Dielectrics 2015 at NPL

1. Giant Dielectric Constant: All an illusion?
Darryl Almond & Chris Bowen
Materials and Structures (MAST) Centre,
University of Bath, UK
Dielectrics 2015, NPL, Teddington

2. Photoinduced Giant Dielectric Constant in Lead Halide Perovskite
Solar Cells
Emilio J. Juarez-Perez,† Rafael S. Sanchez,† Laura Badia,† GermáGarcia-Belmonte,† Yong Soo Kang,‡
Ivan Mora-Sero,*,† and Juan Bisquert*,†,§
†
Photovoltaics and Optoelectronic Devices Group, Departament de Fisica, Universitat Jaume I, 12071 Castello, Spain
‡
Center for Next Generation Dye-sensitized Solar Cells, Department of Energy Engineering, Hanyang University, Seoul 133-
791,
South Korea
§
Department of Chemistry, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
J. Phys. Chem. Lett. 2014, 5, 2390−2394
Quote: Here we report an outstanding and unique physical behaviour in the
photovoltaic classes of materials, a carrier –induced giant dielectric constant.
The large dielectric relaxation is a central element to understand the physical
processes in the perovskite photovoltaic devices and opens up these materials
for unexpected applications.

3. Photoinduced giant dielectric constant
Juarez-Perez et al. J. Phys. Chem. Lett. 2014, 5, 2390−2394

4. Enhanced permittivity of lead zirconate titanate (PZT)
Almond & Bowen, PRL 2004.
22% porous PZT + water
22% porous PZT
relativepermittivity(ε’)
frequency (Hz)

5. Low frequency permittivity of porous water filled PZT,
(J. Phys. Chem. Lett.)
Colossal Dielectric Constant!

6. ac conductivity lead halide solar cells
Electrode polarisation
Conductivity plateau
Anomalous
high frequency
dispersion

7. Log frequency
Log 
Power law (high frequency dispersions)
Slope n
(w)= dc + Awn
0<n<1
Log frequency
Log ’
0<n<1
Slope (n-1)

8. Power law dispersions found in all classes of materials
Polymers
Concrete & cements
Glasses and single crystals
e.g. Li2O
Ionic conductors Concrete and cements

9. Power law dispersions in all classes of materials
Composites Biological materials

10. Microstructural R-C networks
Al2O3 – TiO2 ceramic composite
10m
RTiO2
CAl2O3
R. Uppal and R.Stevens

11. R-C electrical networks
Example electrical network of randomly positioned resistors
(R) and capacitors (C) characterised using circuit simulation
software.

12. 10
2
10
3
10
4
10
5
10
6
1E-7
1E-6
1E-5
1E-4
1E-3
(b)
(a)
slope -0.6
slope 0.4
Network conductivity
Network capacitance (F)
1E-9
1E-8
Conductivity(S)
Frequency (Hz)
Simulation of (a) ac conductivity and (b) capacitance of a 2D
network of 512 randomly positioned components,
60% 1k resistors and 40% 1nF capacitors.
Power law
frequency
dependences
n = capacitor proportion
= 0.4
n-1 = -0.6
R-C network simulation

13. AC conductivity of 256 2D networks randomly filled with 512
components: 60% 1 k resistors and 40% 1 nF capacitors
Power law (w) wn
Network independent property
Percolation
determined dc
conductivity.
Network dependent
Network type
(%R:%C)
Power law fit, n
60:40 0.399
50:50 0.487
40:60 0.594
Repeat simulations (conductivity)
σnet=Cn(1/R)1-ncos(nπ/2)ωn
Slope n=0.4

14. Repeat simulations (permittivity)
Slope =-0.6
Cnet=Cn(1/R)1-nsin(nπ/2)ω1-n
AC conductivity of 256 2D networks randomly filled with 512
components: 60% 1 k resistors and 40% 1 nF capacitors

15. Frequency range of power law (40% capacitors)
1 10 100 1000 10000 100000 1000000 1E7 1E8 1E9
1E-3
0.01
0.1
1
10
60% R, 40% C
NormalisedConductivity
Frequency (Hz)
1 10 100 1000 10000 100000 1000000 1E7 1E8 1E9
1E-9
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
10
wC
R
-1
ACConductance(ohm
-1
)
Frequency (Hz)
Power law
frequency
R-1 ~ wC
Resistor
conductivity = R-1
frequency independent
Capacitor ac
conductivity = wC
frequency dependent
Slope 0.4

16. Percolation path (60% R, 40% C)
Example electrical network of randomly positioned resistors (R) and
capacitors (C) characterised using circuit simulation software.
60% 1kΩ resistors 40% 1nF capacitors
RESISTIVE PERCOLATION PATH

17. -4
-3
-2
-1
0
1
2
3
-5 -4 -3 -2 -1 0 1 2 3 4 5
log component conductivity, k1
logequivalentconductivity,K
k2 = 1
50% k1 , 50% k2
12 randomised cases
30 x 30 array
Slope = 0.5 line
for reference
Slope = 1 line
for reference
Slope = 1 line
for reference
k2 (blue) constant
k1 (purple variable)
Finite element model

