Ab Initio Screening of Lithium Diffusion Rates
in Transition Metal Oxide Cathodes
for Lithium Ion Batteries
ARCHVES
by
MASSACHUSETTS INSTMfrE
OF TECHNOLOGY
Charles J. Moore
MAYS2113
B. S. Materials Science and Engineering
Northwestern University, 2006
L B A R IE
SUBMITTED TO THE DEPARTMENT OF MATERIALS SCIENCE AND
ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF
MASTERS OF SCIENCE IN MATERIALS SCIENCE AND
ENGINEERING AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SEPTEMBER 2012
@2012 Massachusetts Institute of Technology. All rights reserved.
Signature of Author:
Departmdnt of Materials Science and Engineering
August 9, 2012
Certified by:
Accepted by:
Gerbrand Ceder
Professor of Materials Science and Engineering
Thesis Supervisor
-
Gerbrand Ceder
Professor of Materials Science and Engineering
Chairman, Committee for Graduate Students
Ab Initio Screening of Lithium Diffusion Rates in Transition Metal Oxide
Cathodes for Lithium Ion Batteries
by
Charles J. Moore
Submitted to the Department of Materials Science and
Engineering on August 9th, 2012 in Partial Fulfillment
of the Requirements for the Degree of Master of
Science in Materials Science and Engineering
ABSTRACT
A screening metric for diffusion limitations in lithium ion battery cathodes
is derived using transition state theory and common materials properties.
The metric relies on net activation barrier for lithium diffusion. Several cathode materials are screened using this approach: 3'-LiFePO 4, hexagonal
LiMnBO 3 , monoclinic LiMnBO 3 , Li 3Mn(CO 3 )(PO4), and Li 9V 3 (P 2 0 7 )3 (PO4) 2.
The activation barriers for the materials are determined using a combined
approach. First, an empirical potential model is used to identify the lithium
diffusion topology. Second, density functional theory is used to determine
migration barriers.
The accuracy of the empirical potential diffusion topologies, the density
functional theory migration barriers, and the overall screening metric are
compared against experimental evidence to validate the methodology. The
accuracy of the empirical potential model is also evaluated against the
density functional theory migration barriers.
Thesis Supervisor: Gerbrand Ceder
Title: Professor of Materials Science and Engineering
3
Contents
1
Introduction
2
Background
2.1 Transport theory . . . . . . . . .
2.2 Experimentally measuring rate
2.3 Atomistic methods for diffusion
2.4 Similar studies . . . . . . . . .
2.4.1 Layered materials . . . .
2.4.2 Olivine materials . . . .
2.4.3 Other materials . . . . .
8
8
10
12
16
16
18
19
3
Methodology
3.1 Topology Determination . . . .
3.2 Barrier Evaluation . . . . . . . .
22
23
24
4
Screening Results
4.1 )'-LiFePO4 . . . . . . . . . . . . . . . . . . . .
4.2 Hexagonal LiMnBO 3
. . . . . . . . . . . . .
4.3 Monoclinic LiMnBO 3 . . . . . . . . . . . . .
4.4 LiMn(CO 3)(PO 4 ) .. . . . . . . . . . . . . . .
4.5 LixV 3 (P 2 0 7) 3 (PO4 ) 2 . . . . . . . . . . . . . . .
4.5.1 Evaluation of lithiated structure, x = 9
4.5.2 Evaluation of partially lithiated structure, x = 8 . . . . . .
4.5.3 Comparison to experiments . . . . . . . . . . . . . . . .
26
26
29
31
34
37
37
39
41
5
Discussion
46
46
48
52
6
5.1
5.2
Accuracy of Topology Determination . . . . . . . . .
Acuracy of Barrier Evaluation . . . . . . . . . . . . .
5.3
Factors affecting Li diffusivity . . . . . . . . . . . . .
A Empirical Potential Specification
55
References
57
5
1
Introduction
Cathode choice for Li-ion batteries is an extremely active area of research. While
most currently commercialized cells use one of three classes of materials: layered compounds, Spinel, or olivine, interest in potential new cathode materials
spans a much wider range of materials. Some of the recently discovered potential
cathodes include new phosphates [1], borates [2], and oxy-fluorides [3].
One effort at cathode discovery particularly worthy of mention is the highthroughput computational cathode design project at MIT [4] of which this thesis
was a part. This design project sought to accelerate cathode discovery with computational methods. All known inorganic crystals structures from the Inorganic
Crystal Structure Database were modeled using Density Functional Theory [5].
Novel materials were predicted using structure prediction methods [6]. Potential
Li-ion cathode materials were then screened using a variety of computational predictions: Li-voltage, phase stability, thermal stability, and rate prediction. This
thesis contains the rate prediction aspect.
Rate limitations play an important role in determining the viability of a cathode material and high rate cathodes have received much interest [7]. Indeed, one
of the main factors preventing the widespread commercialization of electric vehicles is the limited range-between charges; doubtless, the range-between charges
would be much less of an issue if charging times were under 5 minutes. Realistically, the choice of cathode material for commercial batteries is partially deter-
6
mined by matching the rate-capability of the cathode with the required rate for
the application. High power applications, such as power tools and hybrid electric
vehicles tend to skew towards higher power cathodes such as olivine and spinel
cathodes while low power applications such as portable electronics and fully electric vehicles tend to skew towards layered cathode materials due to the higher
energy density.
Rate limitations in cathode materials may stem from a variety of factors including polaron-based electrical conductivity [8], surface transport [7] and bulk,
solid-state Li-diffusion. This thesis focused soley only on bulk, solid-state Lidiffusion. Since cycling an intercalation cathode involves inserting (and removing) Li ions directly into (from) the crystal unitcell, the Li ions must be able to
migrate to (egress from) each active Li site. Short-circuit diffusion processes such
as grain boundary and surface diffusion do not access the entire active range of
the material and are not sufficient for cycling. Ample solid-state bulk lithium diffusion may be considered a necessary, but not sufficient criteria for a high rate
cathode.
There has recently been a surge in the use of atomistic modeling to try and
evaluate Li-diffusion in electrodes for Li-ion batteries (Section 2.4). This thesis further develops an automated methodology for performing such screenings
by using emirical potentials to first identify the Li+ diffusion topology and then
refining migration barriers with ab initio calculations. The method is applied to
several novel cathode materials and the validity of the the results is used to evalu7
ated the validity of the underlying screening methodology.
2
2.1
Background
Transport theory
The kinetic theory of transport helps to link the inherent diffusivity of Li in a cathode to the rate at which it may be cycled. The solution to the diffusion equation
in one dimension for the diffusivity of Li into a completely delithiated cathode
reveals the following relation:
c(x, t) = co erfc j.......
where c(x, t) is the concentration of Li at a depth of x at a time t, co is the concentration of Li at the surface, and D is the diffusivity of Lithium in the cathode [9].
Given that primary particle size may be as small as 100nm, significant solid state
diffusion is required across at least 50nm during charging and discharging. In order for the concentration at 50nm to reach 80% of the concentration at the surface
2
is required. This is a reasonable
within 2 hours, a diffusivity of 2.7 - 10-14cm
Sec
benchmark for the required diffusivity.
When Li diffusion consists of a series of discrete hops, the diffusivity of Li is
8
related to the hops by the relation:
DLu
=
F (r 2 )ffd
2d
where F is the total hop frequency, (r 2 ) is the mean squared distance Li travels per
hop, d is the dimensionality of the system (ID, 2D or 3D), and
f is a correlation
factor [9]. The correlation factor f is between 0 and 1 and serves to suppress the
diffusivity when the atoms tend to hop back and forth with no net motion. To first
order, the hop frequency for any given hop F' is
=
-AEm/kT
where v is a characteristic frequency, and AEm is the migration barrier [9]. Variance in DLu is dominated by variations in the activation barrier. The exponential
dependance produces a ten fold reduction diffusivity for every 59 meV increase
in migration barrier. As migration barriers may vary by over 500 meV between
different materials, differences in migration barriers may account for nine orders
of magnitude of differences in diffusivities!
With some typical values, it is possible to estimate an upper limit on activation
barrier. A typical value for the attempt frequency is v = 10-1 s- 1 [10-12]. Hop
distance is on the order of atomic lengths, take 3A as a typical value. To obtain a
diffusivity of 2.7. 10-4 csec with these typical values requires a migration barrier
of less than 635 meV. Due to the steep dependance of DLi on activation barrier, it
9
is possible to broadly classify materials with migration barriers less than 450 meV
as facile bulk Li diffusers and materials with migration barriers above 700 meV as
insufficient Li diffusers for Li-ion battery cathodes. Materials with intermediate
activation barriers have indeterminate diffusivity.
2.2
Experimentally measuring rate
A variety of experimental techniques are used to evaluate the rate capabilities of
cathode materials. The majority of these techniques test the material as an electrode in a electrochemical cell, generally a half-cell with a Li-metal anode. These
techniques benefit from both their ease and technological relevance. Galvanostatic cycling at a variety of rates is arguably the most technologically relevant test,
because it directly measures the performance of a cathode as it is to be used. It is
also the most ubiquitous. However, it provides little to distinguish between different potential rate limiting mechanisms. Electrochemical impedance spectroscopy
(EIS) provides time domain separation of rate-limiting processes, but still requires
an equivalent circuit. The physical origins of different elements in this equivalent
circuit are not always entirely clear. Similarly, other techniques such as galvanostatic intermittent titration technique (GITT) [13] and cyclic voltametry (CV) [14]
suffer from the need for knowledge about underlying mechanisms and physical
parameters such as electrode area, and diffusion isotropy/anisotropy before a diffusion constant can be backed out.
