S-D exchange model of zero bias tunneling anomalies observed in meta P-N junction

Appelbaum, J.

English

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Hard copy (HC) 1 L 00 
Department of Physics and Institute for the Study of Metals 
The University of Chicago, Chicago, Illinois 
(CODE1 
e 
E (PAGES) 
2 
2 0 R/ 763YA 
1-A CR OR TUX OR'AD NUMBER) 
Anomalies in the tunneling conductance centered at zero 
bias have been found in several experiments.' 
were performed on a large class of p-n junctions, as well as on 
junctions composed of normal metals separated by an insulating 
oxide layer. 
These studies 
In particular,Wyatt has observed' a peak in the conduc- 
tance, G(V), centered at zero bias in tunneling junctions where 
Ta or Nb was separated from A 1  by a thin oxide layer. 
found that G(V) could be divided into a temperature-independent 
part Go( V)> and a strongly temperature-dependent part AG(V)=G(VkGo(V). 
AG(V)/Go(V) varied as lnleV/kT/ for eV > kT, while A G ( @ ) / G o ( 8 )  
varied with temperature as InT. 
and below the superconducting transition temperature when care 
Wyatt 
The effect persisted both above 
b 
https://ntrs.nasa.gov/search.jsp?R=19660022079 2020-03-16T20:40:50+00:00Z
2. 
was taken to quench the superconductivity with an applied magne- 
tic field of 9 kG. The effect of varying the magnetic field 
between 4 kG and 20 kG was observable only at l.5OK, where a 
broadening of A G ( V >  was observed. Wyatt ass-mled that 
the zero bias anomaly in G(V) was due to a logarithmic singu- 
larity in the density of states at the Fermi surface. 
Anderson has suggested3 that the Wyatt anomalies may be 
4 caused by magnetic impurities, and recent experiments appear 
to corroborate his idea. Stimulated by Anderson's suggestion, 
we have analyzed several microscopic mechanisms; one of these 
is closely related to the scattering singularities discussed 
by Kondo. 5 
We begin by remarking that near zero bias and at low 
temperatures one expects localized states to contribute to the 
tunneling current by serving as a momentum reservoir for the 
tunneling electrons. Such a reservoir is obviously essential 
if the initial and final states of the tunneling electron have 
wave vectors differing by a substantial fraction of a recipro- 
cal lattice vector. This is the case for p-n junctions of Si 
and Ge and is probably the case for the junctions studied 
by Wyatt, considering the nature of the Fermi surfaces of the 
metals involved. 
We shall assume that the localized states (which may be 
associated specifically with impurities or with the metal-oxide 
3 .  
interface) are paramagnetic and are coupled to the conduction 
electrons by an exchange interaction of the following form: 
+ J C S+(ai-ak;+bk-bi+) * 
k, kf w L I . .  - -  
where S z ,  S+ and S- are the spin operators of the localized 
states. Electrons on the left (a) side of the junction which 
have momentum & and spina are described by the creation and 
respectively, while the opera- annihilation operators akb, akb, + - * 
have similar meanings for electrons on the tors bkf - of Y bgf, 6’ 
right ( b )  side of the junction. 
J is an exchange coupling for electrons which remain 
on the same side of the junction after scattering o f f  the lo- 
calized spin. TJ is similarly an exchange coupling, except it 
i 
4. Lr" 
i s  def ined i n  terms o f  e l ec t rons .on  oppos i te  sides of t h e  junc- 
t i o n .  It w i l l ,  consequently, be smaller than  J by a f a c t o r ,  
t h e  order  of the  over lap  of  t h e  exponent ia l  t a i l s  of t h e  wave- 
func t ions  on oppos i te  sides of  the  junc t ion .  The two terms, 
H1 and H2, represent respec t ive ly ,  t h e  two p o s s i b l e  outcomes of 
an  e l e c t r o n  s c a t t e r i n g  off  a l o c a l i z e d  spin:  e i ther  t h e  e lec-  
t r o n  s c a t t e r s  t o  t h e  o t h e r  side of t h e  junc t ion  o r  remains on 
t h e  same side. Thus, t h e  i n t e r a c t i o n  descr ibed  by H i s  confined 
t o  wi th in  a mean free path of  t h e  junct ion.  
