NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHENOMENON IN SEMICONDUCTORS I. A THEORETICAL NON-MARKOVIAN APPROACH

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NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHENOMENON IN SEMICONDUCTORS I. A THEORETICAL NON-MARKOVIAN APPROACH

J. Zimmerman, P. Lugli, D. Ferry

To cite this version:

J. Zimmerman, P. Lugli, D. Ferry. NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHE- NOMENON IN SEMICONDUCTORS I. A THEORETICAL NON-MARKOVIAN APPROACH.

Journal de Physique Colloques, 1981, 42 (C7), pp.C7-95-C7-101. �10.1051/jphyscol:1981709�. �jpa-

00221645�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 -page C7-95

NON-EQUILIBRIUM HOT-CARRIER DIFFUSION PHENOMENON IN SEMICONDUCTORS .1 • A THEORETICAL NON-MARKOVIAN APPROACH

J. Zimmermann , P. Lugli and D.K. Ferry

Colorado State University, Fort Collins, CO 80523, U.S.A.

*Permanent address : C.H.S. and Greco Mioroondes, Universite de Lille 1, Franae

Résumé - Le problème de la diffusion de porteurs chauds dans les semicon- ducteurs est étudié à l'aide d'une Equation de Langevin Retardée (RLE)

appliquée au cas de la réponse transitoire dynamique (TDR) à un champ électrique stationnaire homogène. Une fonction de corrélation des fluctuations de vitesse non-stationnaire est définie et est reliée à un coefficient de diffusion dépendant du temps. Ceci est appliqué au Silicium type-N et les quantités intéressantes sont étudiées en fonction de l'espace et/ou du temps. Le problème de diffusion non-stationnaire est particulièrement important dans les composants à canaux ultra-courts où réponse transitoire dynamique et sur- vitesse se manifestent.

Abstract - The problem of hot carrier diffusion in semiconductors is studied with a non-Markovian Retarded Langevin Equation (RLE) applied to the case of carriers in the transient dynamic response (TDR) to a steady homogeneous elec- tric field. A non-sfationary velocity fluctuation/correlation function is defined and related to a time dependent diffusion coefficient. This is applied to n-type silicon and the parameters of interest are studied as a function of space and/or time. The problem of non-stationary diffusion is particularly important in very short channel devices in which TDR and velocity overshoot occur.

1. Introduction.- In recent years, much interest has centered upon the transient dynamic response of electrons as it impacts carrier transport through small spatial regions of high electric field. With recent improvements in technological fabrica- tion of very-short-channel devices, this problem has become not only of theoretical interest but of practical interest as well. For instance, in the pinch-off region of a short-gate field-effect transistor, the carriers injected at the source move by a combination of drift and diffusion in a very high electric field. Then the transit time of the carriers under the gate can be shorter than, or of the same order of magnitude as, the time needed to establish a steady-state high-field dis- tribution function. In fact, a condition for this to occur is t^at the transit- time in the high-field region be comparable to the momentum relaxation time thus Causing the velocity to increase, but much shorter than the energy relaxation time.

Thus, on average the carriers may transit through a considerable portion of the high-field region with almost their low field mobility even though the applied field corresponds to the saturated velocity range [1]. This is true not only for the first-order moment of the distribution function of the carriers (drift velocity), but also is true for higher order moments, and especially for diffusion (related to the second moment). The diffusion coefficient is one of the most important

+

This work supported in part by the U.S. Office of Naval Research.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981709

C7-96 JOURNAL DF PHYSIQUE

parameters required in modeling semiconductor devices, it is not only necessary for evaluatin~ operating characteristics and frequency characteristics but it provides also fundamental characterization of velocity fluctuations in the system and their contribution to noise in the device [ 2 ] . Diffusion actually is a process depending upon velocity correlation and the relationship between diffusion and drift, as expressed by the Einstein relation, is a steady-state relation [ 3 ] . Indeed, tract- able results for the steady-state hot electron prohlem have only recently been achieved [4-61. The problem in the transient region is complicated by the fact that the random-walk equations governing transient diffusion do not reduce to normal Fick's law behavior on time scales comparable to the relaxation process, a result of the general non-Yarkovian and non-stationary nature of transport on these time scales. In this paper, we address some of these problems with the help of a Yetard- ed Langevin Equation (RLE) in order to aporoach the random walk of the carriers.

