FIELD DEPENDENT SCATTERING AND HOT ELECTRON KINETICS

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FIELD DEPENDENT SCATTERING AND HOT ELECTRON KINETICS

D. Herbert, S. Till

To cite this version:

D. Herbert, S. Till. FIELD DEPENDENT SCATTERING AND HOT ELECTRON KINETICS.

Journal de Physique Colloques, 1981, 42 (C7), pp.C7-277-C7-282. �10.1051/jphyscol:1981733�. �jpa-

00221670�

FIELD D E P E N D E N T S C A T T E R I N G A N D H O T E L E C T R O N K I N E T I C S D.C. Herbert and S.J. Till

Royal Signals and Radar Establishment, Malvern, United Kingdom

Abstract. - A new approach to hot electron transport based on real space Airy functions is outlined. It is shown that the scattering vertex is localised in real space and this allows derivation of simple kinetic equations. The energy and field dependence of the scattering rates are discussed in detail.

Introduction. - The theory of hot electron phenomena in semiconductors relies heavily on the Boltzman transport equation, and accurate numerical solutions of this equation can now be obtained^ '. The value of these solutions is restricted both by uncer- tainties in the material and scattering parameters, and by the restricted validity of the Boltzman equation itself in high electric fields. The technological trend towards VLSI systems, and the use of very high electric fields, for example in avalanche or electroluminescence devices, has led us to consider theoretical alter- natives which take explicit account of quantum effects.

In a k-space formulation, the intra-collisional field effect involves time integ- ration along the electron trajectory, leading to a complicated non-markovian des- cription of transport^ '. In this paper we propose a different approach based on real space Airy functions, which greatly simplifies the scattering vertex in high fields. The theory predicts strong field dependences for scattering rates, and significant scattering into field induced band tails. For very small geometry devices, ballistic transport and tunnelling effects may become important, and a correct theoretical description of these phenomena also requires the real space Airy functions.

A Theoretical Model. - Band structure wave functions in high electric fields have been used to discuss Zener tunnelling^ ', but are complicated for describing real space phenomena. In this paper we are concerned with transport in the regime where electrons can still be associated with particular band structure extrema, and we therefore consider an effective mass Hamiltonian, with eigenstates i|i and eigenvalues E

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-277

Résumé.- Une approche nouvelle au problème du transport des électrons chauds basée sur les fonctions d'Airy en espace réel est décrite. On montre que le vertex de diffusion est localisé en espace réel et que cela permet de trouver de simples équa- tions cinétiques. Les dépendances de l'énergie et du champ de vertex de diffusion sont discutées en détail.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981733

C7-278 JOURNAL DE PHYSIQUE

where m* i s an e f f e c t i v e mass, and F denotes an e l e c t r i c f i e l d i n t h e x d i r e c t i o n . The eigenstates a r e products o f plane waves f o r propagation perpendicular t o t h e f i e l d and A i r y f u n c t i o n s f o r propagation p a r a l l e l t o t h e f i e l d . The A i r y f u n c t i o n can be w r i t t e n i n terms o f Bessel f u n c t i o n s ( 4 ) and t h e asymptotic form o f t h e t r a v e l 1 in g wave A i r y f u n c t i o n becomes

2 3/2

0

-

^{y-1/4 exp}

[-

^{i y}

]

To d e r i v e s c a t t e r i n g r a t e s f o r use i n a k i n e t i c equation we consider m a t r i x elements o f t h e electron-phonon i n t e r a c t i o n V e19'r, which f a c t o r i s e i n t o a planewave and an A i r y f u n c t i o n component q

where 5 i s a v e c t o r i n t h e (yz) plane and t h e l a b e l s R f o r t h e A i r y f u n c t i o n s denote t h e p o s i t i o n o f zero k i n e t i c energy ( F i g . 1 ) . The argument o f the x i n t e g r a t i o n , r e q u i r e d t o evaluate the A i r y f u n c t i o n m a t r i x element i n (3) i s shown i n Fig. 2.

Fig. 1 : S c a t t e r i n g processes between 2 : Argument o f t h e A i r y A i r y f u n c t i o n s . The band edge v a r i e s

P-

u n c t i o n m a t r i x element showing t h e

1 in e a r l y w i t h p o s i t i o n , corresponding r e g i o n o f s t a t i o n a r y phase. The t o a constant a p p l i e d e l e c t r i c f i e l d . abscissa i s i n atomic u n i t s o f energy

eFx. The w i d t h 28-1/2 i n d i c a t e d by H was obtained from (4).

The main c o n t r i b u t i o n t o t h e i n t e g r a l i s l o c a l i s e d a t t h e p o i n t o f s t a t i o n a r y phase.

I f the phase o f t h e m a t r i x element i s expanded about t h e s t a t i o n a r y p o s i t i o n x t h e form a

+

^{~ ( x}

-

x ) 2

+ . . . ,

then t h e h a l f w i d t h can be estimated

'4

a n a l y t i c a l l y using the asymptotic forms ( 2 ) , and i s indicated i n Fig. 2.