H conta ins  t he  i n t e r a c t i o n  of conduction e l e c t r o n s  w i t h  
a s i n g l e  l o c a l i z e d  sp in .  We assume that  we may neg lec t  i n t e r -  
fe rence  e f f e c t s  between loca l i zed  s p i n  states. To o b t a i n  the 
t o t a l  cu r ren t  Jab between sides a and b w e  m u l t i p l y  jab, t h e  
cu r ren t  c a l c u l a t e d  from H by N, the  number of l o c a l i z e d  sp ins .  
We o b t a i n  jab from 
where f ( C k a  ) i s  the  Fermi-Dirac d i s t r i b u t i o n  f u n c t i o n  and 
W 
t e r i n g  from state ( , k , r )  on side a t o  ( & f , c r r )  on side b. 
c 
i s  the t r a n s i t i o n  p r o b a b i l i t y  f o r  an e l e c t r o n  s c a t -  k c  ;&tu f 
c 
T& t h i r d  order  i n  the exchange coupling W i s  given by: 
Hik Hk. Hi.  
+ complex conj .  27r 
'i;j = F[Gi Ei - Ek 
.- 
where Ei and E .  are t h e  energies  of  t h e  i n i t i a l  and f i n a l  states 
of t h e  e l ec t ron - loca l i zed  spin system. We sha l l  be i n t e r e s t e d  
J 
i n  TJ.  We 6Y7 i n  only those terms of Wi which are f irst  order  
; j  
assume t h e r e  i s  a magnetic f ie ld ,  2 present ,  so  tha t  t h e  
l o c a l i z e d  sp ins  have a Zeemamenergy, A equal t o  glMBIR, g i s  
t h e  g- fac tor  of t h e  l oca l i zed  sp in ,MB t h e  Bohr magneton. The 
Zeemannenergies of t he  conduction e l e c t r o n s  were found t o  have 
no e f f e c t  on t h e  cu r ren t .  
Typical s c a t t e r i n g  processes which con t r ibu te  t o  AG are 
shown i n  Fig. 1. Following an-analysis similar t o  that  of 
Kondo? one obta ins  f o r  t h e  processes shown i n  Fig. 1 t h e  f o l -  
lowing: 
where M i s  t h e  component of l o c a l i z e d  s p i n  along t h e  magnetic 
f i e l d  and S i s  the t o t a l  s p i n  of t h e l m a l i z e d  state.  We assume 
the Fermi energy on the  a side has been raised by an energy eV, 
the  applied 
g q u )  = 
p o t e n t i a l .  g ( n ) ( u )  i s  def ined  by: 
I 
. 
6 .  
where t h e  index n s p e c i f i e s  whether t h e  above sum i s  c a r r i e d  out 
over  states from t h e  a o r  b sides. 
i s  t h e  energy of an e l e c t r o n  w i t h  momentum ,k and 
15a E 
s p i n  d b e f o r e  the  extern21 e l e c t r i c  f i e l d  i s  applied,  and t h e  
Fermi energy appearing i n  f ( E  ) i s  the common Fermi energy of k 
sides a and b when V=O. 
We have l i m i t e d  t h e  sum over s ta tes  t o  a narrow energy 
r eg ion  of  width 2Eo centered a t  the Fermi energy. 
t he  f a c t  t h a t  only i n  a narrow energy i s  ou r  assumption v a l i d  
t h a t  the exchange couplings J and TJ appearing i n  Eq. ( l ) - ( 3 )  
are cons tan t .  
t aken  as an a d j u s t a b l e  parameter. 
