Potassium Ordering and Structural Phase Stability in Layered KxCoO2

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Potassium Ordering and Structural Phase Stability in Layered KCoO Michael Yuji Toriyama, Jonas Leif Kaufman, and Anton Van der Ven

ACS Appl. Energy Mater., Just Accepted Manuscript • DOI: 10.1021/acsaem.8b02238 • Publication Date (Web): 07 Mar 2019 Downloaded from http://pubs.acs.org on March 11, 2019

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ACS Applied Energy Materials

Potassium Ordering and Structural Phase Stability in Layered KxCoO2 Michael Y. Toriyama,† Jonas L. Kaufman,‡ and Anton Van der Ven∗,‡ †Department of Materials Science and Engineering and Department of Mathematics, University of Illinois at Urbana-Champaign, Champaign, Illinois 61820, United States ‡Materials Department, University of California, Santa Barbara, Santa Barbara, California 93106, United States E-mail: *[email protected] Abstract The recent surge of interest in K-ion batteries necessitates a fundamental understanding of phase stability and K ordering tendencies in common electrode materials. We report on a first-principles study of phase stability in layered Kx CoO2 (0 ≤ x ≤ 1) in the O1/P3/O3 family of host structures, and identify K ordering preferences within each host. We find that the P3 host is stable at intermediate K concentrations and exhibits a multitude of hierarchical orderings characterized by well-ordered domains separated by antiphase boundaries. We also predict the stability of a new family of layered structures at high K concentrations that are highly distorted and host both octahedrally- and prismatically-coordinated K within each intercalation layer.

Keywords: potassium-ion battery, phase stability, intercalation, layered metal-oxide cathode, antiphase boundaries, potassium ordering

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Introduction The growing demand for large-scale energy storage applications, such as electric vehicles and energy grids, calls for low-cost alternatives to Li-ion batteries. 1–3 Technologies that employ Na and K are possible alternatives, since those species are more abundant than Li in the earth’s crust. While Na-ion batteries are an active field of research, 4–9 less attention is given to K-ion batteries. However, an advantage of K over Na is that K is able to intercalate into graphite, 9–12 an economical, well-studied anode material used regularly in Li-ion batteries. 1,13 Various experimental studies have also demonstrated the potential of K-ion batteries for large-scale energy storage systems. 14–20 Layered intercalation compounds in particular have gained considerable momentum as candidate cathode materials for K-ion batteries due to their flexibility in (de)intercalating large K+ ions. 21–25 Layered K intercalation compounds undergo a variety of phase transitions during cycling that affect diffusion mechanisms and mechanical degradation. In this study, we examine the stability of phases and K ordering tendencies in the O1/P3/O3 family of layered Kx CoO2 using first-principles techniques. At low K concentrations, the O3 and hybrid O1-O3 hosts are predicted to be stable. The P3 phase is predicted to be stable in a wide range of intermediate concentrations. We find that K favors a Devil’s staircase 26 of orderings in P3 characterized by well-ordered domains that are periodically separated by antiphase boundaries. We also discover a new host structure at high K compositions, consisting of highly strained, undulating CoO2 layers that allow for the simultaneous presence of octahedral and trigonal prismatic K sites within each intercalation layer.

Computational Details The Vienna Ab-initio Simulation Package (VASP) 27–29 was used to perform first-principles density functional theory (DFT) calculations. The optB86b-vdW functional 30–33 was used to ensure an accurate description of van der Waals forces that hold the CoO2 sheets together 2


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at low K concentrations. The projector augmented wave (PAW) method 34,35 was employed to describe interactions between valence and core electrons. A 500 eV energy cutoff was used for the plane-wave expansions of the valence electron density. A Gamma-centered kpoint mesh 36 of density 20 Å was used to sample the Brillouin zone. Ionic relaxations were performed with a force convergence criterion of 0.02 eV/Å and Gaussian smearing of width 0.1 eV. A single static calculation using the tetrahedron method with Blöchl corrections 37 was run after each relaxation to produce an accurate relaxed energy. A Hubbard U correction is often used in DFT studies of transition metal oxide intercalation compounds to improve agreement with experimentally measured voltages. 38 However, past studies on layered Nax CoO2 and Lix CoO2 compounds have shown that this can lead to qualitatively incorrect predictions of phase stability due to an electronic transition that occurs above U ≈ 3 eV. 39,40 Rather than use a reduced U value, we chose not to use a Hubbard correction at all in order to ensure predictions of phase stability that are qualitatively accurate. While DFT without U predicts voltages that are lower than those measured experimentally, it has nevertheless been used previously to accurately predict phase stability and to provide fundamental insights about both Lix CoO2 41–43 and Nax CoO2 . 22,39,44 The Clusters Approach to Statistical Mechanics (CASM) code 45–48 was used to systematically enumerate symmetrically distinct K-vacancy orderings in different host structures of Kx CoO2 and was interfaced with VASP to calculate their relaxed energies.