The matrix element ( 3 ) i s used in f i r s t order perturbation theory to derive a s c a t t e r i n g r a t e . The density of s t a t e s N ( E ) f o r Airy functions i s constant a t a given f i e l d , and a normalisation length L i s determined from an e f f e c t i v e band width B

where G i s a reciprocal l a t t i c e vector(3). Here we are assuming that electrons will be localised i n regions of k-space where (1) i s v a l i d , so t h a t the theory should be insensitive t o assumptions on the band s t r u c t u r e f a r from the relevant extrema.

TO obtain an expression f o r the one-electron density matrix, we assume t h a t electrons a r e in wave packets which can be described by weakly localised envelope functions modulating the Airy functions. Then by considering a s t a t i s t i c a l distribution of wave packets, we can define a charge density p ( x

-

R) associated with the Airy function qR. For s u f f i c i e n t l y slowly varying envelope functions, in the absence of s c a t t e r i n g , p s a t i s f i e s the continuity equation

where J denotes the constant current associated with the underlying Airy function and pe(x

-

R) denotes the contribution t o the density from envelope terms. The s c a t t e r i n g contributions t o (6) acting a t the point (x,R) (Fig. 1 ) can be included using the real space l o c a l i s a t i o n of the s c a t t e r i n g vertex (Fig. 2 ) . By introducing densities f o r r i g h t and l e f t propagation, k i n e t i c equations a r e readily derived in terms of the s c a t t e r i n g r a t e s ( x , r ) , r = R'

-

R , between Airy functions. In three dimensions we use cylindrical coordinates and f a c t o r i s e the density matrix t o the form F(k)p(x

-

R), where k denotes the modulus of the wave vector perpendicular t o the f i e l d . A r a t e equation f o r F(k) i n terms of p i s obtained by considering the t o t a l s c a t t e r i n g i n t o a s h e l l (R,k). Details of these kinetic equations will be reported elsewhere and i n t h i s paper we concentrate on the s c a t t e r i n g r a t e s .

Scattering Rates.

-

To obtain the t o t a l s c a t t e r i n g r a t e i t i s necessary t o integrate over the phonon q . We regard each phonon as acting a t the point of stationary phase x I t i s found t h a t back s c a t t e r i n g dominates, when the stationary phase position

q ' 2

can be approximated by x

-

^R

-

^{q /4A,} and integration over q can be converted to x integration using dq = A

9

l 2 / ( x

-

~ ) l / Z d x ~ . This then yields a one-dimensional

'4

C7-280 JOURNAL DE PHYSIQUE

s c a t t e r i n g r a t e s (x,r)dx, where we n e g l e c t any q-dependence o f t h e deformation p o t e n t i a l

.

Computed values o f s(x,r) obtained from exact expressions f o r t h e A i r y f u n c t i o n s are shown i n F i g . 3a f o r several f i e l d strengths. The s t r u c t u r e o f these curves can be understood u s i n g t h e asymptotic wave f u n c t i o n ( 2 ) . The various f i e l d and p o s i t i o n dependent c o n t r i b u t i o n s t o s (x,r) can be i d e n t i f i e d as follows : normal i s a t i o n o f A i r y f u n c t i o n s A ' / ~ / L a ~ ~ / ~ , ~ h o n o n s A1/'/(x

-

R ) " ~ , d e n s i t y o f s t a t e s F-l,

] m a t r i x element12 a A - ~ / ~ ( x

-

R ) - ' / ~ . Combining these f a c t o r s y i e l d s s(x,r) a

( x

-

R ) - I and independent o f F a t f i x e d x. This behaviour i s expected a t l a r g e ( x

-

R) where t h e i m p l i c i t assumptions can be j u s t i f i e d , and may be seen from t h e numerical r e s u l t s i n F i g . 3a. A t s m a l l e r values o f ( x

-

^R), t h e peak i n t h e

s c a t t e r i n g r a t e i s r e l a t e d t o t h e r e a l space

104- w i d t h o f t h e s c a t t e r i n g v e r t e x merging w i t h

la1 Electric Field iv.crn4] t h e band edge, and the r i p p l e s are r e l a t e d

105 -

t o v a r i a t i o n s i n t h e s t a t i o n a r y phase amplitude w i t h x. Band t a i l s have been neglected i n these curves. Under non- b a l l i s t i c c o n d i t i o n s i t i s more u s e f u l t o r e d e f i n e s ( x , r ) by d i v i d i n g t h e curves i n Fig;. 3a by I $ ( x

-

R)

I

2

.

This normalises t h e i n i t i a l s t a t e i n t h e d e f i n i t i o n o f s so

16

-

t h a t t h e t r u e s c a t t e r i n g r a t e i s g i v e n by

0 500 two tsw zwo

x IWI p ( x

-

R)s(x,r). Using (2) again we f i n d

t h a t t h e a d j u s t e d s c a t t e r i n g v a r i e s as One dimensional o p t i c X - ' / ~ F - ' / ~ . I f we d e f i n e a k i n e t i c energy

$%&

:c:;tering r a t e s i n atomic

u n i t s . m = 1

,

hw = 60 meV, d e f o r - ^E by E = eFx, then i n energy v a r i a b l e s the mation o o t e n t i a l = 109 eV/cm. s c a t t e r i n q v a r i e s as E - ' / ~ . and i s i n d e ~ e n - b' scaitering rates as (a) but dent of F-at f i x e d in th; a s m p t o t i c '

normalised t o u n i t d e n s i t y f o r the

i n i t i a l s t a t e t ~ i x

-

^R). Sets o f region. These r e s u l t s can be observed from crosses, c i r c l e s and t r i a n g l e s

r e f e r t o p o i n t s o f equal k i n e t i c t h e numerical r e s u l t s p l o t t e d i n F i g . 3b.

energy (eFx)

.

