Honeycomb Layered Oxides: Structure, Energy Storage, Transport, Topology and Relevant Insights
Godwill Mbiti Kanyolo, Titus Masese, Nami Matsubara, Chih-Yao Chen, Josef Rizell, Ola Kenji Forslund, Elisabetta Nocerino, Konstantinos Papadopoulos, Anton Zubayer, Minami Kato, Kohei Tada, Keigo Kubota, Hiroshi Senoh, Zhen-Dong Huang, Yasmine Sassa, Martin Mansson, Hajime Matsumoto
HHoneycomb Layered Oxides
Structure, Energy Storage, Transport, Topology and Relevant Insights
Godwill Mbiti Kanyolo, a Titus Masese, b , c Nami Matsubara, d Chih-Yao Chen, b Josef Rizell, e Ola Kenji Forslund, d ElisabettaNocerino, d Konstantinos Papadopoulos, e Anton Zubayer, d Minami Kato, c Kohei Tada, c Keigo Kubota, b , c Hiroshi Senoh, c Zhen-Dong Huang, f , Yasmine Sassa, e Martin Månsson d and Hajime Matsumoto ca Department of Engineering Science, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585,Japan. b AIST-Kyoto University Chemical Energy Materials Open Innovation Laboratory (ChEM-OIL), Sakyo-ku, Kyoto 606-8501,Japan. c Research Institute of Electrochemical Energy, National Institute of Advanced Industrial Science and Technology (AIST), 1-8-31Midorigaoka, Ikeda, Osaka 563-8577, Japan.
Email: [email protected] d Department of Applied Physics, Sustainable Materials Research & Technologies (SMaRT), KTH Royal Institute of Technology,SE-10691 Stockholm, Sweden. e Department of Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden. f Key Laboratory for Organic Electronics and Information Displays and Institute of Advanced Materials (IAM), Nanjing Univer-sity of Posts and Telecommunications (NUPT), Nanjing, 210023, China.
The advent of nanotechnology has hurtled the discovery and development of nanostructured materi-als with stellar chemical and physical functionalities in a bid to address issues in energy, environment,telecommunications and healthcare. In this quest, a class of two-dimensional layered materials consist-ing of alkali or coinage metal atoms sandwiched between slabs exclusively made of transition metal andchalcogen (or pnictogen) atoms arranged in a honeycomb fashion have emerged as materials exhibitingfascinatingly rich crystal chemistry, high-voltage electrochemistry, fast cation diffusion besides playinghost to varied exotic electromagnetic and topological phenomena. Currently, with a niche application inenergy storage as high-voltage materials, this class of honeycomb layered oxides serves as ideal peda-gogical exemplars of the innumerable capabilities of nanomaterials drawing immense interest in multi-ple fields ranging from materials science, solid-state chemistry, electrochemistry and condensed matterphysics. In this review, we delineate the relevant chemistry and physics of honeycomb layered oxides, anddiscuss their functionalities for tunable electrochemistry, superfast ionic conduction, electromagnetismand topology. Moreover, we elucidate the unexplored albeit vastly promising crystal chemistry spacewhilst outlining effective ways to identify regions within this compositional space, particularly whereinteresting electromagnetic and topological properties could be lurking within the aforementioned al-kali and coinage-metal honeycomb layered oxide structures. We conclude by pointing towards possiblefuture research directions, particularly the prospective realisation of Kitaev-Heisenberg-Dzyaloshinskii-Moriya interactions with single crystals and Floquet theory in closely-related honeycomb layered oxidematerials. a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec eview Article: Honeycomb Layered Oxides
1. Introduction
Charles Darwin famously described the honeycomb as an engineering masterpiece that is “absolutely perfect ineconomising labour and wax”. For over two millennia, scientists and philosophers alike have found a great dealof fascination in the honeycomb structures found in honeybee hives. These hexagonal prismatic wax cells builtby honey bees to nest their larvae, store honey and preserve pollen are revered as a feat in precision engineeringand admired for their elegance in geometry. The honeycomb framework offers a rich tapestry of qualitiesadopted in myriads of fields such as mechanical engineering, architectural design biomedical engineering etcetera (as briefly outlined in
Fig. 1 ). Figure 1
Schematic illustration of the various realisations of the honeycomb structure found not only in energy storagematerials, but also as pedagogical models in condensed-matter physics, solid-state chemistry and extending to tissueengineering. Specific varieties of fungi ( videlicet , Morchella esculenta ) tend to adopt honeycomb-like structures, whilstinsects such as the fruit flies ( Drosophila melanogaster ) have their wing cells in honeycomb configuration; thus endow-ing them with excellent rigidity. In addition, the honeycomb whip ray (
Himantura undulata ) and the honeycomb cowfish( Acanthostracion polygonius ) have honeycomb patterns on their body that are thought to aid in their facile movement andcamouflage. Section 1 INTRODUCTION
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Figure 2
Illustration of the various sections to be covered in this review. This starts from solid-state chemistry, physics,electrochemistry to solid-state ionics. We finally adumbrate on the challenges and perspectives of these honeycomblayered oxides.
The discovery and isolation of graphene in 2004, not only revolutionised our understanding of two-dimensionalmaterials but also unveiled a new platform for fabricating novel materials with customised functionalities.
Two-dimensional (2D) nanostructures are crystalline systems comprising covalently bonded atom cells in pla-nar arrangement of mesoscopic thicknesses. Due to their small size, this class of materials exhibits highlycontrolled and unique optical, magnetic, or catalytic properties.
By substituting the constituent atoms andmanipulating the atomic cell configurations, materials with remarkable physicochemical properties such as highelectron mobility, unique optical and chemical functionality can be tailor-made for various technological realmssuch as catalysts, superconductivity, sensor applications and energy storage. Assembling such materials intovertical stacks of layered combinations allows the insertion of atoms within the stacks creating a new classof multi-layered heterostructures. A unique characteristic of these structures is that inter-layer bonds holdingtogether the thin 2D films are significantly weaker (Van der Waals bonds) than the covalent bonds within the2D monolayers, which further augments the possibility for emergent properties exclusive to these materials.In electrochemistry, layered frameworks composed of alkali or coinage metal atoms interposed between 2Dsheets of hexagonal (honeycomb) transition metal and chalcogen (or pnictogen) oxide octahedra have foundSection 1 INTRODUCTION
Page 3 eview Article: Honeycomb Layered Oxidesgreat utility as next-generation cathode materials for capacious rechargeable battery systems.
The weakinterlayer bonds between transition metal slabs facilitate facile mobility of intercalated atoms during the bat-tery operation (de)insertion processes, endowing these heterostructures with ultrafast ionic diffusion render-ing them exemplar high energy (and power) density cathode materials. Furthermore, the unique topologicalchanges occurring during these electrochemical processes have been seen to induce new domains of physicsentailing enigmatic optical, electromagnetic and quantum properties that promise to open new paradigms ofcomputational techniques and theories quintessential in the field of quantum material science catapulting thediscovery of materials with novel functionalities.
This has created an entirely new platform of study, en-compassing fields, inter alia , materials science, solid-state chemistry, electrochemistry and condensed matterphysics.
Although layered oxides encompass a broad class of materials with diverse structural frameworks and var-ied emergent properties, in this review, we focus on the stellar properties innate in the aforementioned classof honeycomb layered oxides comprising alkali or coinage metal atoms sandwiched between slabs consisting oftransition metal oxide octahedra surrounding chalcogen or pnictogen oxide octahedra in a honeycomb config-uration. To provide deeper insights, we delineate the fundamental chemistry underlying their material designalong with emergent domains of physics. We further highlight their functionalities for tunable electrochemistry,superfast ionic conduction, electromagnetism and topology, as illustrated in
Fig. 2 . Moreover, we highlightthe unexplored albeit vastly promising crystal chemistry space whilst outlining effective ways to identify re-gions within this compositional space, particularly where potentially interesting electromagnetic and topologi-cal properties could be lurking within the aforementioned honeycomb layered oxides. The looming challengesare also discussed with respect to the governing chemistries surrounding honeycomb layered oxides. Finally,we conclude by pointing towards possible future research directions, particularly the prospective realisationof Kitaev-Heisenberg-Dzyaloshinskii-Moriya interactions in closely-related honeycomb layered oxide materials,and their connection to Floquet theory and fabrication efforts that are not limited to single crystals.Section 1 INTRODUCTION
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2. Materials chemistry of honeycomb layered oxides
The aforementioned honeycomb layered oxides generally adopt the following chemical compositions, takinginto account that charge electro-neutrality is maintained and assuming ordered structures: A + M + D + O − (or equivalently as A + / M + / D + / O − ), A + M + D + O − ( A + M + / D + / O − ), A + . M + . D + O − ( A + / M + / D + / O − ), A + M + D + O − ( A + / M + / D + / O − ), A + D + O − ( A + / D + / O − ), A + A (cid:48) + M + D + O − ( A + A (cid:48) + / M + / D + / O − ), A + M + D + O − ( A + / M + / D + / O − or A + M + D + O − ), amongst others ( Fig. 3a ). Here M denotes transition metal atoms such as Ni , Co , Mn , Fe , Cu , Zn , Cr (including Mg ); D denotes Te , Sb , Bi , Nb , Ta , W , Ru , Ir , Os ; A and A (cid:48) denote alkali atoms such as Li , Na , K or coinage atoms like Cu and Ag (with A (cid:54) = A (cid:48) ). Itis worth mentioning that the honeycomb layered framework is not limited to compositions entailing only onespecies of alkali atoms. Oxides compositions comprising mixed-alkali atoms such as Na LiFeSbO , Na LiFeTeO , Ag LiRu O , Ag NaFeSbO , Ag LiIr O , Ag Li M TeO ( M = Co , Ni ), Ag Li M SbO ( M = Cr , Mn , Fe ) and, morerecently, Li − x Na x Ni SbO as well as oxides with off-stoichiometric compositions such as Li Co . TeO havebeen explored with the aim of merging favourable attributes from multiple species to improve various ma-terial functionalities, for instance, battery performance. Other atypical honeycomb layered oxidecompositions are those encompassing alkali or mixed-alkali species atoms embedded within the transitionmetal slabs. Compositions such as Li x M y Mn − y O ( < x < ; < y < ; M = Li , LiNi ), Na / ( Li / Mn / ) O , Na / ( Li / Mn / ) O , Na / ( Li / Mn / ) O , et cetera have also been considered for use as high-performance bat-tery materials, and can be envisaged to exhibit unique three-dimensional diffusion as they have someof their alkali atoms reside in the honeycomb layers. Although three-dimensional diffusion is covered in a latersection, it is pertinent to note that this review will mainly focus on two-dimensional diffusion of cations inrelation to topotactic curvature evolutions. Other classes of honeycomb layered oxide frameworks with vastlydifferent properties and applications are also beyond the scope of this review. High-temperature solid-state synthesis is often considered an expedient route to synthesise most of the abovementioned honeycomb layered oxides because their initial precursor materials usually require high tempera-tures to activate the diffusion of individual atoms.
