Capturing dynamic cation hopping in cubic pyrochlores
Beverly Brooks Hinojosa,
1,a)
Aravind Asthagiri,
2
and Juan C. Nino
1
1
Department of Materials Science and Engineering, University of Florida, Gainesville, Florida 32611, USA
2
William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University,
Columbus, Ohio 43210, USA
(Received 14 June 2011; accepted 30 July 2011; published online 24 August 2011)
In direct contrast to recent reports, density functional theory predicts that the most stable structure
of Bi
2
Ti
2
O
7
pyrochlore is a cubic Fd
3m space group by accounting for atomic displacements. The
displaced Bi occupies the 96g(x,x,z) Wyckoff position with six equivalent sites, which create
multiple local minima. Using nudged elastic band method, the transition states of Bi cation
hopping between equivalent minima were investigated and an energy barrier between 0.11 and
0.21 eV was determined. Energy barriers associated with the motion of Bi between equivalent
sites within the 96g Wyckoff position suggest the presence of dielectric relaxation in Bi
2
Ti
2
O
7
.
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2011 American Institute of Physics. [doi:10.1063/1.3630005]
The dielectric relaxation commonly observed in pyro-
chlores is often attributed to the highly polarizable A site cat-
ion, chemical substitution, and large ion displacement from
the ideal Fd
3m (space group No. 227) sites of the cubic
pyrochlore structure.
1,2
However, the necessity and/or suffi-
ciency of these contributing factors have been questioned
due to the observation of dielectric relaxation in the CTN
pyrochlore Ca
1.47
Ti
1.47
Nb
1.03
O
7
without the presence of Bi
or a highly polarizable ion in the A site.
2
Additionally,
uncertainty remains on the influence of substitutional disor-
der on the A
2
O
0
versus B
2
O
6
network on these systems.
3
Though there are several bismuth based cubic pyrochlores
without chemical substitution that exhibit small atomic dis-
placement, such as Bi
2
Rh
2
O
7Ày
,
4
Bi
2
Ru
2
O
7Ày
,
5–8
and
Bi
2
Ir
2
O
7Ày
,
7–9
these compounds are non-stoichiometric as
they are deficient in the A
2
O
0
network. Being the only stoi-
chiometric bismuth based cubic pyrochlore reported to ex-
hibit atomic displacement, bismuth titanate presents an
opportunity to isolate the effects of the polarizable A site cat-
ion, chemical substitution, and atomic displacements.
10
The atomic displacement results in multiple energetically
equivalent sites, which facilitate cation hopping. Based on the
infrared active phonon modes of Bi
1.5
Zn
0.92
Nb
1.5
O
6.92
(BZN),
1
Nino and co-workers proposed that the relaxation
observed in BZN is due to ion hopping mechanisms associ-
ated with the A and O
0
atoms jumping between equivalent
positions.
11,12
Ideally, bismuth titanate would be synthesized
and the resulting dielectric properties would be determined
directly through experimental techniques; however, there has
been considerable difficulty in synthesizing large phase
pure Bi
2
Ti
2
O
7
single crystals and thin films or sintering
dense phase pure polycrystals for dielectric property
measurements.
10,13–15
This limitation may be overcome
through simulations using density functional theory (DFT),
where the stoichiometric phase pure Bi
2
Ti
2
O
7
cubic pyro-
chlore has been studied both with and without atomic
displacements.
16–20
Typically, DFT calculations of bulk mate-
rials are restricted to identification of local minima. While this
information provides some insight on cation hopping behav-
ior, for instance, the identification of multiple local minima
for Bi cations displaced from the ideal Fd
3m site, the calcula-
tions do not directly probe the cation hopping process. How-
ever, the use of DFT in combination with nudged elastic band
(NEB) method
21–23
can identify transition states between the
local minima. Though NEB calculations have not been used
to study this type of mechanism before, they have been instru-
mental in identifying transition states in proton transfer, sur-
face diffusion, and surface chemistry.
24–26
In this work, first-principle calculations using the climb-
ing-image nudged elastic band (cNEB) method were per-
formed with Vienna Ab-initio Simulation Package
(
VASP
).
27–30
The calculation details used previously
17
were
repeated within the cNEB calculations, e.g., cutoff energy,
k-mesh, with the exception of the force criteria which was
raised to 0.03 eV/A
˚
in order to reduce the computational
time while maintaining equivalency of both the energy pro-
file and structure intermediates at higher accuracy. The
cNEB calculations were performed on bismuth titanate to
identify pathways to cation hopping and determine the asso-
ciated energy barriers.
21–23
The presence of energy barriers
would suggest the presence of dielectric relaxation in bis-
muth titanate associated with the ions hopping between
equivalent minima. Since the NEB method depends on pro-
viding two connected minima, several different possible
pathways were examined to identify the most relevant hop-
ping mechanisms. The various minima that serve as input to
the NEB calculations are associated with one of the 16 Bi
cations per unit cell moving within the puckered ring of the
96g(x,x,z) Wyckoff position where x ¼ 0.015 and z ¼ 0.964.
Details of the atomic positions and displacements with
respect to the ideal positions observed in Bi
2
Ti
2
O
7
can be
found in the prior publication by the authors.
17
Recent computational work based on first-principles
(DFT) reported lattice instabilities in bismuth titanate
which lead to a monoclinic Cm or orthorhombic Pna2
1
structure.
18–20
Therefore, the initial step in studying the hop-
ping mechanism within Bi
2
Ti
2
O
7
is to confirm the stability of
the cubic Fd
3m space group with atomic displacement. The C
zone centered phonon frequencies and eigendisplacements
a)
Author to whom correspondence should be addressed. Electronic mail:
beverlyhinojosa@gmail.com. Fax: 352-392-7219.
0003-6951/2011/99(8)/082903/3/$30.00
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2011 American Institute of Physics99, 082903-1
APPLIED PHYSICS LETTERS 99, 082903 (2011)