Results Crystallography of Layered Kx CoO2 Layered Kx CoO2 consists of two-dimensional CoO2 sheets made of edge sharing CoO6 octahedra with K occupying sites between the sheets. Though various stacking sequences of the CoO2 sheets are possible in Kx CoO2 , 49,50 we focus only on the O1, O3, and P3 hosts shown in Figures 1a-c. The oxygen anions form close-packed triangular lattices that adopt 3



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an ABABAB stacking sequence in O1, an ABBCCA stacking in P3, and an ABCABC stacking in O3. 51 In O1 and O3, the intercalating K occupy octahedral sites that form simple two-dimensional triangular lattices between CoO2 layers. In P3, K occupy trigonal prismatic sites that collectively form a honeycomb network between CoO2 sheets. Hybrid stackings, which consist of intercalation layers that alternate between the O1 stacking and either the O3 or P3 stacking in the direction perpendicular to the CoO2 layers, 41,42,52 are also examined in this study. The O1 layers in such hybrid stackings, however, are kept unoccupied because octahedrally-coordinated ions in O1 share two faces with positively-charged transition metal polyhedra, reducing the stability of the overall structure through electrostatic repulsion. 22 The hybrid hosts considered in this work are shown in Figures 1d,e. As will be described in more detail, highly distorted layered phases that host a mix of octahedral and trigonal prismatic sites within a single intercalation layer are also found to be stable in Kx CoO2 . An example of such structure is shown in Figure 1f, where undulating CoO2 sheets allow alternating strips of trigonal prismatic and octahedral sites. We label this family of phases as M.

Formation Energy and Voltage The relaxed energies of 147 O1, 531 O3, and 696 P3 symmetrically distinct K-vacancy orderings were calculated using DFT. Additional DFT calculations were performed for 50 and 113 K-vacancy orderings in the O1-O3 and O1-P3 hybrid hosts, respectively. Finally, 73 M structures, which emerged upon relaxing a subset of the O3 and P3 orderings, were identified by visualizing the relaxed structures in VESTA. 53 The formation energy Ef of each configuration is defined by

Ef = EKx CoO2 − xEKCoO2 − (1 − x)ECoO2

4


(1)

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(a) O1

(b) P3

(c) O3

Perspective View

K-Ion Polyhedron

Side View

B A

C B

A C

B A

B A

B A

(d) O1-O3

(e) O1-P3

A C

C B

B A

B A

B A

B A

(f) M

Figure 1: (a-f) The Kx CoO2 host structures investigated in this study. The Co (dark blue) are octahedrally coordinated by O (red). K in trigonal prismatic (purple) or octahedral (light blue) coordination occupy sites between CoO2 sheets. Side views show the stacking sequences of the O layers in (a) O1, (b) P3, (c) O3, (d) O1-O3, (e) O1-P3, and (f) M. The M structure shown is an example of a new type of phase identified in this study. Visualizations were created using VESTA. 53

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Formation Energy (eV/CoO 2)

(a)

O1

O3 O1-P3

M

O1-O3

O3 P3

(b)

+

(V)

4

Voltage vs. K/K

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Experiment 3

2

Predicted 1

0 0.0

0.2

0.4

0.6

x in K xCoO

0.8

1.0

2

Figure 2: (a) Calculated zero-kelvin phase diagram of Kx CoO2 for 0 ≤ x ≤ 1. Convex hulls of each host structure are shown along with the global hull outlined in black. (b) Predicted voltage curve derived from the zero-kelvin formation energies, shown in black. The compositions of three important P3 ground state orderings are indicated. An experimentallymeasured charge curve from 49 is shown in gray.