In this technique, precursors are mixed in stoichiometricamounts and pelletised to increase the contact surface area of these reactants. Finally, they are fired at hightemperatures (over 700 ◦ C) resulting in thermodynamically-stable honeycomb layered structures. The firingenvironment (argon, nitrogen, air, oxygen, carbon monoxide, hydrogen, et cetera ) needs to be adequately con-trolled to obtain materials with the desired oxidation states of transition metals. For example, an inert firingenvironment is demanded for layered oxides that contain Mn + and Fe + ; otherwise oxidised samples contain-ing Mn + and Fe + are essentially formed. Nonetheless, not all high-temperature synthesis processes are asrestrictive, as compositions containing Ni + , such as A Ni TeO ( A = Li , Na , K , et cetera .) can be synthesisedunder air to obtain samples that contain Ni still in the divalent state. Moreover, varying the synthesis protocols(annealing temperature and time, types of precursors, thermal ramp rate, et cetera ) during high-temperaturesynthesis not only aid in enhancing the scalability of the synthesis but also gives rise to the possibility of ob-taining new polymorphs (or polytypes), as has been shown in the high-temperature synthesis of Na Ni BiO , Na NiTeO and Na Ni SbO . The topochemical ion-exchange synthesis route is also possible for honeycomb layered oxides with accessiblekinetically-metastable phases.
Despite the high binding strength amongst adjacent atoms within thehoneycomb slab, the use of cations with higher charge-to-radius ratio such as Li + in LiNO can drive out the Na + atoms present in Na Cu TeO by lowering their electrostatic energy to create Li Cu TeO . Here, the two pre-Section 2 MATERIALS CHEMISTRY OF HONEYCOMB LAYERED OXIDES
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Figure 3
Combination of elements that constitute materials exhibiting the honeycomb layered structure. (a) Choiceof elements for layered oxide compositions (such as A + M + D + O ( A + / M + / D + / O ), A + M + D + O ( A + M + / D + / O ), etcetera .) that can adopt honeycomb configuration of transition metal atoms. Inset shows a polyhedral view of the crystalstructure of layered honeycomb oxides, with the alkali atoms (shown as brown spheres) sandwiched between honeycombslabs (blue). (b) X-ray diffraction (XRD) pattern of K Ni TeO (12.5% cobalt-doped) honeycomb layered oxide. Inset:Slab of layered oxide showing the honeycomb arrangement of magnetic nickel (Ni) atoms around non-magnetic tellurium(Te) atoms. Dashed line highlights the unit cell. (b) adapted from ref. 92 with permission (Creative Commons licence4.0).
Section 2 MATERIALS CHEMISTRY OF HONEYCOMB LAYERED OXIDES
Page 6 eview Article: Honeycomb Layered Oxidescursors are heated together at a moderate temperature (300 ◦ C) triggering the diffusion of Na + and Li + . Otheroxides that can be prepared via the ion-exchange route include Li Co SbO , Ag M SbO ( M = Ni , Co and Zn ), Li Ni TeO , Ag LiGaSbO , Ag LiAlSbO , Ag Ni BiO and Li M SbO ( M = Fe and Mn ). However, itis worthy to recapitulate that there exists exemplars of compounds that can be synthesised topochemically suchas Ag Co SbO and Ag Li D O ( D = Ru , Ir ) that are exceptions to the rule that ion exchange can only happenfrom ions with lower charge-to-radius ratio to those with higher ratios (taking into account Ag has a larger ionicradius than Li ) . The syntheses of these honeycomb layered oxides are typically done at ambient pres-sures; however, high-pressure syntheses routes remain unexplored, a pursuit which may expand their materialplatforms. Equally important is the utilisation of low-temperature routes such as sol-gel and mechanochemicalsynthesis since they do not require apriori high-temperatures to attain thermodynamically-stable phases. Al-though not delved in the scope of this review, honeycomb layered oxides entailing alkaline-metal atoms as the Figure 4 (a) High-resolution transmission electron microscopy (TEM) image of a crystallite of K Ni TeO (12.5% cobalt-doped) honeycomb layered oxide and (b) Corresponding electron diffractograms taken along the [001] zone axis. (c)Visualisation (along the c -axis [001]) of the honeycomb configuration of Ni atoms around Te atoms (in brighter contrast)using High-Angle Annular Dark-Field Scanning TEM (HAADF-STEM). Dashed lines indicate the unit cell. (d) STEMimaging with Ni atoms (partially with Co ) (in green) assuming a honeycomb fashion (as highlighted in dashed lines) and(e) STEM imaging showing Te atoms (in red) surrounded by transition metal atoms. (f) Annular Bright-Field TEM (ABF-TEM) of segments manifesting potassium atoms (in brown) assuming a honeycomb fashion and overlapped with oxygenatoms. Note that some portions of the honeycomb ordering of transition metal atoms slightly appear obfuscated, owingto sensitivity of the samples to long-time beam exposure.(a, b) reproduced and adapted from ref. 92 under CreativeCommons licence 4.0. Section 2 MATERIALS CHEMISTRY OF HONEYCOMB LAYERED OXIDES
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Figure 5
Summary of the various stacking sequences adopted by representative honeycomb layered oxides includingthose that entail pnictogen or chalcogen atoms in the slab layers. Note here that ‘T’, ‘O’ and ‘P’ denote the coordinationof the alkali or coinage metal atoms (sandwiched between the honeycomb slabs) with the adjacent oxygen atoms of thehoneycomb slab, id est , tetrahedral, octahedral and prismatic coordination, respectively. The numbers or digits (‘1’, ‘2’and ‘3’) indicate the repetitive alkali-atom layers per unit cell, as denoted in Hagenmuller-Delmas’ notation. Note thatthe predominant stacking sequences have been shown, for ease of explanation. A more elaborate list of the stackingsequences for other honeycomb layered oxides reported thus far are provided in Table 2. resident cations such as
SrRu O and BaRu O can be prepared via a low-temperature hydrothermal synthesisroute. Finally, despite the scarce exploration of electrochemical ion-exchange of honeycomblayered oxide materials, invaluable results on the K + / Na + ion-exchange process in Na Ni SbO has been re-cently reported, demonstrating this process as a promising route to pursue. To ascertain the crystal structure of honeycomb layered oxides and discern the precise location of the constituentatoms, transmission electron microscopy (TEM), neutron diffraction (ND) and X-ray diffraction (XRD) analysescan be performed on single-crystals or polycrystalline samples. Although the XRD is the most commonly usedcrystallography technique, it is ineffective in analysing oxides composed of lighter atoms such as Li , H , and B due to their low scattering intensity. Also, honeycomb layered oxides with elements of similar atomic numberare difficult to distinguish using XRD because they diffract with similar intensity.To distinguish light elements or elements with close atomic numbers on honeycomb layered oxides, the NDis used because the neutron beam interacts directly with the nucleus, hence the ability to observe light elements.In spite of the high accuracy, the equipment remain very expensive and ND experiments require the use of verylarge sample amounts to obtain high-resolution data- an impediment to materials that can only be prepared ona small scale.Section 2 MATERIALS CHEMISTRY OF HONEYCOMB LAYERED OXIDES Page 8 eview Article: Honeycomb Layered Oxides
Table 1
Stacking sequences adopted by a smorgasbord of honeycomb layered oxides hitherto re-ported.
Coordination of alkali atoms Hagenmuller-Delmas’ notation Honeycomb layered oxide compositionswith oxygen (slab stacking sequence)
Tetrahedral T2 Li Ni TeO Octahedral O1
NaNi BiO − δ , Na RuO , Li MO ( M = Ni , Pt , Rh ), BaRu O O2 Li MnO , Li x ( Li / Ni / Mn / ) O ( x < ), Li x ( Li / Mn / ) O , Na ZrO , Na SnO , Li RuO , SrRu O O3 Li Ni TeO , Li Ni BiO , Na Ni BiO , Na Cu TeO , Na M SbO ( M = Ni , Cu , Cr , Co , Mg , Zn ), Li Cu TeO Li M SbO ( M = Ni , Cu , Co , Zn ), NaRuO , Li MnO , Li ( Li / Ni / Mn / ) O , Na / ( Ni / Sn / ) O , Na x Ni x / Mn − x / O ( ≤ x < / ), Na LiFeSbO , Li MTeO ( M = Co , Ni , Cu , Zn ), Li M SbO ( M = Cr , Fe , Al , Ga , Mn )Prismatic P2 K Ni TeO , Na M TeO ( M = Mg , Zn , Co , Ni ) Na / ( Li / Mn / ) O , Na / ( Mg / Mn / ) O , BaRu O , Na / ( Li / Mn / ) O , Na / ( Li / Mn / ) O , Na / ( Mg / Mn / ) O , Na / ( Ni / Mn / ) O , Li RuO , Na x Ni x / Mn − x / O ( / ≤ x < / ), Na / Ni / Mn / O ,P3 Na / ( Mg / Mn / ) O , Na / ( Ni / Mn / ) O Dumbbell D Ag NaFeSbO , Ag Li M TeO ( M = Co , Ni ),(linear) Ag Li M SbO ( M = Cr , Mn , Fe ), Ag Li M O ( M = Ir , Ru , Rh ), A M D O ( M = Ni , Co , Mg , Zn , Mn ; A = Cu , Ag ; D = Bi , Sb ) Although, XRD analyses accurately validate the precise crystal structure of honeycomb oxides with heavyelements such as K Ni TeO (partially doped with Co ), as shown in Fig. 3b , TEM is used to obtain unequivocalinformation relating to the structure of materials at the atomic scale. A number of studies have reported theutilisation of TEM analyses on honeycomb layers of oxides to determine, with high-precision, the arrangementof atoms within the honeycomb lattice and the global order of atoms within the structure of materials in a hon-eycomb lattice.
Likewise, the honeycomb lattice comprising Te surrounded by transitionmetals in K Ni TeO ( Fig. 3b ) can be seen from state-of-the-art TEM images, shown in
Fig. 4 . It is worth notingthat TEM analyses are expensive to conduct and may lead to damage of samples because of the strong electronbeam. We also note that the sensitivity of samples to the electron beam differ even within slightly the samehoneycomb layered oxide composition. For instance, Cu Co SbO is more susceptible to electron-beam dam-age than Cu Ni SbO , which implies that tuning of the chemical composition of these materials can inducestructural stability necessary to perform intensive TEM analyses. In a notation system promulgated by Hagenmuller, Delmas and co-workers, honeycomb layered oxides canalso be classified according to the arrangement of their honeycomb layers (stackings) The notation comprises aletter to represent the bond coordination of A alkali, alkaline-earth or coinage metal atoms with the surroundingoxygen atoms (generally, ‘ T ’ for tetrahedral, ‘ O ’ for octahedral, or ‘ P ’ for prismatic) and a numeral that indicatesthe number of repetitive honeycomb layers (slabs) per each unit cell (mainly, ‘1’, ‘2’ or ‘3’) as shown in Table1 . For instance, Na M TeO (with M being Mg , Zn , Co or Ni ) possess P2-type structures, the nomenclatureSection 2 MATERIALS CHEMISTRY OF HONEYCOMB LAYERED OXIDES Page 9 eview Article: Honeycomb Layered Oxidesarises from their repetitive two-honeycomb layers sequence in the unit cell with prismatic coordination of Naatoms with oxygen in the interlayer region.