To o b t a i n t h e s c a t t e r i n g r a t e i n t h r e e dimensions, we consider c i r c l e s o f constant

lkl,

when s t a t e s o f given t o t a l energy can be s p e c i f i e d by t h e l a b e l s (R,k). We consider s c a t t e r i n g between s t a t e s (R,k) and (R',kl) where k denotes a l l wave v e c t o r s o f equal magnitude. I n t e g r a t i o n over t h e qs i n (3) y i e l d s

where E denotes t o t a l energy, e i s t h e usual t h e t a f u n c t i o n and %w i s t h e o p t i c phonon energy. The s c a t t e r i n g r a t e between s h e l l s (R,k), ( R ' ,kl) i s given by s ' ( ~ ~ , k ) s ( x , r ) , where k ' i s f i x e d by energy conservation. A f t e r some a l g e b r a i c manipulation, the t o t a l s c a t t e r i n g from s h e l l (R,k) can be w r i t t e n i n the form

where e+ denotes t r a n s i t i o n s i n v o l v i n g phonon a b s o r b t i o n (+) o r emission (-) from t h e i n i t i a l s t a t e (R,k) and n i s t h e phonon occupation number. For s i m p l i c i t y we consider t h e s c a t t e r i n g p r o b a b i l i t y f o r a b a l l i s t i c e l e c t r o n propagating i n t h e f i e l d d i r e c t i o n , a t a given k i n e t i c energy eFxo. T h i s i s d e r i v e d by s e t t i n g F(k) = 6 ( k ) , p ( x ) = s ( x

-

xo), when (8) s i m p l i f i e s t o

The r e s u l t s o f numerical c a l c u l a t i o n s are shown i n F i g . 4.

w: ^dm

s ( x r ) w i t h s ( x r ) as i n F i g

,

m * = 0 . 4 , a n d a b a n d t a i l c o n t r i b u t i o n included. The band t a i l c o n t r i b u t i o n i s shown e x p l i c i t l y f o r the h i g h e s t f i e l d s t r e n g t h s .

JOURNAL DE PHYSIQUE

I t may be seen t h a t a t v e r y h i g h f i e l d s , s c a t t e r i n g i n t o band t a i l s (Fig. 1) becomes appreciable. I f f m s ( x r ) d r i s approximated by x m u l t i p l y i n g t h e one-dimensional r e s u l t , then t h e normalised s c a t t e r i n g i n F i g . 0 4 i s expected t o vary as G ' / ~ F - ' . The t r u e s c a t t e r i n g r a t e i s g i v e n by i m s ( x r ) d ~ , where Er = eFr, and t h i s w i l l remove the F-I dependence making t h e normalised s c a t t e r i n g r a t e approximately independent o f f i e l d as i n one dimension. The F-' dependence i s r e t a i n e d i n F i g . 4 t o show the r e l a t i v e c o n t r i b u t i o n s o f Band T a i l and A i r y - A i r y components. The zero f i e l d l i m i t o f t h i s formalism r a i s e s some d i f f i c u l t y as t h e A i r y f u n c t i o n s a t negative energy tend towards zero wave vector plane waves. The h a l f w i d t h o f t h e s t a t i o n a r y phase r e g i o n diverges as F tends t o zero and we would expect these r e s u l t s f o r s c a t t e r i n g t o apply as l o n g as t h i s w i d t h parameter i s s h o r t e r than a t y p i c a l wave packet l e n g t h . As a rough approximation we might take t h e s o l u t i o n p ( x

-

^{R ) as}

r e p r e s e n t a t i v e o f an average wave packet, b u t t h i s p o i n t w i l l be explored i n more d e t a i l elsewhere i n c o n j u n c t i o n w i t h t h e k i n e t i c equations. The c o n t r i b u t i o n o f band t a i l i n g i s p a r t i c u l a r l y i n t e r e s t i n g and may have i m p o r t a n t i m p l i c a t i o n s f o r t h e o r i e s o f impact i o n i s a t i o n .

References.

1. W . Fawcett (1973) i n "Electrons i n C r y s t a l l i n e Sol i d s " , I n t e r n a t i o n a l Atomic

Energy Agency Vienna, 531-618.

2. J. R. Barker (1973), J. Phys. C

5,

2663-84.

3 . E. 0. Kane (1959), J. Phys. Chem. S o l i d s

11,

181-88.

4. R. H. Fowler and L. Nordheim (1928), Proc. Roy. Soc.

119,

173-81.