Furthermore, we define a non-stationary two-time correlation function for the ve- lacity fluctuations which can be related to a transient diffusion coefficient. The formal solution of the RLE allows us to derive a general expression for the correla- tion function. Then, we deal with the diffusion coefficient itself and show in particular that the transient diffusion coefficient is related to the time deriva- tive of the mean-square displacement of the carriers. However, it is found that in the limit of long times, stationarity and the normal equations for correlation and diffusion are recovered.

2. The Retarded Langevin Equation Approach.- We consider an ensemble of carriers initially at equilibrium with the crystal lattice. The ensemble is represented by a Plaxwell-Boltzmann distribution in v-space (<v>

0 and <v 2 > = 3k T /m with T the

B L L

lattice temperature). At a certain time, referred as t

o, we aoply a macroscopi- cally homogeneous and steady electric field whose amplitude corresponds to hot car- riers. These conditions give risetoa TDRregime in which the system relaxes toward a non-equilibrium, steady-state and often exhibits a velocity overshoot.

The motion of the particles is governed by a Retarded Langevin Equation for the velocity of the carriers, and is written as [7,8]

0

where R(t) is a random force symbolizing the random (non-regular) part of the col- lisions of the carriers with the lattice (no carrier-carrier interaction is consid- ered here), Eo is the exrernal field, and h(t) is the Heavyside function. +(t) is the memory function of the system and is related to the correlation function of the total force applied to the system. For instance in the case of a stationary regime,

2 2

we would have ?(t) = <~(o)Y(t)>/m <v (o)>, in absence of external forces [31.

Equation (1) is a non-Warkovian form of the Langevin equation, since the rate

of change of the velocity at t not only depends on the velocity at that time but

also depends on all past time. Further, it is a non-stationary equation, since the

lower bound i n t h e i n t e g r a l r e f e r s t o t h a t time where t h e d i s t u r b i n g f i e l d was a p p l i e d . It i s g e n e r a l l y admitted by now t h a t o n l y an equation such a s (1) can de- scribe v e r y - f a s t p r o c e s s e s [ 3 , 7 , 8 ] , and i n p a r t i c u l a r t h e TDR regime can be de- s c r i b e d i n t h i s way. I n ( I ) , we have assumed a p a r a b o l i c energy band 191.

Ve may e a s i l y s o l v e (1) u s i n g Laplace transforms. TJe i n t r o d u c e a f u n c t i o n X ( t ) d e f i n e d by i t s Laplace transform, a s

Then, coming back t o time domain, we f i n d

v ( t )

v ( o ) X ( t ) + zc R(t-u) X(u)du ,

m (3)

which i s a g e n e r a l e x p r e s s i o n of t h e e v o l u t i o n of t h e v e l o c i t y of each c a r r i e r under t h e i n f l u e n c e of t h e e x t e r n a l f i e l d and of t h e c o l l i s i o n s . Be g e t X ( t ) by aver- a g i n g ( 3 ) over t h e ensemble (we assume ' R ( t ) >

0 ) . The r e s u l t i s

where v ( t ) i s t h e ensemble d r i f t v e l o c i t y of t h e c a r r i e r s . Therefore X ( t ) r e p r e - d

s e n t s i n f a c t t h e macroscopic a c c e l e r a t i o n of t h e ensemble. However X ( t ) can b e given a n o t h e r d e f i n i t i o n . F?e d e f i n e a n o n - s t a t i o n a r y corre/Fation f u n c t i o n f o r t h e v e l o c i t y f l u c t u a t i o n s a s

M u l t i p l y i n % b o t h s i d e s of ( I ) by v ( o ) , a v e r a g i n : ~ over t h e ensemble (note t h a t

< R ( t ) v ( o ) >

O), Laplace transforming and making u s e of ( 2 ) we o b t a i n a f t e r r e - transforming

