TIME-DEPENDENT CARRIER VELOCITIES IN III-V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL ITERATIVE METHOD

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TIME-DEPENDENT CARRIER VELOCITIES IN III-V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL ITERATIVE METHOD

S.C. van Someren Greve, Th. G. van de Roer

To cite this version:

S.C. van Someren Greve, Th. G. van de Roer. TIME-DEPENDENT CARRIER VELOCITIES IN III- V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL ITERATIVE METHOD.

Journal de Physique Colloques, 1981, 42 (C7), pp.C7-343-C7-348. �10.1051/jphyscol:1981742�. �jpa-

00221679�

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TIME-DEPENDENT CARRIER V E L O C I T I E S IN 1 1 1 - V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL I T E R A T I V E METHOD

S.C. van Someren Greve and Th. G. van de Roer

Eindhoven University of Technology, Department of Electrical Engineering, P.O. Box 513, Eindhoven, The Netherlands

Résumé.

Une méthode semi-analytique rapide est présentée pour la résolution de 1 équation de Boltzmann par la décomposition en polynômes orthogonaux de Legendre.

Cette méthode est basée sur l'introduction d'un terme "self scatterlng".

Cette méthode, si on la compare à l'intégration numérique est directe et ne souffre pas des difficultés dues aux instabilités numériques. Elle permet aussi de calculer la réponse de la répartition des vitesses sur un champ électrique variable dans le temps.

Abstract.

A fast semi-analytical method is described for solving the time-dependent Boltzmann equation expanded in Legendre polynomials. The method is based upon the introduction of a self-scattering term.

Compared with direct numerical integration this method is straightforward and has no difficulties due to numerical instabilities. It also allows calculation of the time- response of the distribution function to a varying electric field.

1. Introduction.

Different methods to calculate the carrier distribution function in semiconductors have been developed. The Monte Carlo method [1], [2] which is based on the simulation of motion of one or a certain number of electrons in space allows complicated band structures and scattering mechanisms to be taken into account. The method however is time-consuming especially when a great number of carriers is considered in order to model time-space dependent phenomena.

Iterative methods are much faster. The iterative method of Rees [3] is based on transforming the Boltzmann equation into an integral form. This method can be modified so that each iteration step becomes equivalent to a time step of the physical system. To reduce the number of points which represent the distribution function Hammar [4] has modified the iterative method by expanding the distribution function in Legendre polynomials. According to Hammar et least 100 terms in this expansion are required to accurately represent the distribution function but a two-term expansion already gives quite an accurate result for the quantities of physical interest. Four or six terms give these quantities with high precision. The set of equations obtained cannot be integrated in a straightforward manner due to numerical instabilities connected with the homogeneous solutions. This problem can however be circumvented by subtracting unwanted solutions in each step. In this paper it is shown that the set of equations upon the introduction of self scattering can be

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981742

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C7-344 JOURNAL DE PHYSIQUE

transformed i n t o an i n t e g r a l form. Each i t e r a t i o n s t e p now will be e q u i v a l e n t t o a time s t e p of t h e p h y s i c a l system. The simple form of t h e i n t e g r a l s makes i t p o s s i b l e t o perform each i t e r a t i o n very f a s t .

2. Basic e q u a t i o n s .

The time and space independent Boltzmann e q u a t i o n i s :

where

v(&) = ! s ( k , k t ) + '

g ( k ) = I s ( & ' , ' f ) f

(5' )E.

The i t e r a t i v e technique proceeds i n t h e f o l l o w i n g way: f i r s t t h e c u r r e n t approximation of f ( 5 ) i s used t o c a l c u l a t e t h e f u n c t i o n g ( k ) . With t h i s f u n c t i o n t h e next approximation of f ( 5 ) i s o b t a i n e d by s o l v i n g e q u a t i o n (2.1) which now i s a l i n e a r d i f f e r e n t i a l e q u a t i o n .

I t can be shown C51 t h a t f ( 5 ) converges t o thestationarydistribution f u n c t i o n . In t h e method of Rees C31 eq. (2.1) i s converted i n t o an i n t e g r a l e q u a t i o n . To s i m p l i f y h i s method Rees i n t r o d u c e d a f i c t i t i o u s " s e l f s c a t t e r i n g " p r o c e s s which l e a v e s t h e s t a t e of a p a r t i c l e unchanged.

Defining t h e s e l f - s c a t t e r i n g r a t e by:

where I' i s a c o n s t a n t , t h e f u n c t i o n g*(&) i s d e f i n e d by g*(k) = !CS(~' ,E) + S ' (F' , % ) I f ( 5 ' )dk.

Equation (2.1) i s t h e n transformed i n t o

e E .

V f ( 5 ) + Tf(k) = g*(k)

.