10
The various methods and assumptions can produce a wide range of measured
diffusivities for the same system. Within a single study of a single sample, measurements of the diffusivity of Li.CoO 2 differed by over 2 orders of magnitude
depending on the technique used [15]. Comparing different studies of LiCoO 2 ,
2
produced diffusion constants ranging from 10-8csec to 10-15 CM
[16]. Perhaps
sec
not surprisingly, it is extremely difficult to obtain reliable quantitative measurements on the fundamental property of bulk lithium diffusivity from experimental
techniques evaluating a system as complex as a complete electrochemical cell.
Measurements of activation barriers from Arrhenius type plots are perhaps
more quantitatively reliable than diffusivities, although identification of the ratelimiting process (surface vs. bulk and electronic vs. ionic) continues to be problematic.
By using the material as a membrane or solid electrolyte through which Li
must diffuse rather than as an electrode in an electrochemical system allows isolation of Li diffusivity. Unfortunately, to date only a few systems have been studied
in this manner. Single crystal samples of LiFePO4 with LiI electronically blocking
electrodes have been tested [17], as have highly oriented graphitic carbon membranes [18]. While these experiments provide valuable results, sample preparation may be difficult or impossible to perform similar studies on other systems,
especially if one wants to distinguish between diffusivity along different crystallographic orientations or evaluate bulk diffusion as opposed to grain boundary
diffusion.
11
2.3
Atomistic methods for diffusion
In contrast to the experimental methods, atomistic methods are extremely specific. The knowledge of the kinetic mechanism being measured is inherent to the
simulation. They do however sometimes lack relevance. It is entirely possible
to evaluate the bulk diffusivity of a material which in practice is limited by electronic conduction, or surface transport. This is both a virtue and a drawback. As
a drawback, it makes the total evaluation of a potential cathode incomplete; high
Li+diffusivity is not sufficient for hight rate capability. As a virtue, it is possible to
study mechanisms which are not rate limiting, allowing study of bulk diffusivity
even when other processes limit the material. It also allows direct investigation
of high energy pathways when lower energy pathways exist. Such information is
extremely difficult to obtain from experimental results.
At the heart of atomistic modeling is the energy evaluation of a given structural
configuration via a Hamiltonian. The optimization of this total energy with respect
to the structural degrees of freedom, and determining the shape of the energy
landscape through which ions must traverse, is the main goal of these atomistic
methods. Two broad groups of Hamiltonians are relevant to this work: those based
on empirical inter-atomic potentials, and those based on density functional theory.
Also relevant is the optimization method of nudged elastic band for determining
the energy at activated sites.
The Buckingham empirical-potential model consists of a repulsive pairwise
12
potential energy function between oppositely charged ions combined with a Coulombic sum over interactions between formal charges assigned to each ion [19]. Together, they serve as a widely used Hamiltonian for ionic systems. The main benefit of such potentials, is the computational ease with which they may be evaluated.
The two main drawbacks are: (1) lack of accuracy (2) lack of generality.
Many empirical-potentials are able to accurately reproduce equilibrium and
near-equilibrium properties such as lattice parameters, phonon modes, and elastic properties. However, their accuracy for activated states relevant to Li migration is less clear. Migration studies in LiFeSO 4 have shown a large discrepancy
between empirical-potential methods and density functional theory predictions.
Results from DFT predict barriers of 208meV for 1D diffusion and 976meV for
3D diffusion [3]. Two studies with different empirical potentials have returned
very different activation barriers. One study [20] show a migration barrier of 360
meV for 1D diffusion, and 440 meV for 3D diffusion, while another [21] obtained
220 meV and 1101 meV respectively. Similar discrepancies have been shown in
olivine-type materials (Table 1).
Furthermore, established libraries of empirical potentials for inorganic materials tend to include only a few dozen elements at most. The library due to Bush
contains only 19 metals + oxygen [19]. The similar Bond-Valence derived Morse
potential library due to Adams et al. [21,24] is much broader based, and was derived explicitly for determining conduction topologies. However, it still remains
dependent on the existence of BondValence parameters, and formal valences.
13
Table 1: Comparison of published DFT and empirical-potential migration barriers
in olivine type materials LiMPO4 . All migration barriers are for [010] direction
and expressed in meV.
Cation M
Mn
Fe
Co
Ni
DFT [22]
250
270
360
130
Empirical [23]
620
550
490
440
Despite these shortcomings, empirical potential methods have proven very effective at determining the topology for lithium diffusion. In the case of LiFeSO4
[3,20,21] and the Olivine materials [22,23], the empirical potential methods and
density functional theory are in concurrence on the migration topology, and at
identifying the lowest energy path. Expanded validation of empirical potentials
to predict Li+ diffusion topology is one of the goals of this study. The results are
discussed in Section 5.1.
Density Functional Theory (DFT) methods are more computationally expensive. However, pseudopotentials are available for the majority of the the periodic
table through commercial software packages. By tracking the spin polarized density throughout the material, rather than assigning formal charges to ions, the
method can handle materials with non-integer valence states and different magnetic configurations. The success of DFT at explaining Li diffusion in the studied materials is best understood by considering materials on a case-by-case basis
(Section 2.4). One of the primary aims of this study is to further validate DFT for
14
evaluating Li+ diffusion. The results are discussed in Section 5.2.
One common extension to DFT is the +U extension. In addition to the standard DFT Hamiltonion, the +U method adds a penalty for delocalized electronic
states [25]. This method is frequently used to force electrons to localize on transition metal ions in oxide calculations. In structures with mixed valence ions,
standard DFT (no +U) typically delocalizes the electrons, resulting in many ions
with identical (or similar) non-integer valences. With the +U self interaction correction, charge ordering at integer valences may be induced.
Given any of these Hamiltonians it is possible to calculate the energy of a
given atomic configuration. Identification of the minimum energy path between
two locally stable atomic configurations can be found using the Nudged Elastic
Band method (NEB) [26, 27]. In the NEB method, a series of images of the
structure are evaluated. Taken in order, the images represent a pathway from
one local minimum to the other. Rather than optimizing the energy of a single
image, the total set of structures is optimized together. The atomic positions of the
intermediate images are constrained such that the images are equally spaced along
the reaction pathway. As the number of images increases, the images converge
to the minimum energy pathway [28]. The NEB method is amongst the most
efficient algorithms available to find the minimum energy path [29]. It's principle
limitation is that both the initial and final states must be known before calculation
can commence.
15
2.4
Similar studies
This thesis is not the first attempt at computationally screening cathode materials
for lithium diffusion. Two classes of cathodes: layered (Section 2.4.1) and olivine
(Section 2.4.2) have been studied in depth. While not studied as in depth as the
first two classes, a few other materials have been screened for diffusivity (Section
2.4.3). The prior studies on Li diffusion have been piecemeal, considering only
a single structural prototype at a time. Most of them, simply calculate activation
barriers, with no attempt at understanding why the barriers are such and little attempt is made at validating the underlying methodology. There is an obvious need
to unify calculations across structural prototypes and validate the computational
methodology.
2.4.1
Layered materials
The first cathode material to be computationally screened for Li diffusion is layered LiCoO 2 [30,31]. The in-plane Li migration barriers were calculated using
DFT and the Li migration barriers were parameterized as a function of neighboring Li-occupation. A kinetic Monte Carlo (KMC) algorithm was used to simulate
diffusion of Li throughout the material and extract the Li diffusivity. Two important results are noteworthy: (1) Li diffusion was dominated by a divacancy
mechanism at high Li-content. (2) the relevant migration barrier varied by nearly
400 meV depending on Li concentration. These two factors combined to cause the
16
overall diffusivity to vary between 10-10 and 10-6
£*.
sec
The maximum diffusivity
occurs near the composition Lio.7CoO 2 . Similar results were found for layered
LiTiS 2 [32].
While these two factors are very important for understanding the diffusivity in
layered compounds, they are also fairly unique. The importance of the divacancy
mechanism can be explained by the unusually close proximity of Li pathways to
adjacent Li sites. Adjacent sites must all be vacant, or the Li-Li interaction effectively blocks the pathway. The necessity of a divacancy suppresses the diffusivity
at high Li concentrations. The change in migration barriers with Li concentration
can be traced to the large change in c-lattice constant associated with removing Li.
Layered structures are unique in that delithiation results in removal of an entire
plane of atoms, causing a drastic contraction in the layer spacing.
Kang et al. [33] studied a variety of layered compounds and identified four factors which influence the migration barriers. First, the Li slab distance, defined as
the distance between oxygen layers surrounding the lithium layer, has a large effect on the migration barriers. This distance is known to vary as occupancy in the
Li layer changes. The same effect has been found in layered TiS 2 [32]. Second,
the migration barriers increase as the valence on the transition metal increases due
to stronger Coulombic interaction. As Li is extracted, the average valence of the
transition metal species increases. The third factor is the choice of transition metal
(LiCoO 2 vs LiTiO 2 , etc.). It is argued that late transition metals allow more elec-
tron density to sit on the oxygen atoms, and therefore provide better Coulombic
17
screening between the Li and the transition metal, lowering the activation barrier.
