10

TWNELING TRCWSPORT I N INSULATORS

U.Da11acasa and C.Paracchini

E l e c t r i c i i e l d s as h i g h as 100 W/cm are present i ns ide i n s u l a t o r s , as i n d i c a t e d by the broaden ing o f the e l e c t r o n paramagnet ic resonance l i n e s 111 and by the t r e n d o f the abso rp t i on edge C21 and o f the ze ro phonon t r a n s i t i o n s t31. I n d i e l e c t r i c s the ex i s tence o f these f i e l d s i s a l l owed by the h i g h e l e c t r i c a l r e s i s t i v i t y o f the m a t e r i a l wh ich i n h i b i t s the screen ing o f the moving c a r r i e r s . Such e l e c t r i c f i e l d s wh ich f rom now on we c a l l i n t e r n a l f i e l d s may be o r i g i n a t e d by charged d e f e c t s o r by phonon induced l a t t i c e d i s t o r t i o n s t2 ,41. They are randomly d i s t r i b u t e d i n space and t ime g i v i n g an o v e r a l l n u l l va lue and no ex te rna l c u r r e n t , b u t they may p l a y an impor tan t r o l e i n the e l e c t r i c t r a n s p o r t p rocesses wh ich depend on the l o c a l d i s t r i b u t i o n s o f the e l e c t r i c p o t e n t i a l .

I n i n s u l a t o r s e l e c t r i c charges move th rough s teps i n l o c a l i z e d l e v e l s . The jump f rom one cen t re t o the o the r may take p lace w i t h the i o n i z a t i o n o f the cen t re produced by c a r r i e r s pass ing i n the conduct ion band, o r by t u n n e l l i n g between cent res . The f i r s t process, known as Poole-Frenkel e f f e c t , i s t he rma l l y a c t i v a t e d and a s s i s t e d by the e l e c t r i c f i e l d through the l c w e r i n g of the t r a p p i n g b a r r i e r , the second process i s e s s e n t i a l l y temperature independent and i t i s favoured by the e l e c t r i c f i e l d w i t h the r e d u c t i o n o f the b a r r i e r th ickness . A f u r t h e r mechanism, wh ich may be cons idered a m i x i n g of the former two, i s the the rma l l y a s s i s t e d t u n n e l l i n g , where e l e c t r o n s ge t a p a r t of t he energy f rom the phonons and tunne l1 across a reduced b a r r i e r . Both processes may take p lace i n the same compound, then a t h ighe r temperatures the the rma l l y a c t i v a t e d process i s favoured, w h i l e the t u n n e l l i n g occurs a t l w e r temperatures C5,61. I n s p i t e o f the exponent ia l dependence o f these e f f e c t s on the a p p l i e d e l e c t r i c f i e l d the c l a s s i c a l t h e o r i e s o f bo th the processes do no t cons ider the c o n t r i b u t i o n o f the i n t e r n a l f i e l d s .

I n p rev ious works the c o n t r i b u t i o n o f the i n t e r n a l f i e l d t o the Poole-Frenkel e f f e c t has been cons idered and a t h e o r e t i c a l model i n good agreement w i t h the exper imenta l r e s u l t s was fo rmu la ted 171. The improvement of t h i s model w i t h respec t t o o t h e r s i s the non requ i rement about the emission of the e l e c t r o n s i n the d i r e c t i o n oppos i te t o t h a t o f the a p p l i e d f i e l d . Moreover the p rev ious models o f the e f f e c t have been r e i n t e r p r e t e d on the b a s i s o f the i n t e r n a l f i e l d model C81.

I n t h i s paper the i n c l u s i o n o f the i n t e r n a l f i e l d i n t u n n e l l i n g e f f e c t s i s taken i n t o cons ide ra t i on .

The p r o b a b i l i t y P(E) o f t u n n e l l i n g i s g i ven by the f o l l c w i n g law t91:

P(E) = C exp ( - A / E)

w i t h A = ~.a;; ("-U0:* / Sie ( 1 )

where U i s the he igh t o f the b a r r i e r t o be tunne l l ed , Uo i s the e l e c t r o n energy and C i s a p r o p o r t i o n a l i t y f a c t o r . F o l l o w i n g a procedure used t o i nc lude the i n t e r n a l f i e l d c o n t r i b u t i o n i n the case o f Pogle-Frenkel e f f e c t 171, one s u b s t i t u t e s the a p p l i e d f i e l d va lue E w i t h i< t L I , where L i s the i n t e r n a l f i e l d vcctor.Then:

U.Dal lacasa and C.Paracchini a re w i t h the D ipa r t imen to d i F i s i c a o f the U n i u e r s i t a ' d i Parma, Parma, I t a l y .