- a k

I t was shown furthermore by Rees t h a t by choosing T much g r e a t e r t h a n t h e v a l u e s of v(&) a t e n e r g i e s a t which a s u b s t a n t i a l amount of e l e c t r o n s i s p r e s e n t , t h e

i t e r a t i o n s model t h e time-dependent d i s t r i b u t i o n f u n c t i o n t h a t develops from t h e i n i t i a l f u n c t i o n f ( k ) . Each i t e r a t i o n s t e p t h e n i s e q u i v a l e n t t o a time s t e p 1

0

-

Following Hammar C41 i t i s assumed t h a t t h e d i r e c t i o n of t h e a p p l i e d e l e c t r i c f i e l d forms a symmetry a x i s f o r t h e d i s t r i b u t i o n f u n c t i o n which t h e n can be expanded i n orthogonal Legendre polynomials:

m

where k =

[ k l

and ^X= cos8, 8 being t h e a n g l e between and

E.

The advantageous p r o p e r t y of S ( k , k l , x ) b e i n g d i a g o n a l remains t r u e f o r S ( k , k ' , x )

+

S 1 ( k , k ' , x ) . g*(%) can be expanded i n t h e same way with t h e c o e f f i c i e n t s g*(k) given by:

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d i f f e r e n t i a l e q u a t i o n s can be o b t a i n e d :

m

a f m =

1

^{{ % A} c g * - r f )--B 1 f 1 .

-

e E m n n n k m n n (2.9)

ak n=o

The v a l u e s of A and B* have been given by Hammar C41.

mn

For p r a c t i c a l e v a l u a t i o n t h e s e t h a s t o be t r u n c a t e d a t a c e r t a i n number N which h a s t o be even, o t h e r w i s e t h e l a s t e q u a t i o n w i l l be decoupled from t h e o t h e r s .

3. The p a r t i c u l a r s o l u t i o n .

The d i f f e r e n t i a l e q u a t i o n s given by (2.9) have b e s i d e s t h e p a r t i c u l a r s o l u t i o n a l s o homogeneous s o l u t i o n s which e i t h e r have a s i n g u l a r i t y a t k = O o r grow e x p o n e n t i a l l y a s k+m. I t i s t h e r e f o r e important t o have a method t o f i n d only t h e p a r t i c u l a r s o l u t i o n . Consider t h e s e t of d i f f e r e n t i a l e q u a t i o n s :

The fundamental m a t r i x ?(k) i s formed by columns which a r e independent s o l u t i o n s of t h e homogeneous s e t :

The p a r t i c u l a r s o l u t i o n t h e n can be w r i t t e n as C61:

where

An a n a l y t i c a l method has been found t o c a l c u l a t e t h e m a t r i c e s

2

and f o r a r b i t r a r y rank N . F u l l d e t a i l s o f t h e method w i l l b e published elsewhere.

T h e i r g e n e r a l form i s :

(n,m,i=O,l,

...,

N-l). The X. a r e r o o t s of t h e equation:

P N

c-)

eEXi

fir

= 0 ; (PN i s N~~ Legendre polynomial).

since

N is even, t h e h . occur i n p a i r s w i t h o p p o s i t e Signs.

L

The o r d e r i n g of t h e homogeneous s o l u t i o n s i s such t h a t :

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C7-346 JOURNAL DE PHYSIQUE

A useful property is then:

The particular solution (3.3) has to obey the boundary conditions:

lim fo(k) = f (0)

,

^(3.9a)

k-to

lim fR(k) = 0 (00)

,

k+o

lim fa(k) = 0

.

^(3.9~)

k-

The condition (3.9a) is fulfilled if the components of the vector _v(ko) obey the relation at k = 0:

It turns out that condition (3.9b) is then fulfilled automatically. The condition (3.9~) can be fulfilled if the components v (0) are chosen in the following,way:

2n N-l

v 2n (0)

= - I 1

Y2n,i(~)g:(~)ds. (3.11)

o i=o Defining :

the components V ~ ~ + ~ ( O ) take the form:

The particular solution can then be expressed as:

By calculating first the integrals:

Ak N-l

,

12,(k) = ex~(-X~~Ak)I~~(k+Ak) + exp(-X2,t)

1

^{'an, i}(k+t)gZ(k+t)dt, (3.15a) o i = o

and Ak N-l

,

I~n+l(~) = exp(-X2nAk)12n+l(k-Ak) + o

l

exp(-iZnt)

1

^i=oY ~ ~ + ~ , i(k-t )g;(k-t)dt, (3.15b)

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.