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where EKx CoO2 is the relaxed energy of the configuration per Kx CoO2 formula unit. The references EKCoO2 and ECoO2 are taken as the relaxed energies of O3 KCoO2 and O1 CoO2 , respectively. The convex hulls (i.e. the common tangent construction applied to the zero kelvin formation energies) for each host are shown in Figure 2a. The global hull is boldened in black, highlighting the ground state orderings at different K compositions. The O1 host is only stable in the absence of K at x = 0. The hybrid O1-O3 host is globally stable at x = 1/6 while the O3 host is stable at x = 1/3 and x = 1. P3 is stable for 7/17 ≤ x ≤ 2/3, and M is stable for 3/4 ≤ x ≤ 9/10. The M phases emerged during energy relaxations of structures that were initialized in O3 and P3. The energy differences between the M phases and the local hulls of P3 and O3 for x ≥ 3/4 are greater than 15 meV/CoO2 , which is well above typical DFT errors. We note, however, that composition ranges in which each host is stable may be slightly different than indicated, due to the constraint that only a finite number of orderings can be considered and to the numerical uncertainty of DFT calculations. The voltage of a K-ion battery is related to the difference in the K chemical potential between the cathode and anode according to 54 µcathode − µanode K V =− K e

(2)

As the reference anode, we use pure K in the body-centered cubic crystal structure. The chemical potential of K in Kx CoO2 at zero kelvin is equal to the slope of the global hull. Figure 2b shows the calculated voltage profile of Kx CoO2 along with an experimental curve measured by Hironaka et. al. 49 Although it is common that voltages are systematically underestimated when using DFT, 38 the calculated and experimentally-measured voltage profiles agree well qualitatively. The steps in the experimental curve are smoother than those predicted by our calculations, as some chemical disorder is likely allowed at finite temperature. The predicted voltage curve of Figure 2b consists of several pronounced steps and a

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multitude of smaller steps, where each step corresponds to a unique ground state ordering. The height of a voltage step for a particular ordering is proportional to the chemical potential window in which that ordering is stable and is therefore a measure of the stability of that ordering. The large step at x = 1/3, for example, arises from a strong preference for the √ √ 3a × 3a ordering within the O3 host (a being the lattice parameter), which maximizes the distance between K+ ions. 44,52 Two pronounced steps appear in the composition range in which P3 is stable, one at x = 1/2 corresponding to the common K ordering shown in Figure 3a, and another at x = 2/3 corresponding to the ordering in Figure 3c. There are many smaller steps spread over the whole composition range in which P3 is stable. These emerge due to the stability of a family of orderings made up of domains of simple orderings periodically separated by antiphase boundaries. A series of steps in the voltage profile are also present at high K concentrations due to the stability of M phases, which are orderings that are distinct from the O1/P3/O3 family of host structures. We describe the P3 and M orderings in more detail in the next two sections.

P3 Ground State Orderings There are three important orderings, illustrated in Figure 3, that characterize the entire family of P3 ground states. The ζ ordering shown in Figure 3a is the ground state at x = 1/2 and emerges as well-defined domains in P3 ground states for 2/5 ≤ x ≤ 4/7. Examples of these ground states are illustrated in Figure 4. The ζ domains in each of these ground states are separated by one of two types of antiphase boundaries (APBs). In the ground states for 2/5 ≤ x ≤ 1/2, domains of the ζ ordering are separated by straight APBs as illustrated in Figure 4a. These APBs have a lower K concentration than the perfect ζ ordering, and their insertion within the ζ ordering leads to a reduction in the overall K concentration. This family of orderings is therefore denoted the ζ − family, with the subscripts denoting the number of K in the unit cell separated by APBs. 44 The ζ domains in the P3 ground states for 1/2 ≤ x ≤ 4/7, in contrast, are separated by zig-zag APBs shown in Figure 4b. 8


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(a)

x = 1/2 ζ (b)

x = 4/7

η

(c)

x = 2/3 Δ

Figure 3: Three special P3 ground state orderings at (a) x = 1/2, (b) x = 4/7, and (c) x = 2/3 that together reflect the structural characteristics of all other P3 ground state orderings. Light and dark blue are used to distinguish K occupancy of the two triangular sublattices of the P3 honeycomb network. Pink boxes indicate the unit cells. The naming scheme is identical to that introduced for the ground state orderings of P3-Nax CoO2 . 44