18. Origin of the power law
R-C network conductivity and permittivity related to
component values by logarithmic mixing rule
Lichtenecker’s rule:
*
NET=(iwC)n(1/R)1-n
Network
complex
conductivity
Capacitor
conductivity
(admittance)
Resistor
proportion
Capacitor
proportion
Re. *
NET = Cn(1/R)1-n cos(n/2) wn
AC
conductivity
Resistor
conductivity

19. Network capacitance
Cnet = Im. *
net /iw
Cnet= Cn (1/R)1-n sin(n/2) wn-1
system ~ (ins0)n(cond)1-n cos(n/2) wn
system ~ (ins0)n(cond)1-n sin(n/2) wn-1
Materials & Composites
Capacitor
proportion
n = fraction of insulating phase
1-n = fraction of conductor
Insulating phase: C = εinsε0A/
Conducting phase: R-1=σcondA’/l’

20. Model experimental system
Insulating phase (C): 22 vol.% porous PZT ceramic
  1500
Conducting phase (R): Water
10-1 S m-1
R-1 ~ wC
 ~ w0 less than 1MHz
Two-phase conductor-insulator composite
Conductivities of two phases similar less than 1MHz

21. Component characteristics
10
2
10
3
10
4
10
5
10
6
(a)
(b)
PZT rel. permittivity
100
1000
Rel.Permittivity
1.0
0.1
Water conductivity
ConductivitySm
-1
Frequency (Hz)
Both phases frequency independent

22. System characteristics: high frequency dispersion
10
2
10
3
10
4
10
5
10
6
slope -0.22
(b)
(a)
PZT + water
rel. permittivity
1000
10000
Rel.Permittivity
0.1
0.01
PZT +water conductivity
ConductivitySm
-1
Frequency (Hz)
system =(PZT0)n(water)1-n sin(n/2) wn-1
system = DC +(PZT0)n(water)1-n cos(n/2) wnDC
PZT = 1500
water = 0.135 S m-1
n = 0.78 (PZT %density)

23. Low frequency dispersions
10-1 100 101 102 103 104 105 106
10-3
10-2
10-1
Slope 0.032
a
ACConductivity(S/m)
Frequency (Hz)
10-2 10-1 100 101 102 103 104 105 106
10-6
10-5
10-4
10-3
Slope 0.066
b
ACConductivity(S/m)
Frequency (Hz)
Porous PZT +water Lead Halide Perovskite
Solar Cell (1 Sun)
high frequency dispersion
low frequency dispersion
n > 0 !

24. Kramers – Kronig relationship (A K Jonscher 1978)
ε”(ω)
ε’(ω)
∝ ωn-1
Log ε
Log ω
0 < n < 0.5
n > 0.5 < 1
Low frequency dispersion
High frequency dispersion

25. Comparison of permittivities and dielectric losses
10-1 100 101 102 103 104 105 106
103
104
105
106
107
108
109
a
e'
e"
RelativePermittivity
Frequency (Hz)
n = 0.032 n = 0.066

26. Phase angle: θ = - tan-1(ε’/ε”)
10-1 100 101 102 103 104 105 106
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
a
Phase(degrees)
Frequency (Hz)
10-2 10-1 100 101 102 103 104 105 106
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
b
Phase(degrees)
Frequency (Hz)
θ = -nπ/2
Lead Halide Perovskite
Solar Cell (1 Sun)
n=0.066, θ=5.9°
Porous PZT +water
n=0.032, θ=2.9°

27. Low frequency dispersion
Propose low frequency dispersion is a simple refinement of the accepted
percolation model for DC conductivity.
In addition to the continuous percolation paths of conducting phase there are
numerous incomplete percolation paths in which small quantities (small n) of a
dielectric are present.
These small amounts of insulating phase block DC conduction but act as
microscopic capacitors that enable these paths to contribute to AC conduction.
As frequency is raised the admittances, iωC, of the small capacitive regions
increase, enabling additional percolation paths to contribute to electrical
conduction, resulting in the shallow increase in conductivity with frequency,
known as low frequency dispersion.

28. “1-d channels periodically blocked by
structural imperfections….”
A.K.Jonscher, Phil. Mag., 38, 587-601 (1978)
Porous PZT
Low frequency dispersion

29. Conclusions
Impedance spectroscopy of lead halide perovskite solar cells do not
indicate the fundamental relative permittivity has been massively
enhanced by illumination. No need to find new physical models to
explain such an effect.
Significant electrode blocking effect evident in the conductivity data of
lead halide perovskite is a concern as it indicates that the electrodes
are inefficient.
Generally, reports of exceptionally high measurements of dielectric
permittivity are for materials with high electrical conductivity.

30. Acknowledgment
The research leading to these results has received funding
from the European Research Council under the European
Union's Seventh Framework Programme (FP/2007-2013) /
ERC Grant Agreement no. 320963 on Novel Energy
Materials, Engineering Science and Integrated Systems
(NEMESIS).
http://pubs.acs.org/doi/abs/10.1021/acs.jpclett.5b00620