4. R e s u l t s and conclusions.

Hammar C41 has found t h a t a t room temperature a two-term expansion a l r e a d y g i v e s q u i t e good r e s u l t s f o r t h e macroscopic q u a n t i t i e s . From our work we conclude t h a t when t h e e l e c t r i c a l f i e l d i s not t o o high t h e same h o l d s t r u e f o r t h e time-dependent

velocity: For i n s t a n c e , f i g . ( 4 . 1 ) shows t h e response of t h e average e l e c t r o n v e l o c i t y i n GaAs t o a f i e l d s t e p from z e r o t o 6 kV/cm and back t o 5 kV/cm. T h e l a t t e r p a r t compares very w e l l t o t h e r e s u l t s of Rees 131. Also, r e s u l t s f o r f i e l d s t e p s u p t o 10 kV/cm show very good agreement w i t h Monte Carlo r e s u l t s of Kaszynski C71.

In f i g . (4.2) a s i m i l a r curve of Ga(0.47)In(0.53)As i s given where p o l a r o p t i c a l phonon, i n t e r v a l l e y and a l l o y s c a t t e r i n g a r e t a k e n i n t o account. The a l l o y s c a t t e r - i n g p o t e n t i a l was put a t 0 . 4 eV. The o t h e r parameters have been t a k e n from

Fauquembergue [ S ] . The s t r i k i n g f e a t u r e o f t h i s curve i s t h e l a r g e v e l o c i t y over- shoot which i n d i c a t e s . t h a t t h i s i s an e x c e l l e n t m a t e r i a l f o r short-channel f i e l d e f f e c t t r a n s i s t o r s .

A t h i g h e r f i e l d s t h e two-term expansion g i v e s a much l e s s s a t i s f a c t o r y d e s c r i p t i o n of t h e time-dependent behaviour. Fig. ( 4 . 3 ) shows t h e response of t h e e l e c t r o n v e l o c i t y i n GaAs t o a f i e l d s t e p from z e r o t o 20 kV/cm u s i n g two- and four-term expansions. Only t h e four-term expansion g i v e s good agreement w i t h Monte C a r l o r e s u l t s C?]. However, i t can be s e e n t h a t t h e two-term expansion g i v e s q u i t e good e s t i m a t e f o r t h e amplitude and time d u r a t i o n of t h e v e l o c i t y o v e r s h o o t , a s w e l l a s f o r t h e s t a t i c v e l o c i t y . I t i s e s p e c i a l l y t h e c e n t r a l v a l l e y which i s d e s c r i b e d i n - s u f f i c i e n t l y by two terms. Here t h e main s c a t t e r i n g p r o c e s s below t h e energy a t which i n t e r v a l l e y s c a t t e r i n g s t a r t s i s p o l a r o p t i c a l phonon s c a t t e r i n g which i s s t r o n g l y dependent on t h e a n g l e between t h e wave v e c t o r s b e f o r e and a f t e r s c a t t e r i n g . This s c a t t e r i n g mechanism l e a d s t o a s t r o n g l y e l o n g a t e d d i s t r i b u t i o n i n t h e f i e l d d i r e c t i o n . The h i g h e r v a l l e y s a r e dominated by i n t e r v a l l e y s c a t t e r i n g which randomizes t h e d i r e c t i o n s of t h e v e l o c i t i e s much s t r o n g e r . I n t h e s e v a l l e y s a two- term expansion g i v e s a good d e s c r i p t i o n even a t high f i e l d s . A f t e r a c e r t a i n time most of t h e c a r r i e r s w i l l occupy t h e s e h i g h e r v a l l e y s , so e r r o r s i n t h e c e n t r a l v a l l e y w i l l be less i n f l u e n t i a l which e x p l a i n s why t h e s t a t i o n a r y d r i f t v e l o c i t y i s c a l c u l a t e d w e l l even u s i n g a two-term expansion.

*

The time s t e p s used i n t h e c a l c u l a t i o n s were around 10-l4 seconds

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C7-348 J O U N A L DD PHYSIQUE

References.

[l] Kurosawa, T., Proc. Int. Conf. Phys. Semicond. Kyoto (1966).

l21 Kurosawa, T., J. Phys. Soc. Japan Suppl.

21,

424 (1966).

C31 Rees, H.D., IBM J. Res. Dev.

13,

537-542 (1969).

C41 Hammar, C.. J. Phys. C.

g,

70-78 (1973).

l51 Vassell, M.O., J. of Math. Phys.

11,

408-412 (1970).

C61 Coddington, A. and Levinson, N., Theory of ordinary d i f f . e q . , M c G r a w - H i l l ( 1 9 5 5 ) . C71 Kaszynski, A . , These Universitb de Lille, 1979.

C81

Fauquembergue, R., private communication.

2 .

1 (ooo)valley all valleys

GaAs 300 K

... N.2 N.4 M. Carlo

ref.171

i ;

D ^: ⁱ ^.

' ( P S )

\ I "

2

i ' ..

c