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Submitted on 1 Jan 1978
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THE HOPPING MODEL OF ZERO-BIAS
TUNNELLING ANOMALIES : MAGNETIC FIELD
EFFECT
T. Ivezić
To cite this version:
JOURNAL D E PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-854
THE HOPPING MODEL OF ZERO-BIAS TUNNELLING ANOMALIES : MAGNETIC FIELD EFFECT
T. Ivezicf
Institute of Physios of the University, P.O.B. 304, Zagreb, Yugoslavia
Résumé.- Le modèle des liaisons fortes pour l'effet tunnel, proposé par Caroli et al./l/, est utilisé dans l'étude de la diffusion des électrons par les impuretés dans une jonction métal-isolant-métal, avec les impuretés magnétiques situées soit dans les électrodes soit dans l'isolant. Les caractéristiques de la conductivitê en présence d'un champ magnétique sont obtenues et leur différence par rapport à la théorie d'Appelbaum /2,3/ discutée. La dépendance des différentes contributions à la conductivitê sur la position des impuretés magnétiques est étudiée.
Abstract.- A hopping model of tunnelling proposed by Caroli et al./l/ is used to study electron-impurity scattering in metal-insulator-metal junctions with magnetic impurities in either the insulator or the electrodes. The conductance characteristics in magnetic field are obtained and their differences with respect to Appelbaum's theory /2,3/ are dis-cussed. The dependence of the different contributions to the conductance on the position of the magentic impurities is considered.
Expressions for the conductance of a ma-gnetically doped tunnel junction are derived in our earlier paper /4/, using the hopping model of zero bias tunnelling anomalies. Here, we consider the same type of junction and obtain the junction con-ductance characteristics in a magnetic field. As shown in our earlier paper the Appelbaum theory /2,3/ does not include all the important contribu-tions to the conductance. This is also true for the tunnel junctions in the magnetic field.
Due to the freezing out of some spin-flip processes in the presence of the magnetic field H there appear a well in the conductance characteris-tic G(V) for e|v|<gv_H. In Appelbaum's theory the total conductance near zero voltage even falls be-low the background for large enough field. However, Wallis and Wyatt /5/ showed experimentally that there is only a local minimum in conductance at zero bias, and in this respect their results did not agree with the Appelbaum theory.
We present general results for the conduc-tance in a magnetic field and simplify these results to the case of impurities in the barrier not far from the interface. Instead of Appelbaum's undeter-mined phenomenological parameters we haye explicit expressions in which the dependence of the conduc-tance on the position of the impurity is taken into account. It is shown that some new terms appear in our theory. These terms may be important when the impurities are just near the interface, which is the usual experimental situation.
Our H a m i l t o n i a n i s / 4 /
H = H„T - J Z C+. b aC..t. + g u _ £ . 8 ( 1 )
T1IM i a a g 16 1 ° B 1 Ctp
The first term H^T M is the pure contact Hamiltonian in the hopping model of tunnelling, the second term is the Kondo type exchange coupling and the third term describes the local spin coupling to the
ma-+ r a gnetic field. We have calculated £~ and £ ' self energy matrix elements up to order J2 and J3, using the Keldysh technique, for the impurities confined to the barrier region and to the electro-des . The second order contribution to the conduc-tance in the magnetic field is the most important term. In contrast to the zero field case it shows the strong temperature and voltage dependence. When the impurities are in the barrier it has the following form
6&2)(VlTlH) _ .._ -. < M > ( Y C2^ ) +
~
( 2 )( H = o ) "
1+2cos2* " IscsTTT [
k Tr*
+ v ,-Q-eVi . , . 2irSin(|) _ 2 <M> cos<j> Y ( kT >} |_l R K+J S(S+1) R K
-(F(Q-eV) - F(Q+eV)) ( 2 )
WhSre F(eVlT) • - JE
- ^ ' - ^ p f d e gf* (3)
Q=gy^H and <M> is average spin polarization in
D
the field H.
G ™ = R eisi and K = T2 p° | G T ° |2 ± T'2p° |G?°|2 11 + a ' xa' 6 ' ib' (see /4/)
e2'2xex-' and 6 ~ ( ~ ! (H=O) i s t h e 6' second y ( x ) = (Imex)2
c-
o r d e r c o n t r i b u t i o n t o t h e conductance i n t h e zero f i e l d . c a s e G ( O ) is t h e conductance f o r an undoped j u n c t i o n . The sum of a l l terms without t h ed
f u n c t i o n cor- responds t o Appelbaum's r e s u l t . The magnetic f i e l d does n o t a f f e c t t h e f i r s t two terms i n ( 2 ) , which a r e none o t h e r than t h e second o r d e r conductance i n t h e absence of magnetic f i e l d G ( ~ ) ( H = O ) . It a f f e c t s 6 ~ ' ~ ) through t h e terms p r o p o r t i o n a l t o<M>. One of them (which does n o t c o n t a i n
d)
i s due t o t h e Cf self-energy and corresponds t o Appel- baum's t h e o r y . It always d e c r e a s e s 6 ~ ' ~ ) n e a r z e r o v o l t a g e . On t h e o t h e r hand, t h e term p r o p o r t i o n a l t o <E9 s i n d, which i s due t o Jm z r l a does n o t appear i n Appelbaum's theory. This term c o n t a i n s t h e e x p l i c i t dependence on t h e o p o s i t i o n of t h e im- p u r i t i e s i n t h e b a r r i e r . For t h e i m p u r i t y n e a r t h e i n t e r f a c e , t h e a n g u l a r b r a c k e t i n (2) can b e w r i t t e n a s 1-
2sin2+.1n t h a t c a s e , t h e i m p u r i t i e s which a r e approximately one atomic d i s t a n c e fromt h e i n t e r f a c e /4/ c a n i n c r e a s e t h e second o r d e r conductance
.