X ( t ) i s t h e reduced n o n - s t a t i o n a r y c o r r e l a t i o n f u n c t i o n c a l c u l a t e c l a t t ' = o. Com- p a r i n g t h i s e q u a l i t y w i t h ( 4 ) g i v e s

t

This r e l a t i o n s h i p e x i s t i n g between t h e f i r s t and a s e c o n d moment of t h e v e l o c i t i e s of t h e c a r r i e r s i s a d i r e c t i n t r i n s i c p r o p e r t y of t h e %LR used t o d e s c r i b e t h e TDR regime. Whether t h i s e q u a l i t y i s met i s one of t h e g o a l s of t h e next paper [ l o ] , b u t ( 7 ) i s a statement of t h e f a m i l i a r Kubo formu'la [ l l ] found i n e a u i l i b r i u m s t a t i s t i c a l mechanics.

To develop a complete e x p r e s s i o n f o r t h e c o r r e l a t i o n f u n c t i o n d e f i n e d i n ( 5 ) ,

we need t o know t h e c o r r e l a t i o n f u n c t i o n of t h e random f o r c e R ( t ) wh.?.ch a p p e a r s when

we p u t (3) i n t o ( 5 ) . I f we assume t h a t t h e c o l l i s i o n s occur i n s t a n t a n e o u s l y i n time

we can w r i t e

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R e l a x a t i o n of t h i s c o n d i t i o n does n o t s i g n i f i c a n t l y a f f e c t t h e r e s u l t s found i n t h e p r e s e n t work. A consequence of (8) i s t h a t o n l y I R l ( t )

I R ( t , t T = t ) i s of i n t e r e s t i n t h e c u r r e n t c o n t e x t and t h i s l a t t e r f u n c t i o n can b e o b t a i n e d from t h e time evolu- t i o n of t h e mean energy of t h e ensemble. Then t h e c o r r e l a t i o n f u n c t i o n 4 can b e

Av put i n t h e form (with 9 L 0)

to

2 2 '

mAv(to,to + e ) =

( ~ ) > x ( t ~ ) x ( t ~ + 0 ) + I ~ l ( t o - y ) ~ ( y ) ~ ( y + , (9) m

and i n p a r t i c u l a r we o b t a i n f o r t h e mean-energy:

t

0

We can sum up t h i s p a r t by s a y i n g t h a t i f , i n t h e c u r r e n t c o n d i t i o n s of t h e TDR regime, t h e e v o l u t i o n s of t h e d r i f t v e l o c i t y and t h e mean-energy a r e known, t h e n t h e f l u c t u a t i o n s of t h e system a r e completely s p e c i f i e d through (9) and (10) t o g e t h e r w i t h ( 4 ) .

3.'The D i f f u s i o n C o e f f i c i e n t . - Using e x p r e s s i o n (5) f o r t h e n o n - s t a t i o n a r y c o r r e l a - t i o n f u n c t i o n of v e l o c i t y f l u c t u a t i o n s , we can d e f i n e a n o n - s t a t i o n a r y o r time-de- pendent d i f f u s i o n c o e f f i c i e n t , which i s a g e n e r a l i z a t i o n of t h e d e f i n i t i o n given i n a s t a t i o n a r y regime [3,6]:

t

D ( t ) = [ O A v ( t ' , t ) d t ' .

Another way t o d e f i n e t h e d i f f u s i o n i s through t h e s p r e a d i n g of a packet of c a r r i e r s d r i f t i n g under t h e i n f l u e n c e of t h e e x t e r n a l f o r c e and s p r e a d i n g due t o t h e f l u c t u a t i o n s of t h e v e l o c i t i e s of t h e c a r r i e r s . This s p r e a d i n g i s c h a r a c t e r i z e d hy t h e mean-square displacement of t h e c a r r i e r s s t a r t i n g from a known i n i t i a l d i s t r i b u - t i o n ( a n i r a c - f u n c t i o n i n space, f o r i n s t a n c e ) . It i s e a s y t o s e e t h a t

0 0

From t h e d e f i n i t i o n ( 1 2 ) , i t i s s t r a i g h t f o r w a r d t o show t h a t t