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These APBs have a higher K concentration than the ζ ordering. This family of orderings is therefore denoted by ζ + , with the subscript again measuring the number of K between pairs of APBs. 44 Notice that the ground states belonging to the ζ + family exhibit local ordering along the APBs similar to the η ordering at x = 4/7 (Figure 3b). The limiting case of the ζ + family is precisely the η ordering, which also defines the underlying motif of the P3 ground states above x = 4/7 (Figure 5). The P3 ground states for 4/7 < x ≤ 2/3 have well-ordered domains that are two orientational variants of the η ordering shown in Figure 5a. Each P3 ground state for 4/7 < x ≤ 2/3 consists of alternating strips of the two variants interleaved by APBs, as shown in Figures 5b-e. This family of orderings is labeled η + with the subscripts referring to the number of K in the unit cell separated by APBs. 44 We found no ground states containing extended regions of the first orientational variant (green), which may indicate that all η + ground states consist of extended regions of the second variant (orange) separated by small strips of the first. The limiting case of the η + family is the ∆ ordering at x = 2/3 (Figure 3c), which is given a separate name due to its unique pattern of triangular islands. A large step exists at x = 2/3 in the voltage profile in Figure 2b, indicating that the ∆ ordering is relatively more stable than other P3 orderings in Kx CoO2 . The relaxed structures of all orderings found to lie on the local hull of P3 are provided in the Supporting Information.

M Ground State Orderings The Kx CoO2 compound also favors a family of highly-distorted phases, denoted as M, for 3/4 ≤ x ≤ 9/10. Ground states belonging to this family are shown in Figure 6. The M phases are distinct from the traditional O1/P3/O3 family of phases and their hybrids in that they consist of undulating CoO2 layers to host alternating strips of octahedrally- and prismatically-coordinated K within a single intercalation layer. The M phases emerge spontaneously during DFT energy relaxations from configurations that start in either O3 or P3 and have periodically spaced rows of vacancies (Supporting 10


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_

(a)

(b) ζ+ Family

ζ Family

(i)

(i)

_

x = 2/5

ζ2

*

(ii)

+

x = 8/15

ζ8

x = 6/11

ζ6

x = 5/9

ζ 4,6

(ii)

_

x = 7/17 ζ 2,2,3 (iii)

+

(iii)

_

ζ3

x = 3/7 (iv)

+

(iv)

_

x = 5/11

+ x = 4/7 ζ 4 /

ζ5

Figure 4: Ground state orderings belonging to (a) the ζ − family or (b) the ζ + family in P3, with (i-iv) showing distinct orderings from each family. APBs that separate strips of the ζ ordering (Figure 3a) are shown in black. Light and dark blue are used to distinguish K occupancy of the two triangular sublattices of the P3 honeycomb network. Pink boxes indicate the unit cells. The naming scheme is identical to that introduced for the ground state orderings of P3-Nax CoO2 . 44 The asterisk indicates that the ordering is above the global hull.

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(a)

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x = 4/7

η

(b)

x = 3/5

η+4,8

(c)

x = 20/33

η+4,4,4,8

(d)

x = 8/13

η+4,4

(e)

x = 7/11

η+4,4,2,4

(f)

x = 2/3

η+2,4 / Δ

Figure 5: The η + family of ground state orderings in P3 can be viewed as alternating strips of two orientational variants of the η ordering (a) separated by APBs. (b-f) show distinct ground state orderings belonging to this family. APBs that separate strips of the η ordering are shown as black lines. Light and dark colors are used to distinguish K occupancy of the two triangular sublattices of the P3 honeycomb network. Pink boxes indicate the unit cells. The naming scheme is identical to that introduced for the ground state orderings of P3-Nax CoO2 . 44

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Information Figure S1). K+ ions that are adjacent to vacant rows relax towards the vacant sites and, in the process, warp the CoO2 sheets until a mix of octahedrally- and prismatically-coordinated K emerges. In spite of the large amplitude undulations of the CoO2 sheets and distortion of the K coordination polyhedra, each K remains coordinated by six O atoms, as the seventh shortest K-O bond for each K is significantly longer than the six shortest K-O bonds (Supporting Information Figure S2). The undulations of the CoO2 sheets are strictly one-dimensional, occurring along the direction perpendicular to the rows of vacancies separating octahedral and prismatic domains. The existence of these undulations in the relaxed structures indicates that the strain energy penalty is outweighed by electrostatic repulsion between K+ ions. The stacking sequence of the oxygen sublattices in the M phases is distinct from those of P3 and O3, as shown by the top-down view of the M phase at x = 3/4 in Figure 7. In contrast to P3 and O3, where the triangular oxygen lattices above and below the K layer either adopt an AA or an AB stacking, respectively, the oxygen stacking in the M orderings is intermediate between the two. An examination of the in-plane K-K distances shows that the octahedrally-coordinated K lie directly in between the neighboring prismaticallycoordinated K (Supporting Information Figure S3), which further suggests that electrostatic repulsion between K drives the formation of the M phases. The symmetry of each M phase is dictated by the periodicity of the octahedral and prismatic domains. The relaxed lattice vectors and atomic positions of the M ground states, as well as an analysis of the various Co-O, K-O, and K-K distances are provided in the Supporting Information. The overall undulations of the CoO2 sheets are achieved via fluctuations of the individual Co-O bond lengths (Supporting Information Figure S4a,b). As shown in Figure 6, an increase in the K concentration leads to an increased wavelength of the undulations, with both the octahedral and prismatic domains becoming wider. At high K content, this results in mostly flat regions of either K coordination (octahedral or prismatic) with larger distortions of the CoO6 octahedra located at the kinks in between (Supporting Information 13