The term p r o p o r t i o n a l t o <M> cosd, which
i s due t o Re z r l a i s an odd f u n c t i o n of b i a s i n t h e magnetic f i e l d . ( A l l o t h e r terms a r e even i n V). I f t h e i m p u r i t i e s were symmetrically p o s i t i o n e d i n t h e j u n c t i o n , t h i s term would v a n i s h i d e n t i c a l l y . Howe- v e r , i f t h e i m p u r i t i e s a r e a s s o c i a t e d s p e c i f i c a l l y w i t h one s i d e of t h e j u n c t i o n , i e c a n n o t b e neglec-
t e d .It becomes p r o p o r t i o n a l t o
<M>
s i n 2d 6 r t h e i m p u r i t y n e a r t h e i n t e r f a c e . Appelbaum / 2 , 3 r o b t a i - ned a s i m i l a r term and i n t e r p r e t e d i t a s a n i n t e r - f e r e n c e term i n v o l v i n g simultaneous p o t e n t i a l a n d exchange t u n n e l l i n g . Our c o n s i d e r a t i o n s show t h a t such a term a r i s e s from t h e p u r e exchange s c a t t e - r i n g p r o c e s s i n t h e second o r d e r of t h e exchange c o u p l i n g c o n s t a n t.
The t h i r d o r d e r c o n t r i b u t i o n t o t h e con- ductance can be w r i t t e n as a sum of even and odd p a r t s
.
6 ~ ( 2 : ~ ~ = ( I
-
2 s i n z d ) 6 ~ " ) (HIVIT) ( 5 )z*
e x p r e s s i o n (5) i s a l r e a d y w r i t t e n f o r t h e u s u a l c a s e of t h e i m p u r i t y n e a r t o one f o t h e e l e c t r o d e s . The odd p a r t , which i s due t o Re ~ ~ ( f o r t h e ~ 1 , same p o s i t i o n of t h e i m p u r i t y , can b e w r i t t e n a s s ~ ( i L , r =
-
--
*
J s i n 2d < M > ~ ( F ( Q + ~ v )-
ITS(S+I) 60(:i (H=O)
I n (6) we approximated t h e double i n t e g r a l s over t h e Fermi f u n c t i o n s by 1n2 terms (E i s t h e c u t - o f f i n t h e i n t e g r a l s ) .
For t h e i m p u r i t i e s i n t h e e l e c t r o d e we ob- t a i n e d t h e same f u n c t i o n a l form f o r z e r o b i a s anomalies a s f o r t h e i m p u r i t i e s i n t h e b a r r i e r . However, t h e s i g n of zero b i a s anomalies due t o i m p u r i t i e s i n t h e e l e c t r o d e i s random. Because of t h a t t h e b a r r i e r i m p u r i t i e s w i l l g i v e t h e main c o n t r i b u t i o n t o t h e conductance.
W r i t t i n g t h e p r o p a g a t o r s i n t h e continuous r e p r e s e n t a t i o n / 4 / and f o r t h e s q u a r e b a r r i e r po- t e n t i a l model we a r e a b l e t o compare our t h e o r y w i t h a v a i l a b l e e x p e r i m e n t a l r e s u l t . This w i l l be r e p o r t e d elsewhere.
We s e e t h a t exchange c o n t r i b u t i o n does n o t always give a conductance i n p a r a l l e l w i t h t h e normal t u n n e l l i n g a s i n Appelbaum's t h e o r y . I n f a c t , t h e exchange c o n t r i b u t i o n h a s a r a t h e r com- p l i c a t e d dependence on t h e p o s i t i o n of t h e i m p u r i t y i n the b a r r i e r .
T h e e ~ p l i c i t form of t h e p o s i t i o n dependence of t h e conductance i s our main r e s u l t .
References
/ I / C a r o l i , C., Combescot, R., Noziares, P. and S a i n t James, D., J.Phys. C : S o l i d St.Phys.5 (1972) 21.
/ 2 / Appelbaum, J . A . , Phys .Rev.Lett.l7 (1966) 91 /3/ Appelbaum, J.A., Phys.Rev. 154 (1967) 154 /4/ I v e z i i , T., J.Phys. C : S o l i d S t . Phys.8
(1975) 3371.
/5/ W a l l i s , R.H. and Wyatt, A.F.G., J.Phys. C : S o l i d S t . Phys. 7 (1974) 1293.