Although the effect is small compared to the other effects. Finally, anion choice
has a large impact. Comparisons between layered oxides and sulfides indicate that
the sulfides have significantly lower ( 200meV) migration barriers. They attribute
this to the larger size of Sulfide ions and the opening of the Li slab spacing. And
improved Coulomb screening by Sulfur ions.
2.4.2
Olivine materials
As mentioned in section 2.3, the olivine class of materials, principally LiFePO 4 ,
has been studied with both DFT [22, 34, 35] and empirical potentials [21, 36].
While significant differences were found with different transition metals, no study
has identified the root cause of these differences.
Most of the work has focused on the commercialized LiFePO 4 . As in the
layered cases, migration barriers are higher at lower Li content. Initially, experimental results showing 3D diffusion conflicted with predictions of extremely fast
ID Li diffusion. This can be explained by the presence of anti-site defects blocking the one dimensional diffusion channels leading to a size-dependant diffusivity
and cross links between the chanenls [37]. This is in agreement with the experimentally observed ultrafast charging at small length scales [7]. Although, ultrafast
rates require a surface coating, presumably to aid in transport of Li into and out
of the diffusion channels. This theory has been supported by recent atomistic
18
simulations [21]. In addition, the material is also known have limited electrical
conductivity [8] which in some cases may limit the rate.
Ultimately, the olivine materials are a good demonstration that atomistic diffusion prediction adds value by helping to understand the root cause of limited
diffusivity. The importance of channel blocking defects, surface coating, and electrical conductivity to the overall rate of the material are crucial to designing a high
rate olivine. Without atomistic modeling, the material may have been dismissed
as inherently low-rate. It remains unclear how frequently other cathode materials are limited by anti-site blocking, surface diffusivity, and/or limited electrical
conductivity.
2.4.3
Other materials
Recently, interest has been shown in the Tavorite-like compounds (e.g. LiFeSO 4 )
[3, 20, 21, 34]. As detailed in section 2.3 there is a discrepancy between some
empirical potential studies and those using DFT. The DFT NEB results predict
very low activation barriers in one dimension. This has yet to be demonstrated
experimentally. More work is necessary to fully understand this material.
The novel cathode Li 2 NiO 2 [7] has been studied in a similar manner. A lowbarrier pathway was identified with DFT. This agrees well with the ability of the
material to be electrochemically cycled.
Two spinel classes: Li-Mn-C [38, 39] cathodes and Li-Ti-O anodes [2,40]
19
have been studied. First, evident from the Ti cases is the importance of Li site
shifting. Li occupies different sites at different compositions. When x in the
formula Lii+xTi2 0 4 is positive, Li occupies octahedral sites. When x is negative,
Li occupies tetrahedra sites. In Li 4 +xTi5 012, some of the Ti sites are filled with Li.
This has a profound impact on the diffusion pathway and the migration barriers.
Site shifting did not show up in the Mn study, because of the limited composition
considered. It is also worth noting that true activation barriers were not calculated
for the Mn case, rather only the difference in site energies of the two potential Li
sites. There is a strong dependance with charge ordering on the Mn sites [38],
but it is unclear how closely the site energy differences calculated represent true
migration barriers.
A study on anatase TiO 2 [41] offers a rare opportunity to compare GGA and
GGA+U functionals and charge ordering effects on migration barriers. The primary expected effect of changing functionals on migration barriers is that the +U
addition localizes charges on the transition metal sites. Most of the relevant NEB
calculations are defect calculations on either a Li+ vacancy, or a single Li+ ion.
To balance the charge, a corresponding electron or hole is also present. In GGA,
this electron/hole tends to be distributed across the material. In GGA+U it tends
to localize on a single transition metal ion.
In TiO 2 [41], activation barriers were calculated with GGA, several charge
orderings with GGA+U, and a charged computational cell with GGA+U (so no
charge ordering is necessary). The spread between different GGA+U charge or-
20
derings was 98 meV, with the lowest and most relevant barrier at 588 meV. The
GGA result was 511 meV. An amazingly similar comparison can be found in
the studies of LiFePO 4 olivine. With GGA [22], the vacancy and Li barriers are
200meV and 250meV respectively. With GGA+U [35], the minimum vacancy
and Li barriers are 190meV and 290meV respectively. Charge ordering effects
were again on the order of 100meV.
At first glance, it may be surprising that while the +U method introduces
charge ordering effects on the order of 100meV, but that the difference in the
net activation barrier was at most 77 meV, and as small as 10 meV. Upon further inspection it is less surprising. Migration barriers are determined by the
difference in energy between the endpoint configuration and the transition state
configuration. Finding the net activation barrier requires a global optimization of
the endpoint over both Li position and charge ordering configurations. Similarly,
finding the net migration barrier requires finding the lowest energy transition state
which allows net Li+ migration. Due to the increase proximity of the diffusing
Li+ and nearby transition metals, the energetic effects are stronger for transition
states than for the endpoints. As long as the energy barrier for charge reordering
is less than the energy barrier for Li+ migration, finding the net migration barrier requires optimizing both the endpoint Li configuration and the Li+ migration
transition state across all charge orderings. Running in pure GGA (without +U)
to some degree allows the electronic configuration to perform this optimization
automatically, whereas the localized charges in GGA+U can easily find the wrong
21
configuration.
Lithium diffusion in Wurtzite-type Li2 FeSiO4 cathodes has been studied via
DFT [42]. Many potential pathways were examined, no pathway with a migration barrier of less than 700meV was found. This may explain why experimental
performance has been somewhat sluggish.
Li diffusion in graphitic anodes has been studied [18]. A careful experimental
diffusivity measurement was performed, treating the graphite as a Li+ conducting
membrane in a in a Devanathan-Stachurski-type cell. A diffusivity of 4.4.10-6
cm2
sec
was measured. Simultaneously,. diffusivity was predicted with DFT activation
barriers and KMC to be in the range of 10~7 to 10-8m.
This is good agreement
sec
considering the ambiguity of the vibrational prefactor needed in KMC.
3
Methodology
Screening of cathode materials was performed in two steps. First, an empirical potential was used to calculate the diffusion topology. Detection of diffusion topologies is coupled with a symmetry analysis of the structure so that symmetrically
distinct calculations are only run once. Once the diffusion topology was known,
potential Li diffusion pathways were formed. A diffusion pathway is a series of
links between Li sites (Li hops) which allows migration of either a Li+ vacancy or
Li+ interstitial to traverse the entire unit-cell. The pathways were sorted based on
22
the empirical-potential barrier energy. The second step was an accurate evaluation
of the barrier energies with DFT and the NEB method.
3.1
Topology Determination
In early works [22,30] pathways for Li diffusion were found through some combination of manual examination and exhaustive testing with DFT. When studying
a large number of structural prototypes and large unitcell structures, this process
is not only wasteful of computational time, but may quickly become unreliable.
Initial studies benefited from structures with high symmetry and simpler topologies than some of the structures of interest. As discussed in Section 4.5, some
diffusion pathways are non-intuitive.
The diffusion topology was detected using an empirical potential model. The
empirical potential model consists of a repulsive Li-Oxygen pair potential and
screened Coulombic electrostatics. All atomic positions are fixed and any Li in
the structure is removed. A potential energy surface is then calculated for Li by
laying a grid across the the entire unitcell and calculating the total energy of the
cell with a Li+ ion at each grid point. After obtaining the potential energy surface,
it is searched for Li sites (local minima) and hops between sites (minimum energy
paths between minima). The resulting network represents a diffusion topology.
This method is similar to a method recently published [21]. The specifics of the
empirical potential are specified in Appendix A.
23
3.2
Barrier Evaluation
To evaluate the migration barrier energies, NEB inputs were generated from the
diffusion topology. The optimal setting for running the calculation was identified by enumerating superlattices and selecting the smallest and most orthogonal
unitcell where the distance between periodic images was above 9 Angstroms. In
practice, this results in supercell of 60 to 110 atoms.
The first step in an NEB calculation is relaxing the endpoints with DFT. The
initial atomic positions for the intermediate images were obtained by taking the
minimum energy pathway from the empirical potential topology and shifting the
atomic positions by an interpolation of the atomic shifts which occurred during
the endpoint relaxation. These smart initial conditions greatly reduced the computational burden of the NEB runs.
Great effort was taken in finding optimal DFT parameters for running these
jobs. The goal was to minimize the computational cost while maximizing robustness, accuracy, and precision. Several complications existed: (1) NEB calculations must be much more strictly converged than calculations for stability and
voltage screening [5]. To obtain the precision within one order of magnitude of the
diffusivity, the NEB calculations must be converged to within 59 meV per unitcell.
This is much stricter than for voltage calculations where an error of 50 meV per
Li is only 0.050 volts. (2) All calculations were run ferromagnetic without the +U
extension. This was chosen primarily to avoid the complexity of charge ordering,
24
but is not expected to dramatically effect the results as discussed in Section 2.4.3.