T h i s  r e f l e c t s  
It i s  d i f f i c u l t  t o  estimate Eo and i t  w i l l  be 
When a l l  terms which con t r ibu te  t o  Wi i f  are evaluated, 
the &-space sums i n  Eq. ( 4 )  performed, and t h e  d e r i v a t i v e s  
w i t h  r e spec t  t o  V taken, one ob ta ins  8-11 f o r  G(v) :  
G ( V )  = Go(V)+AG(V) 
where 
and 
AG1(V) = ( anh2 A- kBT eV+t anh- A+ev)] 2kBT x 
I e V  I+kBT 
I n  
EO 
.- 
2 To order  JTJ , only terms with a s t rong  temperature and vo l t age  
dependence have been re ta ined .  <M > and (M> refer  t o  appropr ia te  
s t a t i s t i c a l  averages of Sz2 and Szy r e spec t ive ly .  
form I n  2 which appear above are i n t e r p o l a t i v e  approxima- 
t i o n s  to t h e  func t ion  F ( w ) ,  where 
2 
Terms of t he  
kT+ cu 
EO 
6 
We have assumed throughout that  +")(E) ,  t h e  d e n s i t y  of 
states,  i s  a slowly varying func t ion  of energy and where it has 
appeared i n  i n t e g r a l s  we have replaced i t  by its value a t e F y  
the Fermi energy. 
For H = 0, ( A  = 0) G(V)  s implif ies  t o :  
whence 
a. 
1 
i . i  
T h i s  i s  p r e c i s e l y  the  temperature and vo l t age  dependence found 
by Wyatt. 
JFa+yb) = 0.012. L i t t l e  s ign i f i cance  should be attached t o  
t he  s m a l l  values  of  Eo and J 
parameters w i l l  be renormalized when, e.g., t he  c u r r e n t  from 
non-magnetic l o c a l i z e d  s t a t e s ,  i s  considered. Notice a l s o  that  
J is  p o s i t i v e ,  which implies  ant i ferromagnet ic  coupling between 
the  conduction e l e c t r o n s  and t he  l o c a l i z e d  spins, and t h a t  
ferromagnetic coupling i n  Eq. (12 )  impl ies  a d i p  i n  G(V) 
i n s t e a d  o f  a peak. 
F i t t i n g  h i s  data w e  o b t a i n  Eo = 10.6 meV and 
obtained from the data, as these 
For #- 0, G ( V )  assumes the  rather complicated form given 
i n  Eqs. (8)-(10) .  For weakmagnetic f ie lds  ( those  f o r  which 
A/2kBT << l), g s h o u l d  have no no t i ceab le  e f f e c t  on G(V) .  
i s  expected t o  be t h e  case f o r  a l l  temperatures s t u d i e d  by 
T h i s  
Wyatt except T = l . 5 ' K .  (We have assumed g = 1 i n  eva lua t ing  A . )  
A t  t h i s  temperature t h e  e s s e n t i a l  e f f e c t  of f;s is  t o  broaden the  
peak. l2 T h i s  can be seen from ( l o ) ,  where t h e  s i n g l e  l o g a r i t h -  
mic peak centered  on zero  bias f o r  E = 0 is  s p l i t  i n t o  three 
peaks, two of which a r e  d isp laced  by A t o  e i ther  s ide of zero 
bias. W e  see, the re fo re ,  t h a t  the  magnetic f i e l d  dependence 
of G(V) i s  c o n s i s t e n t  w i t h  that found by Wyatt. 
I n  t h e  l i m i t  eV, kBT (< A w e  expect H t o  have a more 
no t i ceab le  e f f e c t  on m. T h i s  e f f e c t  w i l l  be d i f f e r e n t ;  
depending on whether t h e  main con t r ibu t ion  t o  Go comes from 
magnetic o r  nonmagnetic s c a t t e r i n g .  I n  t he  former case,  Eq.(13) 
is replaced by 
while i n  t h e  l a t te r ,  one should use i n s t e a d  of Eq. (13) 
i n  t h e  h igh  f i e l d  l i m i t .  
Experimental s t u d i e s  i n  t h e  above regime should s e r v e  to 
t e s t  the  proposed exchange model. 