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Figure S4c).

(a)

(d)

x = 3/4 (b)

x = 6/7 (e)

x = 4/5 (c)

x = 7/8 (f)

x = 5/6

x = 8/9

Figure 6: (a-f) Side views of relaxed M ground state structures at select compositions, showing K in both octahedral (light blue) and trigonal prismatic (purple) coordinations between the undulating CoO2 layers. Structures are periodic into the page. Visualizations were created using VESTA. 53

Discussion Though K-ion batteries show promise as a viable alternative to Li-ion batteries for large scale energy storage applications, 14,16,18,20 very little is known about the thermodynamic and kinetic properties of common electrode materials when intercalated with K. The absence of

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Figure 7: Top-down view of the relaxed M ground state ordering at x = 3/4. Octahedrallyand prismatically-coordinated K are shown as light blue and purple, respectively. Red and orange denote O above and below the K occupation layer, respectively. Visualization was created using VESTA. 53 Jahn-Teller activity and the itinerant nature of the valence electrons of the CoO2 intercalation compounds 54 however makes layered Kx CoO2 an ideal model system with which to understand more complex layered K-intercalation compounds. Our first-principles study has elucidated favorable K ordering tendencies in the Kx CoO2 system as a function of K concentration over the O1/P3/O3 family of host structures. We have also discovered the stability of a new family of layered phases, collectively labeled M, in which the large K+ ions occupy both prismatic and octahedral sites in a single intercalation layer between undulating CoO2 layers. Our results indicate that stable P3 orderings in Kx CoO2 are composed of strips of wellordered domains separated by APBs. Such a description highlights three important P3 orderings (Figure 3). Ground state orderings belonging to the ζ − and ζ + families (Figure 4) for example consist of strips of the ζ ordering at x = 1/2 (Figure 3a) stitched together by APBs. Similarly, stable orderings in the η + family (Figure 5) are made up of strips of two orientational variants of the η ordering at x = 4/7 (Figure 3b) separated by APBs. Many of

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the P3 structures with such well-ordered domains separated by periodically-arranged APBs are stable and break the convex hull. While we have only considered a finite number of orderings due to size constraints, we expect many more ground states spaced closely along the K composition axis that consist of similar well-ordered domains separated by APBs. The particular ordering preferences in Kx CoO2 revealed in this study are expected to have important consequences on the kinetic mechanisms with which K can be inserted and removed from the P3 host. Diffusion in well-ordered phases is usually sluggish as cations prefer to remain locked in their sublattice positions. 55 K mobility within the well-ordered ζ or η domains is, therefore, likely to be low. Diffusion may instead occur along APBs as hypothesized for Nax TiS2 , 52 or it may occur through the collective motion of APBs passing through the crystal in waves. The precise mechanisms have yet to be established in these compounds, however, and will require kinetic Monte Carlo approaches combined with cluster expansion methods. 48,56–58 Ordering preferences in P3 Kx CoO2 are remarkably similar to those exhibited by P3 Nax CoO2 and Nax TiS2 . In particular, the P3 in-plane ground state orderings of Kx CoO2 are identical to those that have been predicted for Nax CoO2 , 44 and the special orderings at x = 1/2 and x = 4/7 are also predicted as ground states in Nax TiS2 . 52 The P3 host, while not energetically favored by small ions such as Li and Mg, is stable at intermediate compositions when intercalated by larger cations such as Na and K. This is in large part driven by in-plane electrostatic interactions, since the P3 honeycomb network offers more intercalation sites than the triangular network of O3. 22 The ζ ordering in P3 is especially favorable and is commonly observed as a stable phase in chemistries that adopt the P3 host. This is clearly evident in Kx CoO2 as reflected by the large voltage step at x = 1/2 in Figure 2b. A similar step height is predicted for the ζ ordering in Nax CoO2 ; however, the more covalent Nax TiS2 compound only exhibits a small voltage step at x = 1/2. In spite of the many similarities between Kx CoO2 and Nax CoO2 , there are also notable differences. Our results show that a new family of phases becomes stable in Kx CoO2 at