All ab initio calculations were performed with the Vienna Ab Initio Simulation
Package (VASP) [43,44]. The NEB relaxations were optimized with the transition
state tools extension to VASP [45]. The PBE GGA functional was used [46]. To
obtain the precision of 50meV per unitcell, the break criteria for the ionic relaxation of endpoints was very strict at 5 - 10IeV. The break criteria for electronic
optimizations was 5 - 10- 6 eV. The NEB calculations were considered converged
when the barrier energy had stabilized and the calculated atomic forces were consistent with the image energies in the NEB profile. The atomic forces were nominally converged to around 0.05eV/Angs. If the atomic forces were not consistent
with the NEB profile, the number of intermediate images was increased and the
relaxation continued. To expedite the NEB calculation and increase reliability, the
total spin of intermediate images was fixed at the value of the converged total spin
obtained during the endpoint relaxations. The final magnetic states were checked
to ensure that similar magnetic states were obtained for each image. The lattice
parameters for both the endpoint and intermediate images were fixed at values
obtained by relaxing the undefected bulk with the GGA+U method. The bulk
relaxation was performed with the "accurate parameters" as specified by Jain et
al [5].
25
4
Screening Results
The screening methodology was applied to a series of structural prototypes. Lithium
vacancy migration barriers were calculated for lithium containing compositions
and lithium interstitial migration barriers were calculated for compositions not
containing Li. The net migration barrier for all structures was compared against
the metric developed for diffusion Cathode materials in Section 2.1, namely that
structures with net barriers greater than 700meV are diffusion limited, structures
with barriers less than 450meV have facile diffusion, and structures with intermediate barriers are indeterminate.
4.1
3'-LixFePO 4
In addition to the olivine form (discussed in Section 2.4.2), LiFePO 4 also exhibits
a high pressure phase, 3'-LiFePO 4 . The crystal structure of 3'-LiFePO 4 is shown
in figure 1. The structure consists of edge sharing columns of FeO 6 octahedra
extending along the b-axis. The empirical potentials indicated a Li+ diffusion
topology consisting of two types of hops. The first hop is primarily in the c-b
plane. These hops connect the central ion in figure 1 (labeled c) to four of the
neighboring Li ions (labeled a). These hops form a 2D network in the c-b plane.
The second hop is primarily in the a-b plane. Each Li ion is connected to two
of it's neighbors (labeled b) by these types of hops. The second type of hop
forms a 1D zig-zag diffusion network between and parallel to the FeO6 octahedra.
26
Both hops taken together form a 3D diffusion network. Both types of hops pass
through a distorted oxygen quadrilateral where two sides are formed by edges of
the FeO octahedra and two sides are formed by the edges of P0 4 polyhedra. Sides
consisting of FeO edges oppose one another, as do the remaining P0 4 edges.
The DFT refined migration barriers are shown in Table 2. In all cases, the
migration barriers were over 950 meV indicating severe diffusion limitations. The
calculated barriers are in agreement with electrochemical results which show no
electrochemical activity [47].
27
FeO*
4
Figure 1: Crystal structure of 3'-LiFePO 4 . Only the structure local to a single Li
ion (labeled c) is shown. The six neighboring Li ions are labeled a and b according
to their relationship to the central Li ion.
Table 2: Calculated DFT migration barrier energies in I'-LixFePO 4 [meV].
2D network ID channel
#'-LiFePO 4
0'-FePO4
1004
961
28
1239
1303
4.2
Hexagonal LiMnB03
Two known phases exist for the compound LiMnB03: hexagonal and monoclinic.
The diffusion pathway detected from empirical potentials for the heaxagonal form
was primarily one dimensional, parallel to the c-axis. The pathway consists of
a single hop between adjacent Li sites (Figure 2). This hop occurs when Li+
migrates across one of the faces of the equilibrium LiO 4 tetrahedral site into an
adjacent empty square pyramidal void. The hop concludes with another face hop
into the final tetrahedral site.
The calculated DFT migration barrier energies are given in (Table 3). With
a net ID migration barrier of 724meV, the lithiated form may be classified as a
diffusion limited material. While the delithiated form may be classifies as indeterminate with a barrier of 529meV, the material is typically synthesized at the
lithiated composition. Therefore, the material is expected to be severely diffusion
limited and not practical as a battery cathode. Diffusion within the a-b plane was
also investigated, but the migration barrier was over 1.5eV indicating no substantial diffusion.
Hexagonal LiMnBO 3 has been electrochemically tested as a potential battery
cathode but no substantial capacity was found [48,49]. The experimental results
are consistent with the prediction that this material will be diffusion limited.
29
L10
4
Figure 2: Crystal structure of hexagonal LiMnBO 3 and low energy Li migration
pathway (gold). Red spheres on the diffusion pathway indicate Li location from
intermediate NEB images.
Table 3: Calculated DFT migration barrier energies in Hexagonal LiMnBO3
[meVI.
I ID channel
LiMnBO 3
MnBO 3
724
529
30
inter channel
1569
4.3
Monoclinic LiMnBO 3
In the calculated stable structure, the Li ions reside on the same side of LiO 5
trigonal-bipyramidal sites. These Li bipyramids form edge sharing columns as
depicted in Figure 3. The diffusion pathway detected from the empirical potentials was one dimensional, parallel to the c-axis. Traversing the unitcell requires
passing through four Li sites. Due to symmetry, only half of the pathway is distinct. The two symmetrically irreducible hops are depicted in Figure 3. In hop A,
the ion first migrates from the bottom to the top of the starting bipyramidal site.
It then jumps directly across the shared edge to the adjacent bipyramid. In hop B,
again, the ion first migrates from the bottom to the top of the starting bipyramid.
It then jumps across one face of the bipyramid into an adjacent tetrahedral site
before hoping across the face of the final bipyramid.
The calculated DFT migration barrier energies are given in (Table 4). The
lithiated and delithiated 1D net migration barriers are 510 meV and 396 meV respectively. In both cases, the limiting step is the edge jump in hop A. The partially
lithiated structure shows a shifting of Li location. The trigonal bipyramids are vacated, and the Li instead resides in the tetrahedral site in the middle of hop B. The
net migration barrier is only 311 meV, the limiting step remains hop A.
This material is predicted to be a facile diffuser at lithium concentrations concentrations below one half. Above one half lithiation, the diffusivity is indeterminate. Inter channel diffusion was also investigated, but the migration barrier were
31
b)
c)
B
C
Ja
Figure 3: Calculated lithium locations and diffusion pathway in monoclinic
LiMnBO 3 . The edge sharing LiO5 bipyramids are shown as (a) opaque, (b)
translucent, and (c) omitted. The calculated Li+ migration pathway is shown
in gold. The four short arrows in the first subplot denote the tetrahedral Li site
through which hop B passes.
Table 4: Calculated DFT migration barrier energies in Hexagonal Li.MnBO 3
[meV].
LiMnBO 3
LiO. 5MnBO 3
MnB0 3
hop B
317
50
240
hop A
510
311
396
32
inter channel
1367
2116
over 1.3eV indicating no substantial inter channel diffusion.
Jae Chul Kim et al. [50] have electrochemically cycled monoclinic LiMnBO3
with up to 1 lOmAh/g capacity (50% theoretical capacity). When the material
was cycled at elevated temperatures, the accessible capacity increased markably.
While it is not entirely clear if this is due to thermal activation of diffusion, the
intermediate barrier energy of the lithiated state is consistent with this interpretation. A subsequent study [12] suggest anti-site blockages of the Li channels may
also be present to further hinder Li diffusion similar to the olivine LiFePO 4 (discussed in Section 2.4.2). It is unclear if the capacity limitation is due to anti-site
defects, general diffusion limitation, or other factors.
33
4.4
LiMn(CO3 )(PO 4 )
An new class of carbonophosphate cathode materias of the composition Li3 M(CO 3 )(PO 4 ),
M=Fe,Mn,Ni,Co. [51] has recently been synthesized. Diffusion screening of the
triclinic Mn form is presented here.
The empirical potential diffusion topology has low-energy one dimensional
channels and intermediate barrier energy crosslinks between these channels. The
low energy channels were investigated in depth. One such channel is shown in figure 4. The channel encompases four distinct hops. However, in Li2 Mn(CO 3 )(PO 4 ),
and LiMn(CO3 )(PO 4 ), one of the Li sites (labeled C in figure 4) is vacant and hops
1 and 2 combine into a single hop.
The calculated activation energies are tabulated in Table 5. The net activation
energy was determined by allowing the Li vacancy to choose the lower of either
hop #1 and hop #2 or hop #3 to transverse the cell. In all cases, hop #4 was required. With net migration barriers of 380 meV and 396 meV respectively, the
Li3 and Lii lithiations are expected to have facile Li diffusion. The intermediate lithiation, with a net migration barrier of 478 meV, is on the low end of the
indeterminate range.
Electrochemical cycling of the material has been shown [51]. The achieved
experimental discharge capacity was 58% of the theoretical capacity indicating
Li+ diffusion across the composition range 2 < x < 3 for x in LiMn(CO3 )(PO 4 ).