The au thor  wishes t o  express h i s  s i n c e r e  apprec i a t ion  to 
J. C. P h i l l i p s  f o r  suggesting t h i s  problem, as well as h i s  
encouragement during i t s  execution. H e  has a l s o  b e n e f i t t e d  from 
discuss ions  wi th  K. Bennemann, H. F r i t z sche  and D. Perm. H e  
fu r the r  thanks J. C .  P h i l l i p s  and M.  H. Cohen f o r  t he i r  c r i t i c a l  
reading of the manuscript. 
10. 
. 
REFERENCES 
$- This work was supported in part by the National Aeronautics 
and Space Administration. 
NASA Predoctoral Fellow rt: 
1. R. A. Logan and J. M. Rowell, Phys. Rev. Letters 9, 404 
(1964); A.F.G. Wyatt, Phys. Rev. Letters 2, 401 (1964); 
John Shewchun and Robert M. Williams, Phys. Rev. Letters 
a, 160 (1965); J. M. Rowell, private communication. 
2. A.F.G. Wyatt, see reference 1. 
3 .  P. W. Anderson, private communication (1964) to J. C. 
Phillips. 
4. L.Y.L.Shen and J. M. Rowell, Bul. Am. Phys. SOC. - 11, 224 
(1966). 
5. J. Kondo, Prog. Theor. Phys. 32, 37 (1964); 2, 204 (1965); 
2, 523 (1965). 
6. D. J. Kim, Phys. Letters - 18, 215 (1965) has considered a 
transfer mechanism based on electron-electron exchange in 
which electrons spin flip scatter from one side of the 
junction to the other with conduction electrons on either 
side of the junction taking up the spin recoil. Kim 
obtains a singular term in the current, but only in third 
. 
11. 
order in the tunneling coupling constant T. His results 
are open to question, however, on a number of points, not 
the least of which is his use of a tunneling "transfer" 
Hamiltonian to calculate lifetimes t o  third order ( a  pro- 
cedure we feel of doubtful validity); currents are then 
obtained by a semiclassical transport argument. Using 
Kim's Hamiltonian we have calculated currents using the 
equation-of-motion method (ref. 7) and have found that 
singular terms appear first in fourth order in T. These 
terms are probably too small t o  be observable. 
7. M. H. Cohen, L. M. Falicov and J. C. Phillips, Phys. Rev. 
Letters - 8, 316 (1962). 
8. As noted by Abrikosov (ref.g), Nagaoka (ref.10) and Suhl  
(ref.ll), the logarithmic singularity appearing in third 
order in the coupling constants J and TJ is removed at 
sufficiently low temperatures by summing to all orders in 
perturbation theory. In our analysis this will a lso  be 
the case, but because TJ, the relevant coupling constant 
in this analysis, is very much smaller than J, we expect 
the temperature at which this occurs to be very much 
smaller than in the Kondo effect. Thus, higher order 
terms in T are probably unimportant in the temperature 
region we are studying. 
J 
12. 
9. A. A. Abrikosov, Physics 2, 5, 61 (1965). 
10. Y. Nagaoka, Phys. Rev. lJ8, A 1 1 1 2  (1965). 
11. H. Subl, Phys. Rev. 138, A515 (1965). 
” 
. . 13. 
FIGURE CAPTIONS 
Fig.  1 - Diagram 1 represents  a second order  s c a t t e r i n g  process  
i n  which a n  e l ec t ron  i n  state k+ s c a t t e r s  i n t o  the  
v i r t u a l  s tate cy. The loca l i zed  sp in  s t a t e ,  repre-  
sented i n  the  diagram by a c i r c l e ,  changes i t s  2- 
component of sp in  b y  one u n i t .  The v i r t u a l  s t a t e  g- 
then  s c a t t e r s  i n t o  t h e  f i n a l  state kl. Such a process  
may a l s o  occur i n  the  reverse order:  e.g. ,  q- first  
s c a t t e r i n g  E-, then $+ s c a t t e r i n g  9-. The double 
v e r t i c a l  l i n e s  represent  the  junc t ion  i n t e r f a c e .  
Diagrams 11-IV a r e  s i m i l a r l y  in t e rp re t ed .  
d f i  + 
$1 
i 


S-D exchange model of zero-bias tunneling anomalies

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