Structures such as O3-type stackings can be found in Na M SbO (here M = Zn , Ni , Mg or Cu ) and Na LiFeTeO , whereas Na Ni SbO and Na Ni BiO reveal O1-type and P3-type stackings during the electrochemical extraction of alkali Na atoms. Note that theaforementioned oxide compositions are representative of the main stackings observed, and is by no means, anexhaustive summary. Ag- and Cu-based honeycomb layered oxides, prepared via topochemical ion-exchange,such as A M D O ( A = Ag , Cu ; M = Ni , Mn , Co , Zn ; D = Bi , Sb ) and related oxides, adopt a linear (dumbbell-like) coordination of alkali or coinage metal atoms with the adjacent two oxygen atoms with an intricatemultiple stacking sequence of the honeycomb slabs. The various stacking sequences exhibited byrepresentative honeycomb layered oxides are detailed in
Fig. 5 and
Table 1 . The stacking sequences of the honeycomb slabs as well as the emplacement patterns of the oxygen atoms play acrucial role in the nature of emergent properties; even minuscule differences in atomistic placements could re-sult in distinct variations of crystal frameworks with an assortment of physochemical properties.
In general, the various manner of stackings observed in honeycomb layered oxides is contingent on the synthe-sis procedure, the content of alkali A atoms sandwiched between the honeycomb slabs and the nature of alkali A cations (that is, Li , Na , K and so forth). Different stacking sequences of the honeycomb slabs are observedin, for example, honeycomb layered oxides that comprise Na and Li atoms. Na atoms, with larger radii, tendto have a strong affinity to coordinate with six oxygen atoms; adopting octahedral (O) or prismatic (P) coor-dination. Li atoms, vide infra , have been found to possess tetrahedral (T) and octahedral coordination, asrecently observed in Li Ni TeO . Further, TEM analyses performed on oxides such as Na Ni BiO , indicate as-sorted sequences of honeycomb ordering. Using high-angle annular dark-field scanning transmission electronmicroscopy (HAADF-STEM) imaging studies, Khalifah and co-workers have broached another labeling schemeto allow the indication of the number of repetitive honeycomb layers. Using their notation, they illustratedthat Na Ni BiO had 6 layers (6L), 9 layers (9L) and 12 layers (12L) of stacking honeycomb ordering sequence(periodicity). Such sequence of honeycomb ordering (and even stacking disorder) can be influenced by thereaction kinetics during the use of various synthesis conditions and higher orders of stacking sequences (4L, 6L,9L, 12L, et cetera .) can be anticipated in honeycomb layered oxides.Section 2 MATERIALS CHEMISTRY OF HONEYCOMB LAYERED OXIDES Page 10 eview Article: Honeycomb Layered Oxides
3. Magneto-spin models of honeycomb layered oxides
As aforementioned, honeycomb layered oxides mainly comprise alkali cations A + sandwiched in a frameworkcontaining layers or slabs of M and D atoms coordinated, octahedrally, with oxygen atoms. M atoms are es-sentially magnetic with a valency of + or + , whilst D atoms are non-magnetic and generally possess valencystates (oxidation states or oxidation numbers) of + , + or + . The M O and D O octahedra assume a honey-comb configuration within the layers; D O octahedra being surrounded by six M O octahedra, as shown in Fig.6a . Note that such an ordered honeycomb configuration of magnetic M atoms around non-magnetic D atomis contingent on their ionic radii. For instance, for honeycomb layered oxides with Te (or even Bi ), asthe D atoms and transition metal atoms such as Ni , M atoms, typically form ordered honeycomb configurationsin oxides such as Na Ni TeO , Na NiTeO , Na Ni BiO and, more recently, K Ni TeO . .However there is slight disorder between Te and surrounding metal atoms within the honeycomb slab as notedin Na Zn TeO . On the other hand, in honeycomb layered oxides with Sb as the D metal atoms, such as Na Ni SbO , disordered honeycomb configurations are often observed. This is due to the movement of Sb ( D ) atoms to the sites of Ni ( M ) atoms, which have similar ionic radii, a phenomenon commonly referred toas ‘cationic site mixing’. Also worthy of mention, is that the ionic radii of the sandwiched A atoms in honeycomblayered oxides has influence on the honeycomb ordering. This has been noted in Li -based honeycomb layeredoxides such as Li NiTeO and Li Ni TeO , whereby Li atoms are located in the sites of Ni atoms. Here-after, magnetism of honeycomb layered oxides with ordered honeycomb configurations of magnetic M atomsaround D atoms shall be discussed, to serve as an entry point to some fundamental models of magneto-spinphenomena that have generated tremendous research interest in recent years.The honeycomb arrangement of magnetic metal atoms ( M ) within the slabs of honeycomb layered oxidesoften leads to fascinating magnetic behaviour. This is due to the interactions generated from the spins innatein the magnetic atoms (what is commonly termed as magnetic coupling). As is explicitly shown in Fig. 6a ,such interactions primarily originate from spins from the adjacent magnetic atoms (Kitaev-type interactions(denoted as J )) within the honeycomb lattice, but they can also be influenced by spins of magnetic atomsfrom adjoining layers in the honeycomb configuration ( id est , Haldane-type interactions ( J )). Spin interactionsemanating from distant atoms may still occur and shall herein be classified as higher-order interactions ( J ). Such magnetic interactions can be of varied fashion, spanning over short distances across the honeycomb lattice(what is termed as short-range interactions) or long distances extending to those of the adjacent honeycombslabs (long-range interactions).
In the case of J (type) interactions, the spin-spin interactions can either be anisotropic (Kitaev) or isotropic(Heisenberg) leading to an interaction Hamiltonian of the form, H = ∑ (cid:104) j , k (cid:105)∈ x K x S xj S xk + ∑ (cid:104) j , k (cid:105)∈ x K y S yj S yk + ∑ (cid:104) j , k (cid:105)∈ z K z S zj S zk + J ∑ (cid:104) j , k (cid:105)∈ γ (cid:126) S j · (cid:126) S k , (1a)where γ = x , y , z , K x , y , z are the Kitaev interaction terms shown in Fig. 6b , J is the Heisenberg interaction termand (cid:126) S j = ( S xj , S yj , S zj ) are the spin matrices. The sum is taken over next-neighbour interaction sites correspondingto J interactions. In a seminal paper, Kitaev showed that eq. (1a) with J = is exactly soluble into aground state of a topological superconductor in terms of a Kitaev-quantum spin liquid (K-QSL). K-QSL isa spin quantum state with long-range entanglement and short-range order of spin moments which continue tofluctuate coherently whilst still maintaining their disordered formation even at low temperatures.Section 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES
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Figure 6
Nature of magnetic configurations adopted by honeycomb layered oxide materials. (a) Fragment of the tran-sition metal slab showing the honeycomb configuration of transition metal atoms and the possible spin interactions withneighbouring magnetic atoms. Here J i (where i = , and ) represents the magnetic exchange interactions (Kitaev,Haldane and higher-order interactions respectively) between an atom and its i -th neighbour. Transition metal atoms (inpurple) are depicted surrounded by oxygen atoms (in red) in octahedral coordinations. (b) A schematic of the realisationof the Kitaev model. K x , K y and K z denote the Kitaev coupling constants for the next-neighbouring transition metal bondsin the axes (depicted in red, blue and green, respectively). W h denotes the plaquette flux operator with links labelledas ‘1’, ‘2’, ‘3’, ‘4’, ‘5’ and ‘6’. The blue ( t , ) or black ( t ) double-headed arrow indicates the hopping path of an electronfrom one transition metal atom to an adjacent one directly or indirectly (via a shared oxygen atom) respectively along thespin-directional bonds of the honeycomb lattice. (c) The solution of the Kitaev model depicted as a phase diagram in theform of a triangle showing the spectrum constraints in the Brillouin zones as either gapped or gapless. The vertices ofthe triangle correspond to | K z | = , | K x | = | K y | = , | K x | = , | K y | = | K z | = and | K y | = , | K x | = | K z | = . In particular, the K-QSL Hamiltonian has a conserved quantity W h = ( S x S y S z S x S y S z ) h for a given (honeycomb)plaquette h , id est as shown in Fig. 6b . Each plaquette satisfies W = ( W h = ± ) and the commutator [ H ( J = ) , W h ] = i ∂ W h / ∂ t = in the Heisenberg picture of quantum mechanics, which immensely simplifiescalculations of the ground state properties such as spin-spin correlation functions. For instance, since W h and H ( J = ) can be simultaneously diagonalised, the ground state corresponds to W h = for all plaquettes (knownas the vortex-free state) whereas flipping any one plaquette to W h = − corresponds to the lowest excited energystate (single vortex state) with a finite energy gap. Moreover, W h commutes with both spin operators S γ j and S γ (cid:48) k Section 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES
Page 12 eview Article: Honeycomb Layered Oxidesat different sites j and k if and only if γ = γ (cid:48) and k = j ± within a given plaquette. Otherwise, one can find W h that commutes with either one of the operators but anti-commutes with the other one. Thus, using the cyclicproperty of the trace (cid:104) abc (cid:105) = (cid:104) bca (cid:105) = (cid:104) cab (cid:105) where a = S γ j S γ (cid:48) k , b = W h , c = W h , one can show that there is no long-range spin-spin correlation in the honeycomb lattice, id est (cid:104) S γ j S γ (cid:48) k (cid:105) = (cid:104) S γ j S γ (cid:48) k W (cid:105) = (cid:104) W h S γ j S γ (cid:48) k W h (cid:105) = (cid:104) S γ j W h S γ (cid:48) k W h (cid:105) = −(cid:104) S γ j W h S γ (cid:48) k W h (cid:105) = −(cid:104) S γ j S γ (cid:48) k W (cid:105) = −(cid:104) S γ j S γ (cid:48) k (cid:105) = . This is the hallmark of a quantum spin liquid.
The energy spectrum of the model is exactly soluble by mapping the spin operators in H ( J = ) to Majo-rana fermions operators by a Jordan-Wigner transformation. (Majorana fermions are uncharged propagatingdegrees of freedom with quantum statistics described by anti-commuting self-adjoint quantum operators thatsquare to ). This yields the phase diagram in the form of the triangle shown in Fig. 6c for half the Brillouinzone, where for | K x | < | K y | + | K z | , | K y | < | K x | + | K z | and | K z | < | K x | + | K y | under the constraint | K x | + | K y | + | K z | = ,the spectrum is gapless whilst the rest of the Brillouin zone is gapped. The vertices of the triangle correspondto | K z | = , | K x | = | K y | = , | K x | = , | K y | = | K z | = and | K y | = , | K x | = | K z | = .In the gapped phase, spin correlations decay exponentially over a length scale inversely proportional to thegap. Thus, fermionic, vortex or quasi-particle (fermion + vortex) excitations do not have long-range inter-actions. However, they can interact topologically as they move around each other (their worldlines in three-dimensional (3D) space satisfy specific braiding rules). Thus, the energy gap favours topological inter-actions which reveal their Abelian anyonic statistics ( id est , the quantum phases acquired by particle exchangeis additive). This carries major implications in topological quantum computing as it suggests the possibility ofachieving the stabilisation of quantum bits (qubits), considering that any such uncharged system is extremelyhard to disturb, exempli gratia by a photon, because photons do not couple to uncharged quasi-particles. More-over, the quantum state is topologically protected by the energy gap which cuts off any other non-topologicalinteractions. Hence, quantum computation can be carried out topologically by particle exchange exploiting thebraiding rules.