It is unknown why the full theoretical capacity was not achieved. The rate of
34
MnOa
P0 4
A
Cos
4
B
2
C
1
A
a-- C
b
Figure 4: Low energy diffusion pathway in LiMn(CO3 )(PO4 ). The diffusion
pathway is shown within the unitcell (left) and in schematic form (right). The
numbering of hops and lettering of Li sites, indicates symmetrically distinct portions of the pathway. The three Li sites through which the illustrated diffusion
pathway does not pass are part of a symmetrically identical pathway.
Table 5: Calculated DFT migration barrier energies in LiMn(CO 3 ) 4 SO 4 . The
"net" column lists the net activation energy required for bulk Li-diffusion. All
values are in meV.
Hop 1 Hop 2
Formula
275
Li 3Mn(CO 3 )(PO4 ) 396
657
Li 2 Mn(CO 3 )(PO4 )
380
LiMn(CO 3 )(P0 4 )
35
Hop 3
721
478
504
Hop 4
277
173
350
Net
396
478
380
cycling was kept relatively low at C/100 and ball milling was required to achieve
this capacity. It is unclear if diffusion is fast enough in this material to allow
higher rates of cycling. Furthermore, the tested samples had a large degree of
off-stoicheometry. Approximately 19% of the Li was replaced with Na, due to
incomplete ion exchange. Regardless, the prediction of facile to indeterminate
diffusion agrees well with the ability of the material to electrochemically cycle at
a low rate.
36
4.5
LixV 3 (P 2 0 7 )3 (PO4 ) 2
4.5.1
Evaluation of lithiated structure, x = 9
The crystal structure of Li 9 V 3 (P 2 0 7 ) 3 (PO 4 ) 2 contains Li-layers and large c-axis
channels that contain only Li [52].
Three distinct Li sites exist: Li(l), Li(2),
and Li(3). The empirical potentials predict 2D Li diffusion within the Li-layers.
A topological depiction of the predicted diffusion network is shown in Figure 5.
The main features are rings of Li(3) sites with an Li(1) site in the center. The rings
are connected to one another via Li(2) sites. In addition to the predicted diffusion
network, we also investigated diffusion of Li down the large c-axis channels which
connect the Li(1) sites to each other along the c-axis. The migration barrier down
the c-axis channel was 740 meV indicating diffusion limitations. The calculated
migration barriers are sumized in Table 6.
An alternate to the c-axis channels, is diffusion of Li across the Li layer. However, endpoint relaxations indicate that Li vacancies are not stable at the Li(3)
sites. The Li in the adjacent Li(1) relaxes into empty Li(3) with no activation bar-
rier. This restricts the number of potential diffusion paths. One plausible diffusion
path within the Li layer is a simultaneous Li(l) -+ Li(3) and Li(3) -+ Li(2) migration of the vacancy so that the vacancy does not rest on an Li(3) site. The total
activation energy of this transition was extremely large at nearly 1.3 eV. Much of
this net barrier can be traced to the fact that a vacancy in the Li(2) site is thermodynamically much higher in energy than a vacancy in the Li(1) site. The difference
37
Figure 5: Schematic of in-layer Li diffusion topology for Li9 V 3 (P 2 0 7 ) 3 (PO4 ) 2 .
The c-axis is normal to the plane of the diagram. The numbers 1, 2, and 3 represent
the Li(1), Li(2), and Li(3) sites, respectively. Connections between sites are drawn
with tapered lines to indicate relative displacement of the Li sites above/below the
Li layer. Sites and transitions are colored according to symmetry
barriers for Li vacancy diffusion in
Calculated migration
Table 6:
LixV 3 (P 2 0 7 )3 (PO4 ) 2 . The net activation energy is the energy difference between
the activated state and the lowest energy Li-vacany configuration at the given composition.
Transition
Nominal
composition
x = 9
x =
Li(l) -+ Li(l)
Li(l) -+ Li(3)
Li(3)
->
Li(2)
Li(2)
-+
Li(3)
8
Li(3)
->
Li(3)
between layers
i
.
5 simultaneous
Migration
barrier (meV)
Net activation
energy (meV)
740
320 (forward)
740
1291
1291 (reverse)
361 (forward)
518
518 (reverse)
between layers
inside
outside
1534
308
588
1691
465
745
s-route
489
646
38
in site energies was calculated to be 978meV.
A third potential path for Li(1) vacancy diffusion is a simultaneous hop involving 4 Li ions: Li(l) -> Li(3), Li(3) -+ Li(2), Li(2) -> Li(3), and Li(3) --
Li(1). While it appears unlikely that four Li ions are migrating simultaneously,
the balance between entering and departing Li from the high-energy Li(1) site
may produce a lower net migration barrier. This complex transition has not been
investigated.
4.5.2
Evaluation of partially lithiated structure, x
=
8
Migration barriers in Li 8 V3 (P20 7) 3 (PO 4 ) 2 were also calculated. Due to the large
site energy preference, the structure of Li8 V3 (P2 0 7 ) 3(PO 4 ) 2 was assumed to be
the same as Li9 V3 (P2 0 7 ) 3 (PO 4 ) 2, but with all Li(1) sites vacant.
Several diffusion paths were investigated for the migration of an Li(2) vacancy
first to the closest Li(3) site, and then to an adjacent Li(3) site. This series of
hops is sufficient to form a percolating diffusion network through the Li layer
(see Figure 5). For the first portion of the diffusion path, Li(2) -+ Li(3) vacancy
migration, we calculated an activation barrier of 518 meV.
For the second portion of the diffusion path, Li(3) -+ Li(3), migration, we
evaluated four different hops. The first hop tested is through the Li channel and
crosses Li layers, similar to Li(1) diffusion in the channel. The remaining three
hops are illustrated in Figure 6. Each Li(3) site is closely coordinated to three
39
oxygen ions which form a ring around the large c-axis channels. The first Li(3) -+
Li(3) migration pathway we tested consists of a Li-ion entering into the channel
by passing between its neighboring oxygen ions, then exiting the channel via an
analogous path (inside path). The second pathway consists of traveling around the
exterior of the channel (outside path). The third pathway follows a S-shape which
is interior to the channel near one Li(3) site and exterior near the other Li(3) site,
transitioning between the two by passing through the oxygen triangle between the
Li(3) site-coordinated oxygen triangles.
Figure 6: Diagram of the calculated Li(2) -> Li(3) hop and the three calculated
Li(3) -+ Li(3) hops.
For the Li(3) -->Li(3) migration across Li layers, the migration barrier of 1534
meV and net activation energy of 1691 meV are extremely high (Table 6). The
high migration barrier is not surprising because it essentially involves promot40
ing a Li(3) vacancy to the Li(1) site and then Li(1) vacancy in-channel diffusion.
Comparing to the fully lithiated case, it is possible to estimate the first part of this
migration to be approximately 442 meV and the latter to be 740 meV (Table 6)
for a total estimate of 1182 meV.
For the three remaining Li(3) -> Li(3) paths, our calculations indicate that
the most favorable transition occurs via the inside path with a low net activation
energy of 465 meV (Figure 7). The S-route and outside migration paths have
higher activation energies of 646 meV and 745 meV, respectively. The energy
profile of the migration path from Li(2) to Li(3) is plotted in Figure 7. Our results
indicate that the rate-limiting step should be the first portion of Li(2) -+ Li(3)
diffusion with an activation barrier of 518 meV.
4.5.3 Comparison to experiments
The calculations on Li 9 V3 (P 2 0 7 )3 (PO 4 ) 2 indicate negligible diffusion within the
Li-layer (1.3 eV barrier) and faster but still diffusion limited inter-layer diffusion
(740 meV). In contrast, our calculations on LisV 3 (P2 0 7 ) 3 (PO4 )2 indicate indeter-
minate diffusion within the Li-layer (518 meV) and negligible inter-layer diffusion
(1.7 eV). The reversal of preferred diffusion orientations can be understood simply from the large difference in site energies. When more than 8 Li are present per
formula unit, vacancies are only expected on the Li(1) sites, and in-layer diffusion
is sluggish because it requires traversing an Li(2) site. When fewer than 8 Li are
41
600-
5-Path
400-
w
200518
Inside
Li(3)
Li(2)
+465
Li(3)
Li-vacancy Location
Figure 7: Calculated NEB profiles for Li(2) -+ Li(3) -+ Li(3) migration in the Li
layer of LisV 3(P2 0 7 )3 (PO4 ) 2. The circles denote each intermediate image. The
connected lines are cubic spines; the slopes at each image were set determined
by the forces calculated with DFT. Three different paths were tested for Li(3) Li(3) migration.
42
present per formula unit, all Li(1) sites are vacant and additional vacancies will
occupy either Li(2) or Li(3) sites. These additional vacancies may move within
the layer, but inter-layer diffusion is sluggish because it first requires promoting a
Li from a Li(3) site into a vacant high energy Li(1) site.
