Competing tunneling and capacitive
channels in granular insulating thin
films: universal response
Montserrat García del Muro, Miroslavna Kovylina, Xavier Batlle and
Amílcar Labarta
Departament de Física Fonamental and Institut de Nanociència i Nanotecnologia,
Universitat de Barcelona, 08028 Barcelona, Catalonia, Spain
Introduction: magnetic granular solids
Nonmagnetic insulating
matrix (ZrO2, Al2O3)
thin
film
FM metallic particles
(Co, Fe, CoFe, FeNi)
J. Phys. D: Appl. Phys. 35, 15 (2002)
Introduction: magnetic granular solids


Fundamental properties of the FM nanoparticles (new phenomena):
finite-size, surface and proximity effects, and interparticle interactions.
Model systems for studying electric transport properties in disordered
media.
• High coercivity films for
CoPt:C xV= 0.7
magnetic storage

Applications
• High permeability and
resistivity films for
applications at high
frequency
• Tunneling
magnetoresistance (TMR)
(magnetic sensors)
M. Yu et al. APL 75 3992
(1999)
Introduction: dc electric transport properties
Regime:
dielectric
x
Co-ZrO2
ρ(T) changes in many orders of
magnitude
In the dielectric regime, ρ(T)
decreases abruptly with the
temperature.
transition
metallic
The slope of ρ(T) becomes
positive in the metallic regime
Introduction: dc electric transport properties
Dielectric regime: quantum tunneling among metallic particles
Coulomb
Blockade
es
-
+
d

  0 exp 2 B / k BT
EC0  e2 / keff d

2
ln (R[k])
Co-ZrO2 (x=0.27)
B  sEC0 2m /
Charging energy:
P.Sheng y B. Abeles, PRL 28, 34 (1972)
Sample preparation
Out-of-equilibrium methods (ultrafast cooling)
Laser ablation
pressure
gauge
substrate
heater
vacuum
chamber
substrate
target
turbopump
laser beam
focussing
lens
motor
targets
holder
Co
ZrO2
Composed
target
Nanotechnology 17, 4106 (2006)
Structural characterization
Low metal content (x<0.2)
Co particles are crystalline and
show sharp interfaces with the
amorphous matrix.
Particle size distribution is
bimodal.
There is a majority of very
small particles through which
tunneling can take place.
APL 91, 052108 (2007)
Structural characterization
Intermediate metal content (x > 0.2 < xp)
x
0.25
0.30
0.35
 The bimodal distribution collapses in a single broader effective log-normal
function.
 Further increase of the metal content: the size distribution shifts to larger
sizes keeping the width almost constant.
 About x=0.35, the size distribution broadens abruptly because of the
massive particle coalescence just before percolation.
Structural characterization
Z-contrast image
HRTEM
Beam
5 nm
2 nm
 Ultra-small glue particles in between the bigger ones are present at
any composition below the percolation threshold
Tunneling magnetoresistance (TMR)
H
H=0
low
resistance
P
D  D
D  D
Fermi energy
high
resistance
TMR in granular solids
Conductance between two particles:
G  G0 (1  P 2 cos)
q
Average over all orientations:

G  G0  g ( ) (1  P 2 cos ) d
0
 M
cos  
 MS
2

  m 2

G(0) 1  G( H ) 1
P 2 m2
T MR 

1
G(0)
1  P 2 m2
Inoue and Maekawa, PRB 53, R11927 (1996)
TMR in granular solids
x=0.27
 TMR in granular metals can be well reproduced by fitting experimental
data to the model of Inoue and Maekawa.
PRB 73, 045418 (2006)
TMR in granular solids

MRmax  P2m2 1  K / T

“cotunneling”
One electron is transferred
between two large particles
through a collective process
involving several small particles
Glue particles
S. Mitani et al., PRL 81, 2799 (1998)
PRB 73, 045418 (2006)
The role of glue particles
Glue particles
tunneling channels
These granular solids are model
systems for studying electric transport
properties in disordered media with
tunneling conduction among particles.
Ac response
ac conduction mechanisms
tunneling conductance
among particles

 t   0 exp 2 B / k BT

capacitance among
particles
 C does not depend on T
C  d2 / s
Ac response
ac conduction mechanisms
tunneling conductance
among particles

 t   0 exp 2 B / k BT

capacitance among
particles
 C does not depend on T
C  d2 / s
Ac conductance: dominant mechanisms
Random competition among tunneling and capacitive channels
throughout the system

  

Tunneling
conductance
1
Rt  T 
e-
e-
i C p
Capacitance
d2
Cp 
d
s
Homogeneity of the sample
at the macroscopic scale
d
s
s
 const.
d
Ac conductance: logarithmic mixing rule
n
ln    ln  i
Simple case: the conductance of both capacitive
and tunneling channels become comparable.
  (1 / Rt ) (iC p )
xr
1 xr

1 xr
i 1
i 
1 xr

1 xr
e
i

2
1 xr 
xr  fraction of tunneling channels
Constant phase
regime for the
impedance
  (1  xr )

Re   (1xr )
Fractional power
law
2
log10(σ/σ0)
0.4
0.3
1  xr
0.2
41 K
0.1
x=0.27
0
60 K
90 K
Ac conductance: modulus of the impedance
PRB 79, 094201 (2009)
Ac conductance: real and imaginary parts
 0 (T )
  1/ R  iC
C  Cr  iCi
   0  Ci  i Cr
 r T ,   0 T    Ci (T ,)
i T ,   Cr (T , )
Ac capacitance: real and imaginary parts
x=0.27
x=0.24
Cr  pF
290 K
Cr  pF
29 K
290 K
Ci  pF
29 K
Ci  pF
Ac capacitance: real and imaginary parts
Cr  pF
x=0.27
  e W/k T
B
W  18 meV
Arrhenius law
Ci  pF
x=0.27
 By using ντ as scaling variable, all the
p=0.5
curves for the real and imaginary (dielectric
loss) parts of the capacitance collapse onto
two master curves.
 This absorption phenomenon imitates the
universal response of disordered dielectrics.
PRB 67, 033402 (2003); PRB 79, 094201 (2009)
Simple model: random R-C network
Tunneling
among small
particles
Capacitance
among large
particles
Rt  1011  1014 
C p'  1021  1019 F
Cp  1019  1018 F
WINSPICE by M. Smith, University of Berkeley
Simple model: random R-C network
Simple model: random R-C network
x=0.34
x=0.29
The average tunneling resistance between
neighboring particles is in qualitatively
agreement with experimental dc resistance
of the samples multiplying it by an arbitrary
scale factor.
x=0.24
PRB 79, 094201 (2009)
Final remarks

The ac transport properties in granular magnetic
thin films originates from the competition between
interparticle tunneling and capacitance throughout
an intricate three-dimensional random R-C
network.

The effective ac behavior mimics the universal
response observed in many disordered dielectric
materials, but at much lower frequencies.

A random R-C network of resistors and capacitors
reproduces very well the overall experimental
behavior.