On the other hand, the quasi-particles in the gapless phase will have long-range interactions. Nonetheless,these can be suppressed by introducing a gap via a time-reversal symmetry breaking term H int = − ∑ γ ∑ j h γ S γ j in Hamiltonian H ( J = ) in eq. (1a) , corresponding to the interaction of the spins with an external magneticfield h γ = ( h x , h y , h z ) . Since this term destroys the exact solubility of the Kitaev model, the model is solvedperturbatively to yield a gap ∆ ∼ h x h y h z / K , where K ≡ K x = K y = K z . Such a gapped phase admits non-Abelian quasi-particles, that are even more robust for quantum computing than the Abelian quasi-particles ( idest , the quantum phase acquired by the wavefunction under the exchange of quasi-particles is non-additive).
Based on the robustness of the ungapped phase ( K ≡ K x = K y = K z ) under a finite magnetic field ( (cid:126) h (cid:54) = ) forquantum computing, it is illustrative to consider the behaviour of this ungapped phase in eq. (1a) with a finiteHeisenberg term ( J ≡ J (cid:54) = ) and no magnetic field ( (cid:126) h = ), H KH = K ∑ (cid:104) j , k (cid:105)∈ γ S γ j S γ k + J ∑ (cid:104) j , k (cid:105)∈ γ (cid:126) S j · (cid:126) S k . (1b)This Hamiltonian has been diagonalised using a 24-site diagonalisation method by Chaloupka, Jackeli andKhaliullin, through a parametrisation κ = √ K + J and J = κ cos ϕ and K = κ sin ϕ , to yield 1) a phase dia-gram of the Kitaev-Heisenberg model with two distinct gapless K-QSL (disordered) phases around ϕ = π / and ϕ = π / where J (cid:39) , and 2) four ordered phases given by Néel antiferromagnetic, ferromagnetic, zigzag andstripy phases, as shown in Fig. 7 . Distinct phase transitions between two phases occur at peaks correspondingto − ∂ E / ∂ ϕ , where E is the ground-state energy of the Kitaev-Heisenberg model, with the smallest peaksSection 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES Page 13 eview Article: Honeycomb Layered Oxides
Figure 7
Various spin configurations that can be realised in honeycomb layered frameworks, described by the Kitaev-Heisenberg Hamiltonian ( H KH ) in eq. (1b) with coupling constants parametrised by angular variable ϕ , entailing magneticatoms (transition metals that are magnetic such as Ni , Fe , Co et cetera ), based on the phase diagram of the Kitaev-Heisenberg model. Phases of the Kitaev-Heisenberg model (ferromagnetic, stripy, zigzag, Néel ferromagnetic, et cetera )are shown with the direction of the electron spins in the honeycomb lattice. The green arrows show the spin-up, whereasthe brown arrows show spin-down alignment of the magnetic moment of the transition metal atoms. occurring at two ordered-disordered phase boundaries, signifying a relatively smoother transition compared tothe typical ordered-ordered phase transitions.Achieving the Hamiltonian given in eq. (1b) in a condensed matter system with J = is considered tobe the Holy Grail of topological quantum computing. Honeycomb layered oxides consisting of alkalior coinage metal atoms sandwiched between slabs exclusively made of transition metal and chalcogen (orpnictogen) atoms, which are the prime focus of this review, typically exhibit Néel antiferromagnetic proper-ties, which suggests that the Heisenberg term ( J ) dominates over the Kitaevterm ( K ). Nonetheless, observing the K-QSL phase in honeycomb layered oxides based on magnetic transitionatoms with 3 d orbitals such as M = Co , has shown great promise due to the localised nature of the magnetic elec-trons which favour the charge-transfer phase of the Mott insulator – a prerequisite for the realisation of theSection 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES Page 14 eview Article: Honeycomb Layered OxidesKitaev-Heisenberg model. For instance Na Co TeO , has been shown to be favoured by the suppression of non-Kitaev interactions with rather moderate external magnetic fields, since external magnetic fields vastly suppressthe Haldane-type and higher-order interactions. Likewise, Na Co SbO has shown a rather promisingroute to the realisation of K-QSL via strain or pressure control. Conversely, it is also possible to suppress theKitaev-type interactions instead by designing the honeycomb lattice to be composed of alternating magneticand non-magnetic atoms, leaving only Haldane-type interactions corresponding to additional interaction termsin the Hamiltonian; effectively attaining a quantum anomalous Hall insulator (or also referred to as a Cherninsulator).
Based on the immense interest generated by the so-called Kitaev materials, we shall en-deavour to briefly digress into this different class of honeycomb layered oxides, namely the honeycomb iridiumoxides (iridates) embodied by A IrO with A = Na , Li or Cu and other related compounds such as RuCl , which also hold promise to realising the Kitaev-Heisenberg Hamiltonian given in eq. (1b) . Thesematerials exist in different phases (polymorphs), often labelled with a prefix α − , β − or γ − signifying the lay-ering of the honeycomb sublattices which facilitate the existence of stacking faults. For instance, Li IrO ispolymorphic with three distinct phases, α − Li IrO , β − Li IrO and γ − Li IrO which adopt the honeycomb,hyper-honeycomb and stripy honeycomb crystal structures, respectively. α − Li IrO exhibitsNéel antiferromagnetic behaviour below their Néel temperature ( K), and is paramagnetic above 15 K.
Itscrystalline structure consists of
IrO octahedra arranged in a honeycomb fashion and separated by Li + cations,with other Li + cations situated within the honeycomb slab. On the other hand, RuCl also exists in two mainpolymorphic phases, α − RuCl and β − RuCl . However, β − RuCl is prepared under specific conditions at lowtemperatures, and irreversibly reverts to the more stable polymorphic counterpart, α − RuCl at approximately500 ◦ C. The
RuCl layers consist of RuCl octahedra linked by van der Waal forces forming interlayers devoidof any cations. The discovery of high-temperature K-QSL (half-integer thermal hall conductivity) in α − RuCl has brought such materials into the forefront of the pursuit of the non-Abelian K-QSL and its relatedapplications to topological quantum computing. In particular, the magnetic spins, originating from the Ir + ions or Ru + surrounded by the ligand O − or Cl − ions respectively, lead to a K < (ferromagnetic) Kitaevinteraction via the celebrated Jackeli-Khaliullin mechanism. To have a qualitative understanding of Jackeli-Khaliullin mechanism, we briefly offer an intuitive picture of therelevant crystal structure, energy scales and interactions that are predicted to give rise to the Kitaev couplingconstant K in eq. (1b) . Whenever the valence electron-electron Coulomb (Hubbard) interaction strength U ina (semi-)metal is greatly larger than their ion-to-ion hopping rate t ( id est , U (cid:29) t (cid:39) ), the valence electronstend to localise proximal to their parent ions forming a Mott insulator. On the other hand, their spins willtransition from a disordered state (in this case, paramagnetic) to an ordered configuration ( viz . , ferromagnetic,stripy, zigzag, Néel ferromagnetic, et cetera ) below a transition temperature, with their magneto-spin dynamicsgoverned to leading order by the Heisenberg term J in eq. (1b) where K (cid:39) . However, the hopping rate t is small but finite ( id est in honeycomb layered materials such as iridates and α − RuCl , U (cid:29) t (cid:54) = ), whichwarrants the modification of the Heisenberg model. In iridates, the Kitaev interaction term is predicted to arise from the d orbitals of the Ir + ion containing 5electrons ( d ). In particular, from a solid-state chemistry perspective, the d orbital of a free Ir + ionis 10-fold degenerate ( d x − y , d z , d xy , d yz and d xz , spin / ). This degeneracy is lifted by crystal field splitting,into a 4-fold degenerate e g orbital ( d x − y and d z , spin / ) and a lower energy 6-fold degenerate t g orbital( d xy , d yz and d xz , spin / ) when a Ir + ion is bonded with the ligand O − ions forming the octahedral structuredepicted in Fig. 6b . Since the ligand O − ions approach the Ir + ions at the centre of the octahedra alongSection 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES Page 15 eview Article: Honeycomb Layered Oxidesthe x , y and z axes, this finite energy difference ∆ between the e g and t g orbitals arises from the fact that theelectrons in the d x − y and d z orbitals have a greater Coulomb repulsion compared to the electrons in the d xy , d yz and d xz orbitals. The triplet t g orbital experiences a further splitting into a ground j eff = / doublet stateand an excited j eff = / singlet state due to spin-orbit coupling. The j eff = / state is composed of a linearcombination of spin / and orbital ( d xy , d yz , d xz ) entangled states.For brevity, spin-orbit coupling (SOC) is a relativistic effect where, in the inertial frame of an electronorbiting a stationary nucleus generating an electric field (cid:126) E at a distance | (cid:126) r | = r with momentum (cid:126) p , there is afinite magnetic field (cid:126) B ∝ (cid:126) p × (cid:126) E = | (cid:126) E | (cid:126) p × (cid:126) r / r , that couples to the spin of the electron, thus generating a magneticmoment proportional to (cid:126) B · (cid:126) S ∝ λ (cid:126) L · (cid:126) S = H SOC where (cid:126) L = (cid:126) r × (cid:126) p is the angular momentum of the electron and λ is the spin-orbit coupling strength. This energy splitting can be written in terms of the total angularmomentum (cid:126) J = (cid:126) L + (cid:126) S as H SOC = λ ( J − L − S ) / , where we have used J = ( (cid:126) L + (cid:126) S ) = L + (cid:126) L · (cid:126) S + S . Note thatthe p orbital splitting (states labelled by total quantum number j ) will differ from the t g orbital splitting (stateslabelled by effective total quantum number j eff ) by the sign of λ . Thus, taking the spin and angular momentumquantum numbers respectively as s = / and l = and using (cid:104) L (cid:105) = l ( l + ) , (cid:104) S (cid:105) = s ( s + / ) and (cid:104) J (cid:105) = j ( j + ) where j = l ± s , the energy splitting between the ground j = / singlet state and the excited j = / doubletstate becomes (cid:104) H SOC (cid:105) = λ / . Thus, electrons ( spin-up and spin-down states) occupy the j eff = / doublet state and the remaining1 electron (spin-up or spin-down) occupies the j eff = / singlet state, leaving a spin-down or spin-up singletstate unoccupied. It is this unoccupied state (hole) which has a finite U > t (cid:54) = hopping rate from one Ir + ion to an adjacent one. Thus, one condition sine qua non for the realisation of the Jackeli-Khaliullin mechanismis the existence of the j eff = / singlet state in a d transition metal ion such as Ir + and Ru + , bonded withligand ions such as O − and Cl − forming an octahedron. Since hopping occurs from the j eff = / state of atransition metal ion directly to an adjacent one (labelled as t , in Fig. 6b ) or indirectly via the p orbitals of aligand ion (labelled as t in Fig. 6b ), the second condition for the realisation of the Jackeli-Khaliullin mechanismconcerns the geometric orientation of adjacent octahedra. In particular, the adjacent octahedra in honeycomblayered materials can either share a vertex (180 ◦ bond) or an edge (90 ◦ bond), which leads to either a single indirect t hopping path or a pair of equivalent t hopping paths respectively. It is the pair of t hopping paths inthe 90 ◦ bond which results in the Jackeli-Khaliullin mechanism. This is because the Kanamori Hamiltonian describing the Mott insulator is perturbed by the hopping Hamiltonian with terms resulting from the pairof t paths, leading to destructive interference of the symmetric Heisenberg exchange terms but constructiveinterference of the asymmetric Kitaev exchange terms. Such a calculation results in a ferromagnetic ( K < )Kitaev term, K = − t (cid:18) U − J H − U − J H (cid:19) (cid:39) − t U J H , where J H (cid:28) U is the so-called Hund’s coupling originating from the repulsion of electrons occupying the sameorbital due to Pauli exclusion principle.Typically, since spin-orbit coupling is greater for large transition metal atoms with d or d orbitals, muchof the search for K-QSL has focused on their j eff = / state. Based on the Jackeli-Khaliullin mechanism,calculations are often conducted via complementary spin-wave analysis and exact diagonalisation, which indeedreveal that Kitaev coupling constant is ferromagnetic ( K < ) for A IrO when ( A = Li , Na ). Unfortunately,the finite direct t , hopping from the j eff = / state have been shown to restore the Heisenberg term as wellas introduce another bond-independent coupling, often labelled by Γ . Thus, these non-Kitaev terms hinderthe prospects of realising K-QSL with the conventional Jackeli-Khaliullin mechanism. Moreover, crystallinedistortions of the octahedra, exempli gratia
Jahn-Teller distortions introduce long-range order that smears outSection 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES
Page 16 eview Article: Honeycomb Layered Oxidesthe K-QSL phase, hence hindering its experimental realisation.