Our calculations are in reasonable agreement with experimental diffusivity
measurements by Poisson et al. [53] on the isostructural LigA13 (P 2 0 7 ) 3 (PO4 ) 2
and LigFe 3 (P 2 0 7 ) 3 (PO4 ) 2 where the inter-layer activation barriers were measured
at 1.22 eV and 1.20 eV, respectively. The in-layer activation barriers were determined to be 660 meV and 690 meV, respectively. Although it is not known
whether samples from Poisson et al might have been slightly Li deficient, the activation barriers they report fall between our calculated values for Li8 V 3 (P2 0 7) 3 (PO4 ) 2
and LigV 3 (P 2 0 7 ) 3 (PO4 ) 2 -
Given the large calculated migration barrier for Li(1) diffusion, it is somewhat
surprising that the material can be electrochemically cycled over 100mAh/g (56%
Theoretical capacity) at a moderate rate of C/10 with a small overpotential [1].
Jain et al. found similar results [52]. However, Jain et al also found an unusually
large discrepancy between the voltage profile calculated with DFT and the the
experimental results, especially in the Lig to Li8 range of lithiation.
Several theories may explain the discrepancy between the predicted Li+ diffusion limitation and the ability of the material to cycle. Four are presented here.
(1) If the experimentally tested materials were Li deficient, then the predicted behavior of indeterminate diffusion would not conflict with the material's ability to
43
electrochemically cycle. (2) An uneven concentration gradient within the material could allow the majority of Li+ diffusion to occur at concentrations below Li8
where the material is not predicted to be diffusion limited. Taken to the extreme,
Li will be first extracted from the surface states via diffusion over very short distances. This produces a surface region nominally at and slightly below Li 8 and
a core region with a composition of Lig. As more lithium is extracted, the core
region shrinks and the surface region grows. If the interface between these two
regions remains sharp, then Li+ diffusion in regions above a composition of Li8
need only occur over very short distances. However, upon insertion of Li+, a surface region of Lig composition forms around a core region of Li 8 . The surface
region would shield the core region, and insertion would stop with the bulk of the
material at a nominal composition of Li 8 . (3) DFT may simply fail to model this
material, especially at the Li9 composition. (4) The actual diffusion mechanism
differs from what was considered here.
Corroborating or dismissing these four theories with the available information
is difficult. Neither Jain nor Kuang reported precise measurement of atomic composition in their experimental samples. Nor is such information trivial to obtain,
making the theory of Li deficiency difficult to test. The sharp gradient model
predicts incomplete Li insertion upon discharge. Taken together, Kuang and Jain
report three electrochemical voltage profiles. Two of the three show larger capacity upon charge than discharge, but the third shows little discernible difference.
Furthermore, It is unclear from the voltage profiles if the lost capacity was at high
44
Li concentration or low Li concentration, partially due to the discrepancy between
observed voltages and Jain's predicted voltages. Of course, this discrepancy lends
credence to the theory that DFT fails to correctly model the system. But, it remains unclear why this would be. Finally, the theory that something has been
overlooked shall be ever-present.
In summary, it is unclear if the theoretical prediction of limited diffusivity in
the compositional range 8 < x < 9 is in conflict with experiments. However,
the good numerical agreement between the calculated values for the V system and
the experimental measurements on the Fe and Al systems, suggests that the DFT
migration barriers have predictive validity.
45
5
5.1
Discussion
Accuracy of Topology Determination
While prior studies have shown that empirical potentials can be effective at identifying diffusion topologies (Section 2.3), the empirical potential applied here is
novel and requires some degree of validation. It has already been shown that such
empirical potential results may differ frequently from expectations based on simple geometric analysis [21]. This was also true in the case of Li9 V 3 (P 2 0 7 ) 3 (P 2 0 4 ) 2 ,
(Section 4.5). Experimental measurements showing Li+ diffusion primarily in the
a-b plane differed greatly from the expected topology. The presence of large, open
c-axis channels, suggested diffusion primarily along the c-axis. The empirical potentials correctly predicted in-plane diffusion, also consistent with DFT activation
barriers.
Experimental topology information is also available for olivine LiFePO4 . The
experimental topology obtained from neutron diffraction data [54] bears remarkable resemblance to to the empirical potential derived topology (Figure 8).
Unfortunately, experimental data on diffusion topology is extremely sparse.
However, the predicted diffusion topologies and the the DFT derived minimum
energy pathway may also be compared. Of the 32 NEB calculations presented in
this study, none of them differed presented a significantly different topology than
was predicted from the empirical potentials. The predicted 1D diffusion channels
46
Empirical
Potentials
Neutron
Scattering
S. Nishimura,
Nature materials.
2008.
Figure 8: Comparison between Li+ diffusion topology by neutron diffraction data
[54] and an empirical potential model for LiFePO 4 . An isosurface for Li+ energy
is shown in yellow for the empirical potential results.
47
in both monoclinic borates were verified with DFT refinement of both intra and
inter channel NEB calculations. The empirical potential predicted aversion of Li+
to the large c-axis channels in the Li3 V3 (P2 0 7 )3 (P2 0 4 )2 was also validated with
DFT.
The relationship between the DFT calculated barrier energies and the empirical potential barrier energies is shown in figure 9. The obvious correlation provides the empirical potential models the ability to accurately determine diffusion
topology. This data suggests that if a migration barrier of less than 700 meV is
required, it is safe to exclude all diffusion pathways with an empirical potential
migration barrier of greater than 1.5eV. At the same time, the data conclusively
demonstrates that the empirical potential model is not sufficient to screen materials without DFT refinement. For example, the low energy diffusion pathways in
the hexagonal LiMnBO 3 are underestimated by nearly 500 meV.
5.2
Acuracy of Barrier Evaluation
This study is the first to apply of DFT and NEB Li+ diffusivity screening to multiple oxide structural prototypes simultaneously. As such, it offers a valuable
opportunity to examine how reliable DFT derived Li+ migration barrier energies
in for oxide materials. As outlined in Section 2.2, experimental determination
of activation barrier is very difficult, making direct quantitative comparison difficult. However, electrochemical testing has been performed on all of the materials
48
3000
2500
4
2000
* B-LiFePO4
S
mHexagonal
1500
Monoclinic
X CarbonoPhosphate
A
A
- 1000
Diphosphate
Ar7
500
0
1000
Olivine
--
2000
3000
Empirical Potential Barrier [meV]
Figure 9: Comparison between DFT calculated migration barriers, and migration
barriers due to the empirical potential model.
49
discussed allowing qualitative comparison.
For each of the materials evaluated, the net activation barrier for Li+ diffusion is listed in Table 7. Also listed, is the expected behavior according to the
metric derived in Section 2.1. Materials with net barriers less than 450 meV are
expected to be facile Li diffusers. Materials with net barriers more than 700 meV
are expected to be diffusion limited and materials with intermediate barriers are
indeterminate. Finally, a summary of the observed experimental behavior is also
listed. For added comparison data from the olivine and layered compounds are
also Included in Table 7.
Most of the experimental observations are in agreement with the DFT diffusion predictions. The layered and olivine cathodes both cycle well. Neither of
the
#'-LiFePO 4
and hexagonal LiMnB0
3
materials cycle. The remaining three
materials all show electrochemical activity of around half of the theoretical capacity. For the monoclinic LiMnBO 3 and the carbonophosphate, it is not clear
why only half of the capacity cycles. However, since many factors besides Li+
diffusion can limit a cathode material, the experimental observations are not in
conflict with the diffusion prediction. The only remaining possible conflict between the theoretical diffusivity prediction and experimental observations is the
diphosphate material. Although DFT predicted that Li 9 V3 (P2 0 7 ) 3 (PO 4 ) 2 would
be diffusion limited, the material exhibits electrochemical activity. As discussed
in Section 4.5.3 several possible explanations may resolve this apparent conflict.
Without any obvious failures, the data strongly suggests that GGA DFT migration
50
Table 7: Summary of DFT Li+ diffusion screening and comparison to observed experimental behavior.
LiO.,5 CoO 2
DFT predicted
Li+ diffusivity
indeterminatet
facile
Ample diffusion. Commercial cath-
CoO 2
615*
indeterminate
ode matenal.
LiFePO 4
FePO 4
LiFePO 4
FePO 4
LiMnBO 3
MnBO 3
270t
200T
1004
961
724
529
facile
facile
limited
limited
limited
indeterminate
Ample diffusion. Commercial high
rate cathode material.
No substantial Li+ extraction.
LiMnBO 3
LiO.5 MnBO 3
MnBO 3
510
311
396
indeterminate
facile
facile
Electrochemically cycles between
LiMnBO 3 and LiO. 5 MnBO 3 at
elevated temperatures.
Li 3 Mn(CO 3 )(PO 4 )
Li 2 Mn(CO 3 )(PO 4 )
LiMn(CO 3 )(P0 4 )
Li9 V 3 (P2 0 7 ) 3 (PO 4 ) 2
Li8 V 3 (P2 0 7 ) 3 (PO 4 ) 2
396
478
380
740
518
facile
indeterminate
facile
limited
indeterminate
beElectrochemically
cycles
tween
Li3 Mn(CO 3 )(PO 4 )
and
Li 2 Mn(CO 3 )(PO 4 ) at low rate.
Electrochemically cycles at 56% of
theoretical capacity.
LiCoO 2
layered
olivine
hexagonal
monoclinic
carbonophosphate
Experimental
behavior
Barrier
[meV]
225*
400*
Material
No substantial Li+ extraction.