P(E) = Po exp(-A / IE t L I )

11

(2)

and t o o b t a i n the average p r o b a b i l i t y one has t o per fo rm the i n t e g r a t i o n of equat ion (2 ) over the p r o b a b i l i t y f u n c t i o n o f L. The parameter A depends on the b a r r i e r h e i g h t and s ince the t u n n e l l i n g may take p lace i n d i f f e r e n t s i t e s o f the i n s u l a t o r the va lue o f A shou ld a l s o be regarded as a l o c a l l y v a r i a b l e q u a n t i t y . Then an a d d i t i o n a l average may be r e q u i r e d over the d i s t r i b u t i o n O f the va lues o f A , I n t h i s case the hopping process may be thought as a t r a n s p o r t mechanism i n which e l e c t r o n hops on d i f f e r e n t s i t e s o f a d i so rde red i m p u r i t y band. I n such s i t u a t i o n phonon emiss ion and abso rp t i on are r e q u i r e d t o overcome the energy d i f f e r e n c e between the s i t e s and i t i s p l a u s i b l e t o assume t h a t the i n t e r n a l f i e l d a r i s e s i n t h i s case f rom the i nvo l ved l a t t i c e d i s t o r t i o n . Accord ing t o the space dependence o f t he parameter A , two cases are here cons idered.

The f i r s t case concerns the t u n n e l l i n g o f b a r r i e r s whose shapes do n o t depend on the s i t e s . Such a case may be thought t o a r i s e when the d i so rde r i s smal l ; w h i l e L i s a f u n c t i o n o f the p o s i t i o n A may be cons idered as a un i fo rm va lue . I n t h i s case one may be suppose t h a t t u n n e l l i n g takes p lace between cen t res e n e r g e t i c a l l y equ iva len t and t h i s p rocess i s the more probab le i n h i g h l y o rdered ,compounds. I f the r e l a t i o n between P and i s g iven by the approx imat ion IE t fl = E + L, as p r e v i o u s l y done i n the Poole-Frenkel case C71, one needs t o c a l c u l a t e :

Q(L) exp(-A/(E+L)) dL ( 3 )

where Q(L) i s the d i s t r i b u t i o n o f L. For mathematical s e m p l i c i t y one assumes a Poisson d i s t r i b u t i o n :

Q(X) = ( l / (X) ) exp(-X/<X)) ( 4 )

where (X) i s the average va lue o f X.

W i th X = E t L , one o b t a i n s E t (L) = (X) and:

<P(E)) = ( l / (X) ) exp(- X/(X>) exp(- AA0 dX l"- and when <E+L) ( < A, t h a t i s <X) < < A:

which i n d i c a t e s t h a t (P(E)) i s p r o p o r t i o n a l t o exp ( -B/vE+L) where B = 2 fl i s cons tan t [IO].

I n the second case an a d d i t i o n a l d i s t r i b u t i o n o f the parameter A must be considered. Here one has a temperature dependent t u n n e l l i n g , where e l e c t r o n s move f rom one cen t re t o another w i t h a s l i g h t l y d i f f e r e n t energy l e v e l . T h i s i s a p rocess more probab ly expected i n amorphous compounds. One can repeat the argument s e m p l i c i t y , the q u a n t i t y 1 / 2 G as a random v a r i a b l e . Then n e g l e c t i n g the A a ( 1 / 4 ) dependence o f the preexponent ia l p a r t o f equat ion (6), one ob ta ins :

assuming,, f o r

<P(E)) = C* e x p { - 2 f i t<A)/(E+(L) l% 1 (7 )

where C* i s a s l o w l y v a r i n g f u n c t i o n o f < A ) and E*<L).

1 2

The ob ta ined r e s u l t s may become u s e f u l when (L) = W T. I n such cases equat ions (6) and (7 ) l e a d t o the w e l l Known fo rmulas f o r the f i x e d range hopping conduct ion C11,121:

<P(E)) = C* e x p c - w )

<P(E)) = C*. exp(- YTr/T) when E )) (L)

when E ( ( (L)

w i t h E* = 4 A and T* = W . o r those f o r the hopp ing w i t h v a r i a b l e range [12,131 :

(P (E) ) = c i exp(- EWE)% when E )) (L)

( 8 )

(P(E)) = C* exp(- TWT& when E ( < <L) ( 9 )

w i t h E* = 64 A and T* = 64 AIW

f i e l d u s e f u l r e s u l t s a re ob ta ined a l s o f o r t he t u n n e l l i n g process. We thus expect t h a t by i n c l u d i n g the c o n t r i b u t i o n o f the i n t e r n a l e l e c t r i c