This has lent further impetus to the consideration of realisation of honeycomb layered materials beyond theJackeli-Khaliullin mechanism, which rely not only on f orbitals but also on d orbitals. In particular,honeycomb layered oxides entailing for instance high-spin 3 d ions such as Co + , Fe + and Ni + are consideredas apposite models. Since Fe + -based honeycomb layered oxides are difficult to stabilise in the chemicalcompositions enumerated in Section 2 , it leaves Co + -based honeycomb layered oxides entailing pnictogen orchalcogen atoms such as A Co SbO , A Co BiO , A Co TeO , A CoTeO (where A can be Li , Na , K , Ag , Cu , Rb , etcetera .), amongst others as promising candidates. As for Ni + compounds, they can be synthesised, for instancevia the (electro)chemical oxidation of Ni + -based honeycomb layered oxides such as A Ni TeO , A Ni SbO , Figure 8
Magnetic transition temperatures of representative honeycomb layered oxide materials. (a) Various magnetictransition temperatures attained in honeycomb layered oxides that entail a change in spin configuration to antiferromag-netic states ( videlicet ., Néel temperature). (b) The magnetic transition temperatures of Na -based honeycomb layeredoxides (that have mostly been subject of passionate research owing to their intriguing magnetism) has been highlightedfor clarity to readers. Section 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES
Page 17 eview Article: Honeycomb Layered Oxides A Ni BiO , A NiTeO ( A = Li , Na , K , Ag , Cu , et cetera ) and so forth. Indeed, honeycomb layered oxides such as Na Co TeO , Na Co SbO and NaNi BiO − δ have generated traction in the search for K-QSL state. For instance, the necessity to include the J and J Heisenberg couplings in the case of the Mott-insulatinglayered iridates A IrO ( A = Na , Li ) into eq. (1b) , thus further generalising the Kitaev-Heisenberg model hasbeen tackled, exempli gratia , by Kimchi and You within the so-called Kitaev-Heisenberg- J - J model. Gen-erally, higher-order interactions are best observable when these layered oxides are cooled down to extremelylow temperatures, where the thermal motion of the spins is suppressed or negligibly small. At a unique mag-netic ordering (transition) temperature, the spins align themselves in specific directions along the honeycombconfiguration in various manners signifying a phase transition into new states of matter, as shown in
Fig.7 . For example, a paramagnetic material transitions into antiferromagnetic when spins align in the same di-rection (parallel) or the opposite directions (antiparallel). Depending on the magnetic phase of matter theytransition into, transition temperatures can be termed as Néel temperature or Curie temperature.
Néeltemperature is the transition temperature where antiferromagnetic materials become paramagnetic and viceversa . We shall focus on Néel temperature since a vast majority of the honeycomb layered oxides display an-tiferromagnetic transitions at low temperatures.
Figure 8 shows the Néel temperatures for most honeycomblayered oxides (incorporating pnictogen or chalcogen atoms), which tend to be at lower temperatures (below40 K).
Another intriguing manifestation of antiferromagnetism is the mannerin which the antiparallel spins align in the honeycomb configuration. The antiparallel spins may assume var-ious conformations such as zigzag ordering or alternating stripe-like (stripy) patterns within the honeycombslab. Zigzag spin structure has been observed in honeycomb layered oxides, such as Li Ni SbO , Na Co SbO , Na Co TeO , amongst others. Correspondingly, competing magnetic interactions on honeycomb lattices may induce both antiferromag-netic and ferromagnetic spin re-ordering, with the latter dominating when an external magnetic field is applied;a process referred to as spin-flop magnetism, observed in oxides such as Na Co SbO and Li Co SbO . More-over, depending on the distance between the spins, spiral-like or helical spin arrangements may result. Compet-ing interactions or ‘frustrations’ may also cause the spins in a honeycomb lattice to orient haphazardly (magneticdisorder), even at low temperatures, leading to a plenitude of exotic magnetic states such as spin-glasses andspin-flop behaviour as has been noted in oxides such as Li Co SbO . Complex magnetic phase diagramsas well as enigmatic interactions (Heisenberg-Kitaev interactions, Dzyaloshinskii-Moriya (DM) interactions, etcetera are discussed in the last section of this review) can be envisaged in the honeycomb layered oxides.
For instance, the Heisenberg-Kitaev model describes the magnetism in honeycomb lattice Mott insulators withstrong spin-orbit coupling. An asymmetric (DM) spin interaction term in the Heisenberg-Kitaev model can beshown to lead to (anti-)vortices-like magnetic nanostructures commonly referred to as magnetic skyrmions thatact as one of possible solutions describing equilibrium spin configurations in ferromagnetic/antiferromagneticmaterials.
The binding of these vortex/anti-vortex pairs over long distances in 2D constitutes a higher-orderinteraction that becomes finite at a certain temperature when these materials undergo a Berezinskii, Koster-litz and Thouless (BKT) transition - an example of a topological phase transition.
The possibility of these(and more) higher-order interactions demonstrates that there is room for both experimentalists and theorists inphysics and chemistry to expand the pedagogical scope of honeycomb layered oxides.Section 3 MAGNETO-SPIN MODELS OF HONEYCOMB LAYERED OXIDES
Page 18 eview Article: Honeycomb Layered Oxides
4. Solid-state ion diffusion in honeycomb layered oxides
High ionic conductivity is a prerequisite for superfast ionic conductors that may serve as solid electrolytes forenergy storage devices. The presence of mobile alkali atoms sandwiched in honeycomb slabs, as is present inhoneycomb layered oxides, endows them with fast ionic conduction not only at high temperatures but also atroom temperature.
Figure 9 shows the ionic conductivity of honeycomb layered oxides reported to date, withthe tellurate-based honeycomb layered oxides exhibiting the highest conductivity so far.
Experimentally, the measurement of ionic conductivity is conducted via linear response techniques with po-larised electromagnetic fields such as electrochemical impedance spectroscopy (EIS).
EIS entails applying alow amplitude low frequency oscillating voltage (current) and measuring the current (voltage) response. Thecurrent-to-voltage ratio determines the inverse of the impedance (admittance) of the material, where the realpart of the admittance Y ( ω ) = σ ( ω ) + i ωε is proportional to the conductivity σ ( ω ) and the imaginary part isproportional to the permittivity ε of the material. Figure 9
Solid-state diffusive properties of typical cations sandwiched between honeycomb slabs of various layeredoxides showing values of ionic conductivity attained in honeycomb layered oxides at room temperature and also at hightemperature (300 ◦ C).
Honeycomb layered oxides based on tellurates generally tend to show high ionicconduction, owing to the partial occupancy of alkali atoms in distinct crystallographic sites that facilitate rapid hoppingdiffusion mechanism. Inset shows the plot of the dependency of (normalised) conductance σ to (normalised) thermalenergy k B T of the cations determined by linear response of the diffusion current to low amplitude, slowly oscillatingvoltage by electrochemical impedance spectroscopy (EIS). Section 4 SOLID-STATE ION DIFFUSION IN HONEYCOMB LAYERED OXIDES
Page 19 eview Article: Honeycomb Layered Oxides
Figure 10
Regarding the honeycomb interslab distance and the size of sandwiched alkali or coinage metal atoms. (a)Correlation of the average interslab distance ( ∆ z ) and the size ( id est , the Shannon-Prewitt ionic radius) of the sand-wiched alkali ion ( A ) in honeycomb layered oxides adopting the following compositions: A + M + D + O ( A + / M + / D + / O ), A + M + D + O ( A + M + / D + / O ), A + M + D + O ( A + / M + / D + / O ), A + M + D + O ( A + / M + / D + / O ), et cetera where M = Fe,Mn, Co, Ni, Cu, Zn, Mg or a combination of at least two transition or alkaline-earth metal atoms; D = Te, Sb, Bi, Nb ; A = Cu, Ag, Li, Na, K. For clarity’s sake, the error bars associated with each data pointare smaller than the size of the data markers. (b) A fragment of a honeycomb layered oxide such as K Ni TeO in the ab plane, showing the two-dimensional (2D) diffusion channels of potassium ions. This figure also shows that apart fromthe type of alkali or coinage metal atom, where profound change in the interlayer distance can be expected, the natureof the transition metal atom M also influences the interlayer distance albeit to a smaller extent in some instances. In order to rationalise the heuristics behind the high ionic conductivity of the honeycomb layered oxides andpredict associated outcomes, it is imperative to introduce a detailed theoretical approach that incorporates thethermodynamics of the cations. In particular, the connection between the ionic conductivity of honeycomblayered oxides and other physical measurable quantities such as the diffusion of solid-state alkali cations atthermal equilibrium and very low frequencies undergoing Brownian motion satisfies the fluctuation-dissipationtheorem.
Here, we showcase this approach based on heuristic arguments that captures the diffusion aspectsof ionic conductivity of the (honeycomb) layered materials.
Ionic conductivity of A cations can be heuristically modeled under a Langevin-Fick framework of equa-tions, − D (cid:126) ∇ ρ ( t ,(cid:126) x ) = (cid:126) j ≡ ρ ( t ,(cid:126) x ) (cid:126) v , (2a) d (cid:126) pdt = − µ (cid:126) v − q (cid:126) ∇ V ( t ,(cid:126) x ) , (2b)where q is the unit charge of the cation, D = D exp ( − β E a ) is the Arrhenius equation relating the diffusion coef-Section 4 SOLID-STATE ION DIFFUSION IN HONEYCOMB LAYERED OXIDES Page 20 eview Article: Honeycomb Layered Oxides
Table 2
Values of ionic conductivity measured using electrochemical impedance spectroscopy (EIS) along with the activa-tion energy ( E a ) attained in representative honeycomb layered oxides at room temperature and also at high temperature(300 ◦ C).
The pellet compactness, amongst other factors, do influence the conductivity ofceramics and thus have been furnished (where possible).