* The barriers for the layered compound were taken from VanDerVen et al. [31]
t The expected behavior for LiCoO 2 is indeterminate due to the required divacancy mechanism (Section 2.4.1)
I The barriers for the olivine compound were taken from Morgan et al. [22]
barriers are an effective screening criteria for Li+ diffusivity in oxides.
In fact, careful parsing of the data suggests that DFT may be able to produce
a more precise prediction of Li+ diffusivity than the simple facile vs. indeterminate vs. limited classification applied here. Notice that the commercial high-rate
cathode material LiFePO4 has the lowest migration barriers of any material. Furthermore, the monoclinic LiMnBO 3 drifts into the indeterminate range and cycles
better at elevated temperatures. However, more experimental data on Li+ diffusion is needed to quantitatively refine the accuracy of this method.
5.3
Factors affecting Li diffusivity
On of the common methods used to estimate the Li+ diffusivity of a material is
an open space metric. The larger the bottleneck through which the Li+ ion must
pass, the lower the migration barrier is presumed to be. It has already been show
that this fails to describe the limited interlayer diffusion in LigV 3 (P 2 0 7 )3 (PO 4 )2
(Section 4.5). The screening methodology applied in this thesis offers a unique
opportunity to examine the general relationship between bottleneck size and migration barrier.
The bottleneck size for each NEB hop was defined as follows. At the transition
state of the NEB calculation, the three closest oxygen ions which formed a triangle
through which the Li ion passed were identified. The corresponding positions of
these oxygen ions in the bulk were taken to represent the constraining bottleneck.
52
The bottleneck size was taken as the distance between the triangle circumcenter
and the vertices. This represents the largest possible minimum lithium-oxygen
distance during Li+ migration if the non-lithium atomic positions are fixed at the
bulk value.
3000
r,
2500
S 2000
E
U
0
0"
e
* B-LiFePO4
* Hexagonal
A Monoclinic
1500
--
X CarbonoPhosphate
1000
- Diphosphate
X
500
04
1.5
0Olivine
x
XX
2.1
1.7
1.9
Bottleneck size in bulk [Angs]
2.3
Figure 10: Comparison between DFT calculated migration barriers, and bottleneck size determined from the bulk structure.
The bottleneck size and the DFT calculated migration barrier is shown in figure 10. Bottleneck size is a very poor predictor of migration barrier for Li+ diffusivity. The diphosphate and carbonophosphate both exhibit low migration barriers
through small bottlenecks. As already discussed, the diphosphate exhibits a large
migration barrier in the large bottleneck c-axis channel. The olivine structure exhibits migration barriers around 400 meV and 2400 meV with bottleneck sizes
53
that are both around 1.9 A. A few other open space metrics were also considered:
bottleneck size at the transition state, minimum Li-anion distance during NEB,
change in Li-anion distance during NEB-none of them proved particularly predictive.
With the failure of the open space metrics, it remains a mystery what physical
factors can be used to design high-rate solid state Li+ diffusers. Resolving this
mystery is beyond the scope of this thesis. However, the computational methods
developed and verified within offer a valuable tool for approaching this problem.
The specificity of atomistic modeling allows examination of many potential influences on Li+ diffusion which are exceedingly difficult to determine experimentally (e.g. the movement of ions adjacent to the migrating ion). If broadly applied
and well examined, this computational approach may offer valuable insight into
underlying physical factors which do dictate the Li+ diffusion rate in solid oxides.
54
A
Empirical Potential Specification
The specific Hamiltonian used was derived by my colleague Tim Mueller and is
as follows. All interactions are pair-wise between ions. Only Li-nonLi pairs must
be considered because only Li is mobile, and only one Li is in the structure at a
time. The Coulombic potential energy as a function of distance C(d) is specified
by:
C(d) =
exp [-d/l]
q
The charges of all ions (q, and qb) were fixed at the values of the original, nondefected, Li containing structures. Therefore, the Hamiltonian is applied to charged
structures. Convergence in the Coulombic interaction is achieved through use of
the exponential screening. A screening length 1of 0.65 Angstroms was used.
The repulsive Li-oxygen potential is of the Buckingham type:
R(d)= A -exp [-d/B]
The pre-exponential value A was set to 38718 eV. The decay length B was set
to 0.1556 Angstrom. Finally, the potential was softened to account for the fact
that fixing all non-Li ions results in overly short Li-oxygen distances. Generally
these ions are able to relax away from a nearby Li ion to some extent. Let S(d)
represent the sum of Coulombic and repulsive interactions S(d) = C(d) + R(d).
The softening of this potential occurs by evaluating this combined potential at
55
a slightly altered distance. The final total pairwise potential E(d) is given by
evaluation of S(d) at a altered distance d:
E(d) = S(d) = C(d) + R(d)
The new length d is chosen as if the fixed (non-Li) anion was attached to it's location by a spring, and allowed to adjust it's position in response to its interaction
with the Li:
S'(d)
~
S"I(d) + k
The function S'(d) is the first derivative of the combined potential S(d), effectively the force due to the potential. The function S"(d) is the second derivative.
The spring constant k was set to 12 eV/A 2 .
56
References
[1] Quan Kuang, Jiantie Xu, Yanming Zhao, Xiaolong Chen, and Liquan Chen.
Layered
monodiphosphate Li9V3(P207)3(PO4)2: A novel cathode material for lithium-ion batteries. ElectrochimicaActa, 56(5):2201-2205, February 2011.
[2] Ling Chen, Yanming Zhao, Xiaoning An, Jianmin Liu, Youzhong Dong, Yinghua Chen,
and Quan Kuang. Structure and electrochemical properties of LiMnBO3 as a new cathode
material for lithium-ion batteries. Journal of Alloys and Compounds, 494(1-2):415-419,
2010.
[3] Tim Mueller, Geoffroy Hautier, Anubhav Jain, and Gerbrand Ceder. Evaluation of TavoriteStructured Cathode Materials for Lithium-Ion Batteries Using High-Throughput Computing.
Transition,2011.
[4] Geoffroy Hautier, Anubhav Jain, Shyue Ping Ong, Byoungwoo Kang, Charles Moore,
Robert Doe, and Gerbrand Ceder. Phosphates as Lithium-Ion Battery Cathodes : An Evaluation Based on High-Throughput ab Initio Calculations. Chemistry of Materials, pages
3495-3508, 2011.
[5] Anubhav Jain, Geoffroy Hautier, Charles J. Moore, Shyue Ping Ong, Christopher C. Fischer,
Tim Mueller, Kristin a. Persson, and Gerbrand Ceder. A high-throughput infrastructure for
density functional theory calculations. ComputationalMaterialsScience, 50(8):2295-23 10,
June 2011.
[6] Geoffroy Hautier, Christopher C. Fischer, Anubhav Jain, Tim Mueller, and Gerbrand Ceder.
Finding Natures Missing Ternary Oxide Compounds Using Machine Learning and Density
Functional Theory. Chemistry of Materials,22(12):3762-3767, June 2010.
[7] Byoungwoo Kang and Gerbrand Ceder. Battery materials for ultrafast charging and discharging. Nature, 458(7235):190-3, March 2009.
57
[8] Thomas Maxisch, Fei Zhou, and Gerbrand Ceder. Ab initio study of the migration of small
polarons in olivine Li-xFePO-4 and their association with lithium ions and vacancies. Physical Review B, 73(10):1-6, 2006.
[9] Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. Kinetics ofMaterials. John Wiley
& Sons Inc., Hoboken, New Jersey, 2005.
[10] A Van Der Van der Ven and G Ceder. First Principles Calculation of the Interdiffusion
Coefficient in Binary Alloys. PhysicalReview Letters, 94(4):1-4, February 2005.
[11] Kisuk Kang and Dane Morgan. First principles study of Li diffusion in I-Li. {2} NiO. {2}
structure. PhysicalReview B, 79:014305, 2009.
[12] Dong-Hwa Seo, Young-Uk Park, Sung-Wook Kim, Inchul Park, R. Shakoor, and Kisuk
Kang. First-principles study on lithium metal borate cathodes for lithium rechargeable batteries. PhysicalReview B, 83(20):1-8, May 2011.
[13] C. Delacourt, M. Ati, and J. M. Tarascon. Measurement of Lithium Diffusion Coefficient in
LiyFeSO4F. Journalof The Electrochemical Society, 158(6):A741, 2011.
[14] D Singh, W Kim, V Craciun, H Hofmann, and R Singh. Microstructural and electrochemical
properties of lithium manganese oxide thin films grown by pulsed laser deposition. Applied
Surface Science, 197-198:516-521, September 2002.
[15] J Xie, N Imanishi, T Matsumura, a Hirano, Y Takeda, and 0 Yamamoto. Orientation dependence of Liion diffusion kinetics in LiCoO2 thin films prepared by RF magnetron sputtering.
Solid State Ionics, 179(9-10):362-370, May 2008.
[16] Myounggu Park, Xiangchun Zhang, Myoungdo Chung, Gregory B. Less, and Ann Marie
Sastry. A review of conduction phenomena in Li-ion batteries. Journal of Power Sources,
195(24):7904-7929, June 2010.
