K and 300 K is quite small while the variation of the mean free path is almost a factor of 100. This can be understood simply. Indeed, in high fields, electrons reach almost ballisticaly the energy of the optical phonon (fiw -- 30meV in Si) very
0
P
rapidly. In this energy range, inelastic scattering prevails, which leads to short
Fig. 2. - Electron drift velocity Fig. 3. - Calculated relaxation times as a function of distance from vs electron mean energy in pure Si electric field discontinuity. with scattering parameters as in RUCFr ( 2 relaxation times, almost independent on temperature (Fig. 3).
On the other hand, there is a high difference in relaxation times between 30 K and
,. 1 : -
300 K for low energy thermal carriers, where scattering mostly stems from acoustic
/
I S1
/ 20 kV/cm
I I
phonons in high purity Si Fig. 3 . Since the mean free path is clearly not a good
T = 300K
0
-
0
. . . . . , . .
. a . . B . . ,. , , . , , , . , , , . , , , , .
,, . , .,
0 00 0'05 0'10 0'15 0 20 l C+ ld-'
DISTANCE ( p m 1 MEAN CARRIER ENERGY CeVl
concept for predicting the ballistic behaviour of carriers in small devices, the simplest way to synthesize our calculation results is to introduce a "ballistic mean free path"
Abal ' We will define it at as the distance for which, starting from v d = 0 at X = 0, the electron gas reaches a velocity v dsuch that :
This is a crude approximation of the mean free flight of the electron gas before
collision. Figure 4 shows the ballistic mean free path versus electric field
deduced from our model in high purity silicon at 30 K and 300 K. It is clear that
in Si at room temperature, ballistic transport is a much negligible effect on
nowadays devices. This is not so at 30 K where Abal may reach .5 pm. Note that, for
vanishing electric field, Abal converges to zero and not to the thermal mean free
path : this points out the difference between the two notions.
The g e n e r a l f e a t u r e of Fig. 4 can be p h y s i c a l l y understood i n t h e f o l l o w i n g way.
F o r c a r r i e r s i n low f i e l d w i t h concequently low mean energy, a c o u s t i c phonons p r o v i d e t h e main s c a t t e r i n g mechanism. The c a r r i e r s s t a y a long time i n t h e d e v i c e c h a n n e l , l o n g enough t o be s c a t t e r e d by a c c o u s t i c phonons s o t h a t t h e b a l l i s t i c mean f r e e p a t h i s simply g i v e n by :
( d o t t e d l i n e i n f i g . 4 ) . On t h e o t h e r hand, f o r h i g h f i e l d s , a s soon a s t h e c a r r i e r mean energy approaches t h e o p t i c a l phonon energy hw , t h e s c a t t e r i n g
=' P p r o b a b i l i t y i n c r e a s e s s h a r p l y so t h a t :
f o r an i n f i n i t e s c a t t e r i n g e f f i c i e n c y . I n Fig. 4 , we have drawn t h e corresponding s t r a i g h t l i n e . It t u r n s o u t t h a t t h i s v e r y crude approach p r o v i d e s a r a t h e r good q u a n t i t a t i v e e s t i m a t i o n of A bal . Indeed, t h e d i s c r e p a n c y i n t h e h i g h f i e l d r e g i o n i s due t o t h e f a c t t h a t e q u a t i o n (14) supposes t h a t t h e r e l a x a t i o n time v a n i s h e s a s s o o n a s t h e c a r r i e r energy r e a c h e s
v i s i b l e on Fig. 3.
ELECTRIC FIELD (V/cm J
T h i s is o b v i o u s l y too s t r i n g e n t
CHANNEL L E N m (pm1