Compound σ / S cm − σ / S cm − E a / eV Pellet compactness ( ◦ C (
K)) ( ± ◦ C ( ± K)) % K Mg TeO . × − ∼ − . ∼ Ni TeO . × − . ( ∼ K) Li Co . TeO . × − Li Cu SbO . × − Li . Zn . BiO . . × − . ( ∼ K) Li . Fe . Te . O . × − . CrSbO . × − . ( ∼ K) Li FeSbO . × − . ( ∼ K) Li MnSbO . × − . ( ∼ K) Li AlSbO . × − . ( ∼ K) Li . Cr . TeO . × − . ( ∼ K) Li . Mn . TeO . × − . ( ∼ K) Li . Al . TeO . × − . ( ∼ K) Li . Fe . TeO . × − . . Ni . Fe . TeO . × − × − . ( ∼ K) Zn TeO ( . ∼ . ) × − × − ∼ Co TeO . × − Na Co TeO ( . ∼ . ) × − ( . ∼ . ) × − . ( ∼ K) ∼ Mg TeO . × − . × − Mg TeO . × − . ( ∼ K) . NiFeTeO ∼ . × − . ∼ . ( ∼ K) Na NiFeTeO . × − Na NiMgTeO . × − . ( T < K) Na MgZnTeO × − . ( T < K) Na Zn TeO ( . ∼ . ) × − Na Zn TeO ∼ × − Na Zn TeO . × − Na − x Zn − x Ga x TeO ( x = . ) ( . ∼ . ) × − Na Zn − x Ca x TeO ( x = ∼ . ) . × − ( x = . ) Na Zn TeO (Ga-doped) . × − Na Ni TeO ( . ∼ . ) × − ( ∼ ) × − . ( ∼
623 K ) . ∼ . Ni TeO × − (
323 K ) ∼ . ( ) ( T ≥ K), . ( T < K) 90 Na LiFeTeO × − . ∼ . ( ∼ K) ficient to the activation energy (per mole) of diffusion ( E a ), D is the maximal diffusion coefficient, β = / k B T is the inverse temperature, T is the temperature at equilibrium, k B is the Boltzmann constant, ρ ( t ,(cid:126) x ) is theconcentration of alkali cations, (cid:126) v is their velocity vector and V ( t ,(cid:126) x ) is a time-dependent voltage distributionover the material. Imposing charge and momentum conservation, − (cid:126) ∇ · (cid:126) j = ∂ ρ / ∂ t = and d (cid:126) p / dt = re-spectively, and assuming the ionic concentration satisfies the Boltzmann distribution at thermal equilibrium, ρ ( T , t ,(cid:126) x ) = ρ exp ( − β qV ( t ,(cid:126) x )) (where ρ is the ionic density at zero voltage) leads to the ionic conductivity σ = q µρ proportional to the mobility µ of the alkali cations, which satisfies the fluctuation-dissipation relation µ = β D first derived by Einstein and Smoluchowski to describe particles undergoing Brownian motion (diffu-sion). Based solely on eq. (2) , the ionic conductivity of the alkali cations of honeycomb layered oxides,as summarised in
Fig. 9 and
Table 2 , is related to the equilibrium temperature of the materials.In particular, the ionic conductivity computes to σ ( T , t ,(cid:126) x ) = q µρ ( T , t ,(cid:126) x ) = qD ρ β exp ( − β { E a + qV ( t ,(cid:126) x ) } ) .Section 4 SOLID-STATE ION DIFFUSION IN HONEYCOMB LAYERED OXIDES Page 21 eview Article: Honeycomb Layered OxidesPlotting the ionic conductivity versus the normalised temperature k B T / ( E a + qV ) , we find that the ionic conduc-tivity scales with the equilibrium temperature in the regime k B T / ( E a + qV ) ∼ k B T / E a , which is always satisfied inEIS measurements. For k B T / E a < , raising the temperature increases the thermal motion of the cations, whichin turn raises the ionic conductivity. Figure 9 displays the ionic conductivity attained in honeycomb layeredoxides at room temperature (25 ◦ C) and also at high temperature (300 ◦ C), which showcases the increase ofionic conductivity with temperature as expected.
Moreover, classical motion of the alkali cations and other electromagnetic interactions along the z directionare precluded, since these materials often satisfy the condition ∆ z (cid:29) r ion , where ∆ z is the interlayer/interslabseparation distance and r ion is the ionic radius of the alkali cations. This condition effectively restricts theelectrodynamics in these layered materials to two dimensions (2D), and is almost always satisfied since theionic radius of the alkali cations is correlated with interslab distance, as shown in Fig. 10a . For instance, alkalications with large ionic radii such as K in the layered oxide K Ni TeO are restricted to the two-dimensional(2D) honeycomb diffusion channels in the ab plane ( Fig. 10b ), as has also been shown in Na Ni TeO . Thus, the large interslab separation, together with the
TeO octahedra acts as a barrier preventing inter-channelexchange of the alkali cations.Other factors that affect the ionic conductivity of the cations include the ionic radius of the A cations inrelation to the M atoms. For instance, in the case of Li Ni TeO (where M = Ni ) in comparison to Na Ni TeO and K Ni TeO , Ni atoms act as impurities in the diffusion dynamics of A = Li since the interlayer separationdistance in Li Ni TeO is vastly smaller compared to Na Ni TeO and K Ni TeO . Electrochemically, collisionswith such impurities in these honeycomb layered oxides are suppressed by the larger interslab distance in con-junction with the greater sizes of Na and K atoms relative to Li , which ensures their facile mobility within thetwo dimensional planes. In contrast, the smaller interslab distance and the equivalent atomic sizes of Li and Ni atoms, which lie at close proximity to the honeycomb slabs leads to the interchange of their crystallographicsites (commonly referred to as Li / Ni ‘cationic mixing’). Consequently, the mobility of Li is obstructed throughthe collisions with Ni atoms within the 2D honeycomb surface that act as impurities, a process quantum me-chanically referred to as scattering. Within our heuristic approach, scattering of Li ions through collisions with Ni costs more (activation) energy than in the case of Na or K , E Lia (cid:29) E Naa , E Ka ). Thus, the Einstein-Smoluchowskirelation µ = D / k B T together with Arrhenius equation D = D exp ( − E a / k B T ) leads to a smaller ionic mobility inthe case of A = Li ions. This trend is indeed shown in Fig. 9 and
Table 2 , wherein Li -based honeycomb layeredoxides, regardless of the temperature, still show inferior ionic conductivity compared to those with Na or K .Section 4 SOLID-STATE ION DIFFUSION IN HONEYCOMB LAYERED OXIDES Page 22 eview Article: Honeycomb Layered Oxides
5. Electrochemistry of honeycomb layered oxides
The layered structure consisting of highly oxidisable 3 d transition metal atoms in the honeycomb slabs segre-gated pertinently by alkali metal atoms, renders this class of oxides propitious for energy storage. In principle,classical battery electrodes rely on the oxidation (or reduction) of constituent 3 d metal cations to maintaincharge electro-neutrality, thus facilitating the extraction (or reinsertion) of alkali metal cations, a process re-ferred to as ‘charge-compensation’. In principle, the constituent pentavalent or hexavalent d cations (suchas Bi + , Sb + , Te + , W + and so forth) do not participate in the charge-compensation process during batteryperformance. However, the highly electronegative WO − (or TeO − , WO − , BiO − , SbO − , RuO − , et cetera )anions lower the covalency of the bonds formed between the oxygen ( O ) atoms and 3 d transition metal ( M ) re-sulting in an increase the ionic character of M - O bonds within the honeycomb layered oxides. Consequently,the energy required to oxidise M cations increases, inducing a staggering increase in the voltage of relatedhoneycomb layered oxides within the battery. This process is commonly referred to as ‘inductive effect’. Forclarity, the electronegativity trend (based on the Pauling scale) is generally as follows: W > Ru > Te > Sb > Bi .Honeycomb layered oxides such as Li Ni + TeO , and more recently, Li Ni + TeO , manifest higher voltages(over V) in comparison to other layered oxides or compounds containing Ni + . This is rationalised by
Figure 11
High-voltage electrochemistry of honeycomb layered oxides. (a) Molecular orbital calculations of the voltageincrease arising from the ‘inductive effect’ that alters the covalency of Ni–O bonds, due to the presence of more elec-tronegative Te atom surrounded by a honeycomb configuration of Ni atoms. (b) Voltage-response curves (technicallyreferred to as cyclic voltammograms) of honeycomb layered compositions ( A + M + D + O ( A = Li, Na, K)), showing theirpotential as high-voltage cathode materials for rechargeable alkali cation batteries. Technically, these cyclic voltammo-grams (voltage-response curves) were plotted under a scan rate of 0.1 millivolt per second. Part of the data in (b) wasadapted from ref. under Creative Commons licence 4.0. Section 5 ELECTROCHEMISTRY OF HONEYCOMB LAYERED OXIDES
Page 23 eview Article: Honeycomb Layered Oxidesconsidering Te + as a TeO − moiety, which being more electronegative than anions such as O − , increases thevoltage necessary to oxidise Ni + (or technically as redox potential) through the inductive effect (as succinctlyshown in Fig. 11a ). Voltage increase due to this inductive effect has, in particular, been noted in polyanion-based compounds when ( SO ) − are replaced either by ( PO ) − or ( PO F ) − anion moieties. Therefore, theinductive effect seems to be typical and represents a crucial strategy when tuning the voltages of honeycomblayered oxides. Indeed, besides Li Ni TeO and Li NiTeO , analogues consisting of Na (such as Na Ni TeO and Na NiTeO ) and K alkali atoms (such as K Ni TeO and K NiTeO ) also exhibit high voltages surpassing thoseof layered oxides in their respective fields. Theoretical insights regarding the high voltage innate in the aforementioned honeycomb layered oxides arevalidated by experimental investigations. Typical electrochemical measurements performed include: cyclicvoltammetry, which assesses the voltages during charging and discharging at which the 3 d transition metalredox processes occur as well as other structural changes and galvanostatic charge/discharge measurements,which principally determine pivotal battery performance metrics, inter alia , (i) the amount of alkali atomselectrochemically extracted or inserted ( id est , capacity) during charging and discharging, (ii) how fast the alkaliatoms can be extracted or inserted (referred to as rate performance), (iii) voltage regimes where alkali atomsare dominantly being extracted or inserted and (iv) the nature of the extraction or reinsertion process of alkaliatoms, for instance, whether it occurs topotactically (referred to as a single-phase, solid-solution or monophasicbehaviour) or as a multiple phase (referred to as a two-phase or biphasic behaviour). Figure 11b illustratesthe cyclic voltammograms of A + Ni + Te + O ( A = Li , Na and K ), depicting voltage peaks/humps around Vwhere the redox process of Ni (in this case Ni + / Ni + ) occur during electrochemical extraction/insertion ofalkali atoms. It is noteworthy that the larger the ionic radius of A is, the more pronounced the minor voltagehumps are seen. This is indicative of structural changes (phase transitions) occurring, details of which shall beelaborated in a later section. Usually the voltage response curves (cyclic voltammograms) should nicely mirroreach other (taking the line where the current density is set as zero to be the mirror plane in Fig. 11b ). However,due to some electrochemical issues (such as inherently slow alkali-ion kinetics), the voltages at which the redoxprocesses or structural changes occur deviate from each other as seen in
Fig. 11b . This is technically referred toas ‘voltage polarisation’ or ‘voltage hysteresis’.
Voltage polarisation can significantly be decreased by partialsubstitution (or doping) of the constituent 3 d transition metal atoms with isovalent metals. For instance, partialdoping with Zn , Mn or Mg in Na Ni SbO leads to lower voltage polarisation compared to that of the undoped Na Ni SbO . Furthermore, doped oxides present higher voltages; depicting doping as another feasibleroute to increase the voltages of these honeycomb layered oxides.