I n o rder t o o b t a i n a suppor t o f t he proposed model one can es t imate the va lue o f the i nvo l ved parameters i n the low f i e l d and i n the h i g h f i e l d cases f r a n equat ion ( 9 ) and compare i t w i t h the exper imenta l r e s u l t s . I n the h i g h f i e l d case the va lue o f U i s lowered by the e l e c t r o n energy Uo which can be assumed dependent on the mean f r e e g a t h 1 as Uo = eEl . W i th 1 = 1000 f i and E = 10 W/cm one o b t a i n s Uo = 0.1 eU so t h a t , assuming f o r exap le U-Uo = 6 eU, one ob ta ins E* = 1.5 W/cm. Such a r e s u l t i s i n agreement w i t h the measurements 04 the dependence o f t he r e s i s t i v i t y on E % a t h i g h e l e c t r i c f i e l d s performed a t low temperature i n doped CdF2 c r y s t a l s as shown i n Fig.2. On the o the r hand the low f i e l d e s t i m a t i o n o f U-Uo about 0.1 eU leads t o the va lues o f W about 1 W/cm K i n o rde r t o reproduce the usual va lues o f T* (10-100 MK) a t a temperature o f 10 K as i n d i c a t e d i n Fig.1. One then o b t a i n s a va lue o f (L) which i s about 10 W/cm, i n agreement w i t h t h a t eva lu ted f o r the f i e l d a s s i s t e d thermal i o n i z a t i o n process C71 and elsewhere i n d i c a t e d C1,21.

L e t us cons ider more c l o s e l y the r e l a t i o n between the hopp ing conduct ion i n terms o f i n t e r n a l f i e l d and the usual approach o f phonon-assisted t u n n e l l i n g . I n t h i s case the hopping p r o b a b i l i t y P can be w r i t t e n as:

P = C’ exp (-aR w / k T ) (10)

where C‘ i s a p r o p o r t i o n a l i t y f a c t o r , a i s the inverse o f the l o c a l i z a t i o n r a d i u s o f the cen t re , R the average lengh t of hops and w i s t he a c t i v a t i o n energy o f hops between two n e a r l y e q u i e n e r g r t i c cen t res . F o l l o w i n g a proposal o f M o t t C131, the maximum p r o b a b i l i t y i s ob ta ined f o r the minimum va lue o f the exponent i n (10) and assuming w = 6/x. N R3 , where N i s the d e n s i t y o f s t a t e s a t the Fermi l e v e l , one has:

P = C’ exp (- (11)

where l e a d t o the i d e n t i f i c a t i o n :

B = 2.74 a3 / ( I T N # . The comparison o f t h i s r e l a t i o n w i t h equat ion ( 9 )

eL = 6.7 6 (U-UO?’~TI: N/6 4 k T am3 (12)

On m u l t i p l y i n g bo th s ides o f t h i s equat ion by R, one ge ts :

eLR = j kT (13)

13

w i t h j = 6.7 @ (U-Uo?"7CR "6% a3 . I f one assumes c m o n va lues f o r these q u a n t i t i e s , l i k e : U = 0.1 eV, a = 5 107 l/cm, N = 1 0 2 1 l/eV and R = 1000 A, one f i n d s j E! 1 so t h a t r e l a t i o n (12) g i v e s the conserva t i on o f the energy i n the hops sus ta ined by the i n t e r n a l f i e l d .

As f a r as the ac tua l dependence o f L on T i s concerned, we can argue t h a t the i n t e r n a l f i e l d i s a r e s u l t o f l a t t i c e d i s t o r t i o n d u r i n g the hops, as p r e v i o u s l y proposed C71. I f an e x t r a energy has t o be absorbed o r e m i t t e d i n a hop fran e l e c t r o n s whose energy E i s Cl05e t o the Fermi l e v e l E f : E = Ef , t h i s e x t r a energy has t o be d i s s i p a t e d th rough the l a t t i c e and as a r e s u l t a d i s t o r t i o n U i s produced. The va lue o f U can be es t imated by equa t ing the energy 2mVn re leased i n a c o l l i s i o n w i t h an i on o f mass M t o the p o t e n t i a l energy Ep = K * u.2 /2 o f the ion., K* b e i n g the e l a s t i c cons tan t . One ge ts :

U = v- and the r e l a t i v e d i s t o r t i o n du i s :

du = 4m (E f t kT ) / K* M

(14)

= KT v 4m / M K* Ef (15)

and the i n t e r n a l f i e l d is : L = N e du where N i s the t o t a l d e n s i t y o f the ions, so t h a t L = W T w i t h W = N e k v m .