This shows t h a t t h e d i f f u s i o n c o e f f i c i e n t d e f i n e d i n (11) i s r e l a t e d t o t h e time- d e r i v a t i v e of t h e average-square displacement. The u s u a l d e f i n i t i o n of t h e d i f f u - s i o n i n a s t e a d y - s t a t e i s i n f a c t a l i m i t i n g c a s e of r e l a t i o n (13). T*%en t h e time t ' a t which t h e i n t e g r a t i o n of ( t ' , t ) b e g i n s i s g r e a t e r t h a n t h e time t needed

Av s

f o r t h e system t o r e a c h s t a t i o n a r i t y , 4 becomes an even f u n c t i o n of t - t ' o n l y , Av

t h e n

t >

A s t

", JI ( t ) t e n d s t o a c o n s t a n t f i n i t e l i m i t D and

S

l i m <Ax ( t j 2 >

2 t Ds

Therefore (11) d e f i n e s a g e n e r a l d i f f u s i o n c o e f f i c i e n t which i s v a l i d i n b o t h s t a - t i o n a r y and n o n - s t a t i o n a r y regimes, and a s we d i d above f o r +Av we can d e r i v e an e x p r e s s i o n f o r <Ax ( t ) > : 2

2 m 2 2

<Ax ( t ) >

7j-7 <v (0)'

( t ) + - I _R1 ( t - u ) v d (u)du ² .

Eo

The e x p r e s s i o n of D ( t ) f o l l o w s immediately u s i n g (13).

4. A p p l i c a t i o n and Discussion.- I n t h e c a s e of n-type S i < l l l > , t h e c a r r i e r s behave a s i f t h e energy band was unique w i t h a n i s o t r o p i c e f f e c t i v e mass m . There, t h e concepts d e s c r i b e d above a r e e a s i l y a p p l i e d .

i ) The o s c i l l a t o r y n a t u r e of t h e v e l o c i t y overshoot s t r o n g l y s u g g e s t s t h a t t h e memory f u n c t i o n y ( t ) (and q ( t ) , a s w e l l ) i s a n e x p o n e n t i a l . So we s p e c i f y y ( t ) v i a t h e a n s a t z [12]

l / r ^{i s} t h e time needed f o r t h e system t o r e a c h s t a t i o n a r i t y and a s such i s equiva- l e n t t o t h e energy r e l a x a t i o n time.

i i ) To have an i n s i g h t i n t o t h e time dependence of I R l ( t ) , we can make u s e of (10) and t h e r e s u l t s g i v e n by a Vonte Carlo method [ l o ] f o r < ~ ( t ) > . ^{It i s t h u s}

p o s s i b l e t o show t h a t

i s a good approximation f o r t h e c o r r e l a t i o n f u n c t i o n of t h e random f o r c e R ( t ) . IR- i s given by t h e s t a t i c d i f f u s i o n c o e f f i c i e n t

and IRo << I The v a l u e of I does n o t a f f e c t s i g n i f i c a n t l y t h e r e s u l t s s i n c e a t

R-* Ro

s h o r t times t h e second term i n t h e RHS of (16) i s a n o r d e r of magnitude s m a l l e r t h a n t h e f i r s t term.

Taking (17) and (18) i n t o account, + ( t l , t ) and V ( t ) a r e e a s i l y d e r i v e d analy- Av

t i c a l l y and computed. We p r e s e n t h e r e some computed r e s u l t s o b t a i n e d w i t h E

50

kV/cm. The n o n - s t a t i o n a r y c o r r e l a t i o n f u n c t i o n i s d i s p l a y e d i n F i g . 1, and i t s

e v o l u t i o n i s s t u d i e d f o r d i f f e r e n t i n i t i a l times to from 0 t o 0 . 5 p s ( t h e l a t t e r

corresponds t o t h e s t e a d y - s t a t e ) . The f a c t t h a t t h e c o r r e l a t i o n f u n c t i o n i s s t e e p e r

when to i s l o n g e r can b e e x p l a i n e d by t h e i n c r e a s e of t h e s c a t t e r i n g freauency a s a

f u n c t i o n of time corresponding t o t h e h e a t i n g of t h e c a r r i e r s . I n o t h e r words, a t

s h o r t t i m e s t h e c o r r e l a t i o n f u n c t i o n i s dominated by t h e i n i t i a l d i s t r i b u t i o n of t h e

v e l o c i t i e s of t h e c a r r i e r s , w h i l e a t l o n g e r times, when t h e s t e a d y - s t a t e i s

C7-100 JOURNAL DE PHYSIQUE

approached, the system has been completely randomized by the collisions and the correlation function is dominated by the random force (i.e. IRl(t)).