Precise and adequate evaluation of the voltage responses of high-voltage cathode materials, such as the afore-mentioned A Ni TeO , demands the utilisation of stable electrolytes that can sustain high-voltages. Conven-tional electrolytes consisting of organic solvents are prone to decomposition during high-voltage operations;rendering them unsuitable for the high-voltage performance innate in such honeycomb layered oxides elec-trodes. Ionic liquids, which comprise entirely of organic cations and (in)organic anions, are a growing classof stable and safe electrolytes exhibiting a plenitude of distinct properties; pivotal amongst them being theirgood electrochemical and thermal stability, low volatility and low flammability. These attributes assureimproved safety for batteries utilising ionic liquids. Matsumoto and co-workers were amongst the first to showexemplary performance of layered oxides such as
LiCoO with the use of ionic liquids. Honeycomb lay-ered oxides, such as K Ni TeO and their cobalt-doped derivatives, have been shown to display stableSection 5 ELECTROCHEMISTRY OF HONEYCOMB LAYERED OXIDES Page 24 eview Article: Honeycomb Layered Oxides
Figure 12
Illustration showing a selection of electrolytes (in particular ionic liquids) which guarantee the stable electro-chemical performance of honeycomb layered oxides. In principle, ionic liquids consist of organic cations (pyrrolidinium,imidazolium, piperidinium, et cetera ) coupled with organic or inorganic anions (such as BF − , PF − , ClO − , et cetera .) Organiccations are shown in black, whilst organic or inorganic anions are in yellow. Purity of the salts, solubility and compatibilitywith the honeycomb layered oxide cathode materials, amongst other factors are necessary to consider when obtainingsuitable ionic liquids for high-voltage operation. Readers may further refer to the literature for more details regarding theionic liquids. performance at high-voltage operation in electrolytes comprising ionic liquids plausible candidates of which aredepicted in Fig. 12 . Fast kinetics of alkali ions within an electrode during electrochemical extraction/insertion is a crucial parametricthat influences the rate performance of battery performances. For instance, previous reports have attributed thegood rate performance and excellent cyclability of Na Ni SbO cathode material ( Figs. 13a , and ) tothe fast interlayer kinetics of Na + within the highly-ordered Na Ni SbO . Honeycomb layered oxides suchas Na Ni SbO also exhibit preponderant rate capabilities (as shown in Figs. 13d and ) and can sustainfast Na -ion kinetics upon successive Na-ion (de)insertion ( Fig. 13f ). Pertaining to structural stability, theSection 5 ELECTROCHEMISTRY OF HONEYCOMB LAYERED OXIDES
Page 25 eview Article: Honeycomb Layered Oxides
Figure 13
Electrochemical performance of representative honeycomb layered oxide cathode materials. (a) Voltage-capacity plots of Na Ni SbO showing the initial (dis)charge curves under a Na -ion (de)insertion rate (current density)commensurate to . C . Technically, nC rate denotes the number of hours ( / n ) necessary to (de)insert alkali-ions to thefull theoretical capacity (alkali ion occlusion capacity per formula unit). (b) Rate capability of Na Ni SbO . (c) Capacityretention of Na Ni SbO at (de)insertion rates of . C and C (inset). (d) Voltage-capacity plots of Na Ni BiO at acurrent density equivalent to . C . (e) Corresponding rate performance, showing Na Ni BiO to also sustain fast Na -ionkinetics. (f) Capacity retention of Na Ni BiO at various rates. (a), (b) and (c) were reproduced and adapted from ref.58 with permission. (d), (e) and (f) were reproduced and adapted from ref. 83 by permission of the Royal Society ofChemistry. manner in which 3 d transition metal atoms (for example Ni atoms in Na Ni SbO ) are arranged in a honeycombconfiguration endows it not only with good thermal stability but also structural stability to sustain repeatablealkali atom extraction and insertion ( Na atoms in this case). Besides the high-voltage and facile alkali-ion kinetics, this class of honeycomb layered oxides can ac-commodate ample amounts of alkali atoms depending on the choice of both d (4 d or 5 d ) cations and 3 d transition metal atoms. The increase of the amount of alkali atoms accommodated within the interlayersof the honeycomb slabs implies an increase in the energy storage capacity, indicative of a high energy den-sity. For instance, more of alkali atoms can be extracted from honeycomb layered oxide compositions suchas A + Ni + Sb + O ( A = Li , Na and K ) or A + Ni + Bi + O than in A + Ni + Te + O , despite the higher molar massof A + Ni + Bi + O compared to A + Ni + Te + O . The voltage-capacity plots of representative honeycomb lay-ered oxides that can be utilised as cathode materials for rechargeable alkali-ion batteries are shown in Fig.14 . Note that the capacities of these oxides have been calculated based onthe manifold oxidation states of the constituent transition states that can be allowed to facilitate maximumextraction of alkali atoms from the layered structures. In principle, the theoretical capacity ( Q (mAh g − )) ofa material is determined by the molar mass ( M (g mol − )) and the number of electrons ( n ) involved duringSection 5 ELECTROCHEMISTRY OF HONEYCOMB LAYERED OXIDES Page 26 eview Article: Honeycomb Layered Oxides
Figure 14
Voltage-capacity plots of various honeycomb layered oxides encompassing mainly pnictogen orchalcogen atoms, showing their potential as high-energy-density contenders for high-voltage alkali-ion batter-ies.
The error bars represent the upper and lower limits of the voltages attainedexperimentally. Note also that the theoretical capacities have been calculated based on the change in oxidation states oftransition 3 d metals as charge-compensation cations. alkali-ion extraction (charging) or insertion (discharging), in accordance with the following equation, Q = n × N × eM = n × F . M = . × × nM , (3)where N is the Avogadro constant ( . × mol − ), e is the elementary charge ( . × − C) and F theFaraday constant ( . C mol − ). It is apparent that these honeycomb layered oxides exhibit competitiveenergy storage capacities to justify them as high-energy-density contenders for rechargeable batteries.Another point of emphasis is the nature of the redox process occurring within these honeycomb layeredoxides. During the charge compensation redox process, the constituent Ni cations ( videlicet ., Ni + / Ni + ) arecompletely utilised in oxides such as A + Ni + Te + O ( A = Li , Na and K ) ensuring full electrochemical extractionof the alkali atoms. However, for oxides such as A + Ni + Te + O , it is impossible to fully extract all the alkali A atoms relying on the redox process of constituent Ni atoms ( Ni + / Ni + ) alone.Section 5 ELECTROCHEMISTRY OF HONEYCOMB LAYERED OXIDES Page 27 eview Article: Honeycomb Layered Oxides
To fully tap the capacity (hence energy density) of such oxide compositions, the redox process of anions suchas oxygen also have to be utilised, besides the redox process of 3 d transition metal cations. Formation of ligandholes, peroxo- or superoxo-like species are expected to occur in the oxygen orbital when anionic redox processestake place, and sometimes oxygen ( O ) may be liberated leading to complete structural collapse; thus affectingthe cyclability/performance durability of such oxides when used as battery materials. Anionic redox processes provide a judicious route to utilise the full capacity of electrode materials and hasbeen a subject that has attracted humongous interest in the battery community in recent years.
Apartfrom facilitating an increase in the redox voltage of honeycomb layered oxides, the presence of d cations (suchas W + , Te + , Sb + , et cetera .) also helps stabilise the anion-anion bonding that accompanies the oxygen redoxchemistry. For example, the existence of highly valent W + (5 d ) cations in Li NiWO strongly stabilises the O - O bonds, thereby averting the formation of gaseous O following anion oxidation. Just like Li NiWO ,other honeycomb layered oxides such as Li FeSbO have also been found to manifest good oxygen-based re-dox reversibility, but it generates a large voltage hysteresis in the process. Further investigations on theoxygen-based redox reversibility are still ongoing in this field to uncover the factors underlying the large voltagehysteresis and determine ways to minimise it. What is emerging with these honeycomb layered oxides is that thepresence of high-valency d (4 d or 5 d ) is a crucial condition to produce not only high redox (and in some casesparadoxical) voltages, but also invoke oxygen redox chemistry aside from 3 d cationic redox processes. More-over, the possibility to expand the materials platform of these honeycomb layered compounds through partialsubstitution with isovalent or even aliovalent 3 d transition metals, renders them as apposite model compoundsto study numerous electrochemical aspects.Section 5 ELECTROCHEMISTRY OF HONEYCOMB LAYERED OXIDES Page 28 eview Article: Honeycomb Layered Oxides
6. Topological phase transitions in honeycomb layered oxides
Honeycomb layered oxides are susceptible to undergoing structural changes (phase transitions) upon electro-chemical alkali-ion extraction. The presence of divalent transition metals ( M + ) in the honeycomb slabs plays amajor role in inducing these transitions during alkali-ion reinsertion process. In principle, when alkali atoms areelectrochemically extracted, the valency state (oxidation state) of the transition metal atoms residing in the hon-eycomb slabs increase and vice versa during the reinsertion process; earlier defined as the charge-compensationprocess. Voids or vacancies created during alkali atom extraction leads to enhanced electrostatic repulsionbetween the metal atoms residing in different slabs; leading to an increase in the interslab distance. Evolution of the structural changes upon alkali-ion extraction and reinsertion can readily be discerned us-ing X-ray diffraction (XRD) analyses. During alkali-ion extraction, ( z ) Bragg diffraction peaks that reflectthe honeycomb interslab planes shift towards lower diffraction angles indicating the expansion of the inter-slab distance/spacing. The reverse process occurs during alkali-ion reinsertion, as has been exemplified in K Ni TeO upon potassium-ion extraction and reinsertion (as shown in Fig. 15a) . Apart from overall peakshifts observed during topotactic alkali-ion extraction and reinsertion (which technically manifests a single-phase (monophasic/solid-solution) behaviour), peak broadening or asymmetric peaks can be observed alongwith the disappearance of peaks and the emergence of new ones (reflecting a two-phase/biphasic behaviour).
The phase transition behaviour of honeycomb layered cathode oxides during alkali-ion extraction (charging)and reinsertion ( id est , discharging), entails intricate structural changes that affects the coordination envi-ronment of alkali atoms. For instance, electrochemical sodium ( Na )-ion extraction from Na Ni BiO and Na Ni SbO during charging process leads to a sequential change in the bond coordination of Na, namely fromthe initial octahedral ( O ) coordination to prismatic ( P ) and finally to an octahedral ( O ) coordination. Further, the manner of stacking of repetitive honeycomb slabs per unit cell changes from to . Thus, the phasetransition of Na Ni BiO during charging process can be written in the following Hagenmuller-Delmas’ nota-tion as previously described: O3 → P3 → O1 stacking mode. However, phase transitions can influencecrucial electrochemical performance parametrics such as the rate capabilities of related oxides when used asbattery materials. As such, crucial strategies have been sought to suppress the intricate phase transformationprocesses, for example, through doping or partial substitution of the transition metal atoms in the honeycombslabs ( exempli gratia , Na Ni . M . BiO (where M = Mg , Zn , Ni , Cu )) or even the alkali atoms in for instance Na . Sr . Ni TeO . Multiple phase transformations observed in honeycomb layered oxides during alkali-ion extraction and rein-sertion have a profound effect on their electrochemical characteristics such as rate performance and nature ofthe voltage profiles. These intricate phase transitions lead to the appearance of staircase-like voltage profilesas is often observed in the voltage-capacity profiles of most of the reported honeycomb layered cathode oxidematerials.
