SCATTERING OF CHARGE-CARRIERS BY DISLOCATIONS

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SCATTERING OF CHARGE-CARRIERS BY DISLOCATIONS

E. Gerlach

To cite this version:

E. Gerlach. SCATTERING OF CHARGE-CARRIERS BY DISLOCATIONS. Journal de Physique Colloques, 1979, 40 (C6), pp.C6-47-C6-49. �10.1051/jphyscol:1979610�. �jpa-00219026�

JOURNAL DE PHYSIQUE CoZZoque C6, s u p p Z h e n t au n06, tame 40, j u i n 1979, page CG-47

SCATTERING OF CHARGE-CARRIERS BY DISLOCATIONS

E. Gerlach

I . PhysikaZisehes I n s t i t u t der RWTH Aachen, F. R. C.

Resume.- La methode de " l l @ n e r g i e perdue" e s t u t i l i s e e pour t r a i t e r l a d i f f u s i o n dc e t dynamique p a r des d i s l o c a t i o n s .

A b s t r a c t . - The energy-loss-method i s used t o t r e a t dc and dynamical s c a t t e r i n g by d i s l o c a t i o n s .

1. I n t r o d u c t i o n . - ^A common method t o s t u d y i r r e g u l a - r i t i e s i n c r y s t a l s such as d i s l o c a t i o n s , p o i n t d e f x t s e t c . .

.

These, however, a r e i n t e g r a l q u a n t i t i e s a v e r a g i n g o v e r a d i s t r i b u t i o n o f t h e e l e c t r o n s i n momentum spa- ce. I n f o r m a t i o n about t h e s c a t t e r i n g mechanism, how- ever, i s r e t a i n e d , i f parameters as t h e temperature T, t h e frequency u o r a magnetic f i e l d can be v a r i e d .

I n a c r y s t a l d i s l o c a t i o n l i n e s a r e arranged i n accordance w i t h t h e c r y s t a l s t r u c t u r e t h u s f o r m i n g a g e o m e t r i c a l l y c o m p l i c a t e d s c a t t e r i n g system. I n a d d i t i o n , t h e s c a t t e r i n g p o t e n t i a l u s u a l l y i s s t r o n - g l y a f f e c t e d by f r e e c a r r i e r s . Thus i n o r d e r t o des- c r i b e t h e s e phenomena we need a t r a n s p o r t t h e o r y which i s a b l e t o cover t h e f o l l o w i n g p h y s i c a l s i t u a - t i o n s :

F i g . 1 : Moving screened p o i n t charge i n an e l e c t r o n gas.

R e t u r n i n g now t o t h e t r a n s ? o r t ~ r o b l e m , we r e - verse t h e frame o f r e f e r e n c e , t h a t i s we c o n s i d e r t h e p o i n t charge t o be f i x e d i n space and p u l l t h e e l e c t r o n gas w i t h t h e v e l o c i t y y o v e r i t . The s c a t - t e r i n g o f t h e e l e c t r o n s leads t o a J o u l e h e a t j.E - - which i s i d e n t i c a l t o t h e energy l o s s c a l c u l a t e d above

a ) Geometrical l y complicated s c a t t e r i n g c e n t e r s -

b ) Screening b y f r e e c a r r i e r s

c) Dynami c a l ( U-dependent) t r a n s p o r t p r o b l ems d) Plasmon e x c i t a t i o n

e) E x t e r n a l magnetic f i e l d s

F u r t h e r , o f course, t h e t h e o r y s h o u l d be conceptual- l y and m a t h e m a t i c a l l y simple.

2. The "Energy-loss" method. - A t r a n s p o r t t h e o r y f u l - f i l l i n g t h e s e requirements has r e c e n t l y been s t u d i e d i n a wide range o f semiconductor p h y s i c s . ( F o r a r e v i e w see / 3 / ) . I n o r d e r t o i l l u s t r a t e t h e i d e a o f t h i s method we c o n s i d e r t h e s c a t t e r i n g by a p o i n t charge o f d e n s i t y pf.

We s t a r t f r o m an energy l o s s arrangement i n w h i c h a charged partic1.e i s moving w i t h c o n s t a n t ve- l o c i t y t h r o u g h a s o l i d ( F i g . 1).

The energy l o s s o f t h e p a r t i c l e i s g i v e n by a s i m p l e f u n c t i o n a l F o f t h e moving charge d e n s i t y pf, o f i t s v e l o c i t y 1 and t h e d i e l e c t r i c p r o p e r t i e s E

( k , ~ )

t h e s o l i d /4/

(N c o n c e n t r a t i o n of s c a t t e r i n g c e n t e r s , h e r e assumed t o be indeoendent o f each o t h e r , p r e s i s t i v i t y , n c a r r i e r c o n c e n t r a t i o n ) . From t h i s i d e n t i f i c a t i o n t h e dc r e s i s t i v i t y of t h e c a r r i e r s s c a t t e r e d b y p o i n t charges o f d e n s i t y p f i s i m m e d i a t e l y obtained. (The f u n c t i o n a l F i s an i n t e g r a l . The d r i f t v e l o c i t y y i s c a n c e l s ) . The s u b t l e t i e s o f t h e s c a t t e r i n g a r e con- t a i n e d i n t h e wavevector- and frequency dependent d i e l e c t r i c f u n c t i o n o f t h e e l e c t r o n gas which we con- s i d e r i n t h e Lindhard-(RPA)-approximation /3/,/5/

( F i g . 2 ) . T h i s can e a s i l y be extended t o n o n - s p e r i c a l cases.