[17] Ruhul Amin, Palani Balaya, and Joachim Maier. Anisotropy of Electronic and Ionic Transport in LiFePO[sub 4] Single Crystals. Electrochemicaland Solid-State Letters, 10(1):A13,
2007.
58
[18] Kristin Persson, Vijay a. Sethuraman, Laurence J. Hardwick, Yoyo Hinuma, Ying Shirley
Meng, Anton van der Ven, Venkat Srinivasan, Robert Kostecki, and Gerbrand Ceder. Lithium
Diffusion in Graphitic Carbon. The Journalof Physical Chemistry Letters, 1(8):1176-1180,
April 2010.
[19] Timothy S Bush, Julian D Gale, Richard A Catlow, and Peter D Battle. Self-Consistent
Interatomic Potentials for the Simulation of Binary and Ternary Oxides. JournalofMaterials
Chemistry, 4(6):831-837, 1994.
[20] Rajesh Tripathi, Grahame R Gardiner, M Saiful Islam, and Linda F Nazar. Alkali-ion Conduction Paths in LiFeSO 4 F and NaFeSO 4 F Tavorite-Type Cathode Materials. Chemistry
of Materials,pages 2278-2284, 2011.
[21] Stefan Adams and R. Prasada Rao. High power lithium ion battery materials by computational design. Physica Status Solidi (a), 208(8):1746-1753, August 2011.
[22] D. Morgan, A. Van Der Ven, and G. Ceder. Li Conductivity in Li[sub x]MPO[sub 4]
(M=Mn,Fe,Co,Ni) Olivine Materials. Electrochemical and Solid-State Letters, 7(2):A30,
2004.
[23] Craig A J Fisher, Veluz M Hart Prieto, and M Saiful Islam. Lithium Battery Materials Li M
PO 4 ( M = Mn, Fe, Co, and Ni): Insights into Defect Association, Transport Mechanisms,
and Doping Behavior. Chemistry of Materials,20(18):5907-5915, September 2008.
[24] S. Adams and R.P. Rao. Transport pathways for mobile ions in disordered solids from the
analysis of energy-scaled bond-valence mismatch landscapes. Phys. Chem. Chem. Phys.,
11(17):3210-3216, May 2009.
[25] Vladimir I. Anisimov, Jan Zannen, and Ole K. Andersen. Band theory and Mott insulators:
Hubbard U instead of Stoner I. Physical Review B, 44(3):943-954, 1991.
[26] Gregory Mills, Hannes J6nsson, and Gregory K. Schenter. Quantum and Thermal Effects in
H2 Dissociative Adsorption: Evaluation of Free Energy Barriers in Multicimensional Quantum Systems. Physical Review Letters, 72(7):1124-1127, 1994.
59
[27] H. J6nsson, G. Mills, and K. W. Jacobsen. Nudged elastic band method for finding minimum
energy paths of transitions. In B. J. Berne, G. Ciccotti, and D. F. Coker, editors, Classical
and Quan- tum Dynamics in Condensed Phase Simulations, page 385. World Scientific,
Singapore, 1998.
[28] Daniel Sheppard and Graeme Henkelman. Letter to the Editor Paths to which the Nudged
Elastic Band Converges. Journalof computationalchemistry, 32(8):1769-1771, 2011.
[29] Daniel Sheppard, Rye Terrell, and Graeme Henkelman. Optimization methods for finding
minimum energy paths. The Journalof chemicalphysics, 128(13):134106, 2008.
[30] A. Van der Ven and G Ceder. Lithium Diffusion in Layered Li. MaterialsScience, 3(7):5-8,
2000.
[31] A Van der Ven and G Ceder. Lithium diffusion mechanisms in layered intercalation compounds. Journalof Power Sources, 97-98(1-2):529-531, July 2001.
[32] Anton Van der Ven, John Thomas, Qingchuan Xu, Benjamin Swoboda, and Dane Morgan. Nondilute diffusion from first principles: Li diffusion in LixTiS2. Physical Review B,
78(10):1-12, September 2008.
[33] Kisuk Kang and Gerbrand Ceder. Factors that affect Li mobility in layered lithium transition
metal oxides. PhysicalReview B, 74(9):94105, 2006.
[34] Zhaojun Liu and Xuejie Huang.
Structural, electronic and Li diffusion properties of
LiFeSO4F. Solid State Ionics, 181(25-26):1209-1213, August 2010.
[35] Gopi Krishna Phani Dathar, Daniel Sheppard, Keith J. Stevenson, and Graeme Henkelman.
Calculations of Li-Ion Diffusion in Olivine Phosphates.
Chemistry of Materials,
23(17):4032-4037, September 2011.
[36] M Saiful Islam, Daniel J Driscoll, Craig A J Fisher, and Peter R Slater. Atomic-Scale Investigation of Defects, Dopants, and Lithium Transport in the LiFePO 4 Olivine-Type Battery
Material. Chemistry of Materials, 17(20):5085-5092, October 2005.
60
[37] Rahul Malik, Damian Burch, Martin Bazant, and Gerbrand Ceder. Particle size dependence
of the ionic diffusivity. Nano letters, 10(10):4123-7, October 2010.
[38] Bo Xu and Shirley Meng. Factors affecting Li mobility in spinel LiMn204A first-principles
study by GGA and GGA+U methods. Journal of Power Sources, 195(15):4971-4976, August 2010.
[39] Robert Darling. Dynamic Monte Carlo Simulations of Diffusion in Li[sub y]Mn[sub 2]O[sub
4]. Journalof The ElectrochemicalSociety, 146(10):3765, 1999.
[40] Jishnu Bhattacharya and Anton Van Der Ven. Phase stability and nondilute Li diffusion in
spinel Li_{1+x}Ti_{2}0.4}. PhysicalReview B, 81(10):27-30, March 2010.
[41] Benjamin Morgan and Graeme Watson. GGA+U description of lithium intercalation into
anatase TiO..2}. Physical Review B, 82(14):1-11, October 2010.
[42] Anti Liivat and John 0. Thomas. Li-ion migration in Li2FeSiO4-related cathode materials:
A DFT study. Solid State Ionics, 192(1):58-64, June 2011.
[43] G Kresse and J Furthmiller. Efficient iterative schemes for ab initio total-energy calculations
using a plane-wave basis set. Physical review. B, Condensed matter, 54(16):11169-11186,
October 1996.
[44] G Kresse and D Joubert. From ultrasoft pseudopotentials to the projector augmented-wave
method. PhysicalReview B, 59(3):1758-1775, January 1999.
[45] http://theory.cm.utexas.edu/vtsttools/.
[46] John P Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized Gradient Approximation
Made Simple. PhysicalReview Letters, 77(18):3865 LP - 3868, October 1996.
[47] 0. Garcia-Moreno, M. Alvarez-Vega, J. Garcia-Jaca, J. M. Gallardo-Amores, M. L. Sanjuin,
and U. Amador. Influence of the Structure on the Electrochemical Performance of Lithium
Transition Metal Phosphates as Cathodic Materials in Rechargeable Lithium Batteries: A
61
New High-Pressure Form of LiMPO 4 (M = Fe and Ni). Chemistry ofMaterials, 13(5):15701576, May 2001.
[48] V. Legagneur. LiMBO 3 (M 5 Mn, Fe, Co): synthesis, crystal structure and lithium deinsertion/insertion properties. Solid State Ionics, 139:37-46, 2001.
[49] JL Allen, K Xu, SS Zhang, and TR Jow. LiMBO sub 3(M= Fe, Mn): potential cathode for
lithium ion batteries. Symposium V&quot; Materialsfor Energy Storage,, 730:1-6, 2002.
[50] Jae Chul Kim, Charles J. Moore, Byoungwoo Kang, Geoffroy Hautier, Anubhav Jain, and
Gerbrand Ceder. Synthesis and Electrochemical Properties of Monoclinic LiMnBO[sub 3]
as a Li Intercalation Material. Journalof The ElectrochemicalSociety, 158(3):A309-A315,
2011.
[51] Hailong Chen, Geoffroy Hautier, Anubhav Jain, Charles Moore, Byoungwoo Kang, Robert
Doe, Lijun Wu, Yimei Zhu, Yuanzhi Tang, and Gerbrand Ceder. Carbonophosphates: A New
Family of Cathode Materials for Li-Ion Batteries Identified Computationally. Chemistry of
Materials,24(11):2009-2016, June 2012.
[52] Anubhav Jain, Geoffroy Hautier, Charles Moore, Byoungwoo Kang, Jinhyuk Lee, Hailong Chen, Nancy Twu, and Gerbrand Ceder.
A Computational Investigation of
Li9M3(P207)3(PO4)2 (M = V, Mo) as Cathodes for Li Ion Batteries. Journalof The ElectrochemicalSociety, 159(5):A622, 2012.
[53] S Poisson, E Bretey, and P Berthet. Crystal Structure and Cation Transport Properties of the
Layered Monodiphosphates: Li 9. Journalof Solid State Chemistry, 40(138):32-40, 1998.
[54] Shin-ichi Nishimura, Genki Kobayashi, Kenji Ohoyama, Ryoji Kanno, Masatomo Yashima,
and Atsuo Yamada. Experimental visualization of lithium diffusion in LixFePO4. Nature
materials,7(9):707-11, 2008.
62