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high guest cation concentrations. Whereas P3 transforms directly to O3 with increasing Na concentration in Nax CoO2 , 6 a unique family of highly distorted M orderings are predicted in Kx CoO2 . The stability of such orderings, where K occupy both trigonal prismatic and octahedral sites in a single plane, is likely due to the large ionic radius of K, which increases the interlayer slab spacing. Electrostatic interactions are therefore expected to be stronger in Kx CoO2 than in Nax CoO2 , as screening by the negatively charged oxygen anions is limited by the large interlayer spacing between CoO2 sheets. 41 We also note that the kinds of vacancy row orderings that give rise to the M phases in Kx CoO2 are the predicted ground states in O3 Nax CoO2 , where they also lead to significant distortions of the host structure. 44 Our results show that the M phases are stable relative to O3 and P3 at high K concentrations. Layered O3 KCoO2 , however, has to our knowledge never been observed experimentally, and nonlayered structures have instead been synthesized at that composition. 59 Further studies are necessary to establish whether the M phases are globally stable relative to other phases not considered in this work. This would require a detailed exploration of phase stability over a vast, but still unchartered, domain in the ternary K-Co-O composition space. The positive voltages predicted for the M phases in this work (Figure 2b), however, indicate that they could form as metastable phases at room temperature during electrochemical K insertion into K2/3 CoO2 . The ability to access metastable phases electrochemically by removing or inserting mobile cations is not uncommon among intercalation compounds in which the kinetic barriers to decomposition into globally stable phases are sufficiently high. 54,60 The large strain modulations within the M phases suggest that their formation from P3 upon K insertion should be accompanied by large coherency strains. Cycling through the P3-M two-phase region is therefore likely to result in a large voltage hysteresis between charge and discharge. 61 Our results suggest that x = 2/3 is close to the composition at which the P3 → M transition will begin upon K insertion. While our study has been restricted to Kx CoO2 , we expect similar ordering tendencies in other K containing layered intercalation compounds such as Kx MnO2 21,23 and Kx CrO2 . 24,25

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Additional complexities may arise in the presence of Jahn-Teller active transition metal species such as Mn, Ni, and Fe due to the coupling between K ordering and the strain fields generated by cooperative or non-cooperative Jahn-Teller distortions. For the M phases, the presence of Jahn-Teller active cations may even be beneficial, as they could potentially mediate the large strains required to generate the undulating metal-oxide sheets to accommodate both octahedrally and prismatically coordinated K.

Conclusion We have performed a first principles study of phase stability and ordering preferences in the O3/P3/O1 family of Kx CoO2 layered intercalation compounds. While several host structures are found to be stable in this compound, P3 dominates the intermediate composition range. Our study reveals a multitude of P3 ground state orderings characterized by domains of highly-symmetric K orderings that are separated by antiphase boundaries. Three orderings are identified as especially important: the ζ ordering at x = 1/2, the η ordering at x = 4/7, and the ∆ ordering at x = 2/3. Our calculations also predict the stability of a highlydistorted family of phases, labeled M, at high K concentrations in Kx CoO2 . Phases belonging to this family of structures consist of undulating CoO2 sheets that result in a combination of both octahedrally- and prismatically-coordinated K sites within each intercalation layer. The stability of M phases is likely a result of the large ionic radius of K, which increases the slab distance between CoO2 sheets and thereby limits the degree with which oxygen anions can screen electrostatic interactions within the intercalation layers. We expect similar ordering tendencies in other layered compounds intercalated with K.

ASSOCIATED CONTENT Supporting Information Available: Additional figures detailing the various Co-O, K-O, and K-K distances within select M phases (PDF). Relaxed structure files for orderings on the 18


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local hull of each host structure (ZIP).

Acknowledgement This research was supported by the FLAM program of the National Science Foundation under Award no. DMR-1460656. J.L.K. acknowledges support from the United States Department of Energy through the Computational Science Graduate Fellowship (DOE CSGF) under grant number: DE-FG02-97ER25308. We acknowledge support from the Center for Scientic Computing from the CNSI, MRL: an NSF MRSEC (DMR-1720256).

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Potassium Ordering and Structural Phase Stability in Layered KxCoO2

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