I n t h e t h e o r y d e s c r i b e d so f a r no p a r t i c u l a r use was made of t h a t pf was a p o i n t charge. The quan- t i t y can be t a k e n t o be a charged l i n e o r a d i p o l e l i n e r e p r e s e n t i n g s i m p l e models of d i s l o c a t i o n s o r p f may be any c o m p l i c a t e d s c a t t e r i n g system. Scree- n i n g e f f e c t s a r e a u t o m a t i c a l l y i n c l u d e d b y t h e func- t i o n E (&,LO)

.

F u r t h e r , i n s t e a d of c o n s i d e r i n g an energy-10s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979610

C6-48 JOURNAL DE PHYSIQUE

arrangement with constant velocity one may study an o s c i l l a t i n g charge density pf thus leading t o a complex frequency dependent r e s i s t i v i t y , which also includes plasmon-effects.

Fig. 2 : Real and imaginary part of the Lindhard die- l e c t r i c function f o r a model semiconductor

( n = 1015cm-3, mg = 0,2m0, ^E~ = 36, T = 80Kl

For a charged monopole l i n e and a dipole l i n e , respectively, the r e s u l t s are given by /6/

m

(q charge per 1 ength, N m density of monopole 1 ines)

( y = dipole moment per length and elementary charge, Nd density of dipole l i n e s ) . These expresssions, of course, have t o be completed be a term originating from the i n e r t i a of t h e electrons (w plasma frequen-

P CY)

pi = - i ^W

E omL

P

The interaction of two neighboring dislocation l i n e s was studied i n ieference /7/.

We now l i s t a number of r e s u l t s obtained f o r s c a t t e - ring by monopole l i n e s and by dipole l i n e s .

a) charged monopole 1 ine / 3 / , /6/

Nondegenerate c a r r i e r d i s t r i b u t i o n P(W=O) = nlA Nm q2 mmm%g(p)

Z9h (n E ~ ~ e kT ) ~ A g(p) = ( 1 + 2p) eP e r f c plA - 2p'h //.rr1/2

p = h2kiH/8 mX kT ;

k i H

E~ d i e l e c t r i c constant of t h e host material.

Degenerate c a r r i e r d i s t r i b u t i o n

1 1

( p a ) = a r c tan p' - -

1 + p"

p' = 2 kF/kFT ; kFT = (m* e 2 / h 2 ~ , ~ 0 ) (3n/+')'h

b ) Dipole 1 ine / 3 / , / 6 / , nondegenerate

(dipole moment perpendicular t o current) pll = 3 pJ-

P ' ( W ⁺ m) ^Q w-I

Figures 3and 4 show the s c a t t e r i n g of f r e e electrons by a monopole l i n e and a dipole l i n e , res- pectively, f o r the model semiconductor of figure 2.

The plasma resonance a t w /GL separates a constant P

behavior from a power law.

The method has also been successfully applied t o deformation potential s c a t t e r i n g and t o piezo e l e c t r i c s c a t t e r i n g /3/.

I f the dislocations a r e arranged i n a layer s t r u c t u r e the dynamical r e s i s t i v i t y under c e r t a i n conditions shows geometrical resonances f o r frequen- c i e s which correspond t o Bragg-reflection, of the f r e e electrons and of the pl asmon, respective1 y , p' (w + m) Q w-5/2 (Gerlach E , Paier K , unpublished).

In conclusion we would l i k e t o mention t h a t a large number of the r e s u l t s obtained by the energy loss method have been successfully applied t o iden- t i f y the various types of s c a t t e r i n g centers i n semi conductors /3/.

The dc-limit i s immediately obtained by taking the l i m i t w -. o.

References

/6/ Gerlach, E. and Rautenberg Y., Phys. Status S o l i d i ( b ) ,

67

/7/ Doukhan, J.C., Drope, R., Farvacque, J.L., Gerlach, E., Grosse, P., Phys. Status S o l i d i ( b ) 64, (1974) 237.

Fig. 3 : Dynamical r e s i s t i v i t y f o r s c a t t e r i n g by a monopole l i n e versus frequency (- r e a l p a r t , ---

imaginary p a r t ) .

/1/ DUster, F. and Labusch, R., Phys. Status S o l i d i (b)

60

/2/ PBdor, B., Phys. Status S o l i d i

16

/3/ Gerlach, E. and Grosse, P. , Festkorperprobleme, Advances i n S o l i d S t a t e Physics, Volume X V I I , p. 157, J. Treusch ed, Vieweg, Braunschweig (1977)

/4/ Pines, D., Elementary E x c i t a t i o n s i n S o l i d s (W.A. Benjamin,New York) 1964, p.126,144.

/ 5 / Lindhard, J. , Kgl. Danske, Videnskab Sel skab. ,

Yat. Fys. Yedd

8,

F i g . 4 : Dynamical r e s i s t i v i t y pL f o r s c a t t e r i n g by a d i p o l e l i n e versus frequency (- r e a l p a r t ,

--- imaginary p a r t )