The mean-square displacement and diffusion coefficient are displayed in Fig. 2 as a function of the distance travelled by the carriers. This shows the spatial

0

extension over which the system is in a non-stationary regime, - 401) A in the present case. These results are compared to what would occur in case of a pure ballistic regime. In fact, the calculated D ( x ) departs from the ballistic trend even at very short times (and distances) meaning that no ballistic resine exists for transient diffusion. On the contrary at longer times D(x) goes through a maximum and then decreases to its stationary value. This is, of course, related to the oscillatory nature of the correlation functions and these oscillati-ons are essen- tially the consequence of the combination of momentum and energy relaxation in the resolvent X(s) of the RLE. In the present example, this resolvent is a rational fraction which has two complex conjugate poles.

Possible extensions of the work could be: i) the application of the present technique to multivalley semiconductors; ii) to take into account spatial varia- tions of the electric field and then get a more precise picture of what occurs in a short-channel device; iii) to try a more physical and less phenomenological approach using quantum statistical mechanics [13].

In summary, we have obtained here a consistent definition of the diffusion coefficient in terms of the velocity auto-correlation function. This definition is valid in the transient and non-stationary regime and reduces to the normal expres- sion as steady-state is approached. The application of this to the retarded, non- stationary Langevin equation yields expressions for the velocity correlation func- tion and diffusion coefficient which have excellent qualitative agreement and satis- factory quantitative agreement to results obtained by a Monte Carlo method r10].

References

1. H. Kroemer, Sol. State Electron., 21 (1978) 61.

2. J.-P. Nougier, "Noise Diffusion in Semiconductors,:' in Physics of Non-Linear Transport in Semiconductors, Edited by D. K. Ferry, J. R. Barker, and C. Jaco- boni (Plenum Press, NewYork) 1979, pp. 415-464.

3. R. Kubo, "Response, Relaxation and Fluctuation," in Lectures Notes in Physics 31:

Transport Phenomena, Edited by J. Ehlers, K. Hepp, and H. A. weidenmiller (Springer Verlag, Berlin) 1974, pp. 74-126.

4. C. Jacoboni, C. Canali, G. Ottaviani, and A. Alberigi-Ouaranta, Sol. State Electron., 20 (1977) 77.

5. R. Fauquembergue, J. Zimermann, A. Kaszynski, and E. Constant, J. Appl. Phys.

51 (1980) 1065.

6. D. K. Ferry, and J. R. Barker, J. Appl. Phys., 52 (1981) 818.

7. R. Zwanzig, J. Chem. Phys, 40 (1966) 2527.

8. H. Mori, Prog. Theor. Phys., 33 (1965) 423.

9. We can take into account non-parabolicity of the energy band by dealing with

the momentum p(t) instead of the velocity v(t), and the formula v(t) = p(t) (1 + 201 (t)/mo)-1i2/mo for example. In this case, the derivations would be 2 much more involved.

10. P. Lugli, J. Zimmermann, and D. K. Ferry, these proceedings.

11. R. Kubo, J. Phys. Soc. Japan, 12 (1957) 570. 'See also H. J. Kreuzer, Nan- Equilibrium Thermodynamics and its Statistical Foundations (Oxford Univ. Press Oxford) 1981.

12. Y is the inverse momentum relaxation time at thermal euuilibrium and a

+ ^{Y is}

the inverse momentum relaxation time when non-equilibrium steady-state is reached.

13. D. K. Ferry, Invited Paper, these proceedings.

Figure 1.

Transient Reduced Correlation Functions Calculated At

t =O (-) ; t =O. 02ps (---) ; t =O. 50s (- . ^-)

Figure 2.

Mean-Sq-uare Displacement (right scale) and Diffusion Coefficient (leftscale) as a Function of Distance. Dashed Curves Stand for a Pure Ballistic

Trend