By u s i n g a r e l a t i o n v= Vs Qd, where Us i s the v e l o c i t y o f sound i n the band Vs = U f V f the Fermi v e l o c i t y and Od the Debye v e c t o r , one can es t imate fi* = Qd U f 6. Wi th the c m o n va lues f o r narrow i m p u r i t y band conduct ion E f = 1 0 - 2 eV and U f = 6 106 CWS, one ge ts f l * = 18. Thus assuming N = 10 8'2 cm-3 and m/ti = 0.01, one has W = lKU/cm K, as ob ta ined fran the exper imenta l da ta shown i n F i g . 1.

Al though a d e t a i l e d s tudy o f the temperature dependence o f <L) needs t o be performed and the o r i g i n o f the i n t e r n a l f i e l d i s s t i l l a ma t te r o f specu la t i ons , one n o t i c e s t h a t q u i t e d i f f e r e n t s i t u a t i o n s , l i k e the Poole-Frenkel e f f e c t and t u n n e l l i n g phcnaena , c a l l f o r a u n i f y i n g p i c t u r e o f a l l the bu lk l i m i t e d e l e c t r o n i c t r a n s p o r t processes i n i n s u l a t o r s i n terms o f i n t e r n a l e l e c t r i c f i e l d .

--------------- Acknowledgements - T h i s work i s suppor ted by the Gruppo Na t iona le d i S t r u t t u r a d e l l a M a t e r i a (C.N.R.) and by the Cent ro I n t e r u n i v e r s i t a r i o d i S t r u t t u r a d e l l a M a t e r i a (M.P.I.).

References:

1. Mims W.8.and G i l l e n R., 1966, Phys.Reu. , 438-443. 2. Oow J.D. and R e d f i e l d O., 1972, Phys.Rev.B, 2 , 594-610. 3. Hughes A.E. and Runciman W.A., 1965, Proc.Phyr.Soc. , 615-627. 4. Stoneham A.M., 1966, Proc.Phys.Soc. , 909-921.

Stoneham A.M., 1969, Reu.Modern Phys. , 82-108. 5. Paracch in i C . , 1981, S o l i d S ta te Camnun. 2 , 1263-1267. 6 . Dal lacasa V . , Paracch in i C. and DeStab i le S., 1988, J.Phys.C, 2 , L567-572.

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7. Dal lacasa U., and Paracch in i C., 1986, Phrs.Rev.B, a , 8967-8970. Dal lacasa U. and Paracch in i C., 1987, IEEE Trans. E lec . I n s u l . EI-22 , Dal lacasa U. and Paracch in i C., 1986, Proc.2nd 1nt.Conf. on Conduct ion and

Breakdown i n S o l i d D i e l e c t r i c s . 21-25. 8. Dal lacasa U. , Paracch in i C. and DeStab i le S., 1988, J.Phys.C, 2 , 9 Landau L.D. and L i f s h i t z E.M., 1958, 'Quantum mechanics', Pergamon press . 10. Gradshteyn I . S . and Ryzhik I.M., 1996, 'Tables o f i n t e g r a l s , s e r i e s and products ' , Accademic press . 11. E f r o s A.I. and Shk lousk i i B.I., 1975, J.Phys.C, 8 , L49-51. 12. P o l l a k M. (ed.), 1987, 'Nonc rys ta l l i ne semiconductors ' , CRC p ress . 13. Mot t N. F., 1969, Phil.Mag. , 835-852.

467-472.

151-161.

M o t t N. F., 1979, Festkoerperprobleme, vo l 19, Braunschweig U ievcg.

10-5

10-6 c

a .- I O - ~

Ira

IO-^

F ig .1 - R e s i s t i v i t y VI. Temperature F i g . 2 - Cur ren t v s A p p l i e d f i e l d dependence f o r two CdF2 doped dependence f o r a CdF2 c r y s t a l a t c r y s t a l s . The da ta a rc p l o t t e d d i f f e r e n t temperatures: 24, 20, 15 accord ing equat ion (9). The s lopes o f and 12 K ( f rom up t o down). The da ta the s t r a i g h t l i n e s g i v e s i n the two a re p l o t t e d acco rd ing equat ion (9) cases2 T* 1.2 10'7 and 1.3 10.8 K. and the s lopes o f t he s t r a i g h t l i n e s

q iue : E* = 0.48, 0.60, 1.5 and 2.4 W/cm