Shifting of the honeycomb slabs during electrochemical alkali-ion extractionand reinsertion, or what is commonly termed as interslab gliding, has been rationalised to occur as the al-kali atoms rearrange their occupying positions (alkali atom ordering). Such a complex phase transformationprocess can be envisioned through successively removing blocks from a complete ‘
Jenga wooden blocks set’,as shown in
Fig. 15b . Assuming that the ‘blocks’ are the ‘alkali atoms’, removal of these wooden blocks willlead to rearrangement of the whole
Jenga set to avoid structural collapse either by sliding (gliding) or rota-tion (shear) of the blocks (slabs). A mechanism akin to this
Jenga -like mechanism, which is further discussedSection 6 TOPOLOGICAL PHASE TRANSITIONS IN HONEYCOMB LAYERED OXIDES
Page 29 eview Article: Honeycomb Layered Oxides
Figure 15
Phase transitions of honeycomb layered oxides. (a) Increase/decrease of the interslab distance ( ∆ z ) of hon-eycomb layered oxide K Ni TeO with charging (K + extraction)/ discharging (K + reinsertion). (b) Crystal structural evo-lution of K Ni TeO upon charging and discharging, showing the occurrence of intricate phase transition mechanism. (c)Broadening and shifting of the (002) and (004) Bragg diffraction peaks that are sensitive to alkali-ion extraction/reinsertionduring discharging/charging. (d) Rendition of the phase transition in these classes of layered oxides that entails complexphase transitions (mono- and bi-phasic, and amongst others), akin to a process of successively removing blocks from acomplete ‘ Jenga wooden blocks set’ which can account for the Devil’s staircase-like voltage profiles typically observed forhoneycomb layered oxides. (a-c) Reproduced from ref. under Creative Commons licence 4.0. below, can account for the Devil’s staircase-like voltage profiles typically observed for honeycomb layered ox-ides. Section 6 TOPOLOGICAL PHASE TRANSITIONS IN HONEYCOMB LAYERED OXIDES
Page 30 eview Article: Honeycomb Layered Oxides
Phase transformation behaviour observed upon electrochemical alkali-ion extraction and reinsertion can spurenigmatic structural changes, like the aforementioned ‘
Jenga -like’ transitions. A comprehensive analysis of thismechanism calls for a deeper understanding based on a more comprehensive theory. Nonetheless, we here-after highlight an approach based on heuristics founded on geometry, topology and electromagnetic considera-tions.
Readers may find it prudent to revise topics on tensor calculus, index notation, Einstein convention and other widely useful concepts in applied mathematics such as Gaussian curvature (Gauss-Bonnet theorem)in 2D, the Levi-Civita symbol and Chern-Simons theory.
Here, we use units where Planck’sconstant and the speed of the massless photon in the crystal are set to unity: ¯ h = c = . Amongst some of the configuration of alkali atoms in a two-dimensional (2D) lattice of honeycomb layeredoxides is shown in
Fig. 16 . Note that such a configuration has also been observed for some potassium atomsin K Ni TeO through XRD and electron microscopy studies (see Fig. 4 ). Potassium extraction (as is the casewhen a voltage is applied during charging process), for instance, leads to a non-sequential interslab distanceincrease; rendering the alkali cations to move in an undulating 2D surface (technically exhibiting a Gaussiancurvature). Charge conservation in such a 2D undulating surface implies that the charge density vector j a =( ρ , j x , j y ) , satisfies ∂ a j a = which has the solution j a = ε abc ∂ b A c (where A c = ( V , A x , A y ) is the 2D electromagneticvector potential and ε abc is the totally anti-symmetric Levi-Civita symbol), hence leading to a Chern-Simonsterm, ε abc ∂ b A c . In turn, the honeycomb lattice introduces further constraints on the electrodynamics ofthese cations. In particular, since the cations (absent the applied voltage) form a 2D honeycomb lattice wherethe (free) alkali cations that contribute to the diffusion current j a are extracted from the 2D lattice by thepotential energy qV of the applied voltage, the total number of these free alkali cations ( g ∈ integer ) are relatedto the quasi-stable configurations displayed in Fig. 16 that we shall refer to as 3 (leaf)-clover configurations.We shall consider each configuration as a g -torus where g ∈ integer is the genus of an embedded 2D surfacelinked to the diffusion heuristics applied earlier in the review. Note that each g -torus supplies a unit charge q of a single alkali cation, and thus determines the total charge density ρ of the alkali cations which is related tothe diffusion current j i = q µρ E i = ρ v i along ab plane of the honeycomb slabs. Consequently, these ideas can besummarised by a useful set of equations consistent which also contain the diffusion approach already tackled inthe previous section of the review (also illustrated in Fig. 17a and ), ∂ a ln ρ ( t , x , y ) = q β ε abc ∂ b A c , (4a) q m (cid:90) ∂ M d (cid:126) x · ( (cid:126) n × (cid:126) E ) = (cid:90) M K d ( Area ) = π χ = π ( − g ) , (4b)where m is the mass of the cations, the interlayer (separation) distance, (cid:126) n = ( , , ) is the normal vector to the ab plane, g is (approximately) the number of free cations, ρ ∝ exp ( − β E a ) is the ionic charge density with E a theenergy of the cations and (cid:126) ∇ · (cid:126) E = πρ / q , β = / k B T is the inverse temperature, K is the Gaussian curvature of acurved closed intrinsic surface M and ∂ M is the boundary of M representing the diffusion pathways of g numberof cations which form honeycomb lattice on M displayed in Fig. 16 . Thus, the integral equation is simply thewell-known Gauss-Bonnet theorem.
In the special case of static equilibrium when the ionic density ρ is strictly time-independent ∂ ρ / ∂ t = and the electromagnetic vector potential is given by A c = ( V ( x , y ) , , ) , the Chern-Simons term reduces to ∂ a ln ρ ( t , x , y ) = q β ε abc ∂ b A c → ∂ i ln ρ ( x , y ) = q β ε i j ∂ j V ( x , y ) which yields ρ (cid:126) v = q µρ ≡ σ (cid:126) E with the ansatz E a ( x , y ) = (cid:82) ∂ M d (cid:126) x · ( (cid:126) n × µ − (cid:126) v ) , where ε i j is the 2D Levi-Civita symbol and µ = D / k B T is the mobility of the cations. Hence,Section 6 TOPOLOGICAL PHASE TRANSITIONS IN HONEYCOMB LAYERED OXIDES Page 31 eview Article: Honeycomb Layered Oxidesthe energy evaluated over a closed loop E a ( (cid:72) ∂ M ) = m ( g − ) over the honeycomb surface conveniently counts themissing mass of cations within the loop, as shown in Fig. 16 . Equivalently, this corresponds to the (activation)energy E a = ( g − ) mc needed to render the cations mobile, where c = is the speed of the massless photonin the crystal. This means that the quasi-stable configuration with g = requires no activation energy to createand can be considered as a ground state of the system. However, since the other configurations are shifted bya constant energy E a = mg from this ground state, the system contains an additional g − number of stableconfigurations. Figure 16
Atomic rearrangement triggered by extraction of alkali-atoms in honeycomb layered oxides such as K Ni TeO ,where g ∈ integer is the number of alkali-atoms extracted by applying an external voltage in the ab ( x – y ) plane. The toridenote the various geometrical objects with holes denoted as g (for genus). The tori can be mathematically mapped tothe various configurations of the honeycomb lattice with ionic vacancies also denoted as g . Section 6 TOPOLOGICAL PHASE TRANSITIONS IN HONEYCOMB LAYERED OXIDES
Page 32 eview Article: Honeycomb Layered Oxides
Figure 17
Topological transitions of honeycomb layered oxides. (a) A two-dimensional (2D) field theory relating aChern-Simons term to the ionic concentration ρ ( t , x , y ) (charge density of the cations) where q is the unit chargeof a single cation, β = / k B T is the inverse temperature and A c = ( V , A x , A y ) is the 2D electromagnetic potential (see eq.(4a) ). (b) A Gauss-Bonnet theorem relating the applied electric-field (cid:126) E ≡ E i = ε ibc ∂ b A c to the Euler characteristic χ = − g and Gaussian curvature K of the honeycomb surface M , where ∆ z ∼ λ c = π / m is taken to be the order ofthe Compton wavelength of the cations and (cid:126) n = ( , , ) is a vector normal to the honeycomb surface (see eq. 4b). χ ( M ) is applied to estimate the transitions from the complete g = honeycomb configuration to g ∈ integer quasi-stableconfigurations such as the three-clover atomic arrangements depicted in Fig. 16 . On the other hand, according to eq. (4a) , the ionic density is time-dependent, ∂ ρ / ∂ t (cid:54) = , when a magnetic field B z = ∂ x A y − ∂ y A x is present. Since g ∈ integer corresponds to the aforementioned 3-(leaf) clover configurationson the honeycomb lattice, magnetic fields drive the system out of one configuration to the next via extractionof cations from the honeycomb lattice. We shall refer to this mechanism of adiabatic extraction of the alkalications from the honeycomb surface accompanied by introduction of time-varying electromagnetic fields as Jenga mechanism, in analogy with the game of the same name.Similar to the total collapse of the pieces in
Jenga at the end of the game, this process of extraction of alkalications and subsequent restabilisation cannot continue indefinitely since eq. (4a) and eq. (4b) remain validonly around equilibrium and the conditions of adiabatic perturbations around equilibria g values. Whence,the transformation of the complete honeycomb structure into a predominantly -clover configuration shouldinduce a phase transition. One possible approach to a theoretical treatment of such transitions is to apply theBerezinskii-Kosterlitz-Thouless model of phase transitions to the magnetic fields (or fluxes) introduced dur-ing this dynamical Jenga phase. Another approach is to consider the phase transitions that may be triggered bysound waves in the crystal arising from rapid (non-adiabatic) extractions of the cations from the honeycombsurface. Geometrically, this entails periodically time-varying Gaussian (curvature) metric analogous to gravita-tional waves in the space-time geometry. When quantised, these sound waves are phonons that can mediate aweak attractive force between the positively charged fermionic cations (forming Cooper-pairs) and hence maySection 6 TOPOLOGICAL PHASE TRANSITIONS IN HONEYCOMB LAYERED OXIDES
Page 33 eview Article: Honeycomb Layered Oxideslead to superconductivity.
In contrast, an idealised approach to the dynamics of the cations has been pro-posed in ref. , where bosonic cations form a Bose-Einstein condensate below the critical temperatureand their dynamics are consistent with eq. (4a) and eq. (4b) . Above the critical temperature, unpaired chargedvortices appear representing diffusion channels of the cations under small curvature perturbations around g (cid:39) .Of course, further research of the physics of the Jenga mechanism including other non-adiabatic phenomenatesting the validity or failures of eq. (4a) and eq. (4b) will certainly be the focus of frontier research in thecoming years.Section 6 TOPOLOGICAL PHASE TRANSITIONS IN HONEYCOMB LAYERED OXIDES
Page 34 eview Article: Honeycomb Layered Oxides
7. High-precision measurement of diffusion and magneto-spin properties
In the previous section(s), we discussed the physics and electro-chemistry of the diffusion of cations withinthe honeycomb layers. However, we neglected their magneto-spin interactions with the inter-layers which inturn can substantially affect their mobility and hence, their solid-state alkali-ion diffusion properties. Thisapproximation is valid since the alkali cations ( exempli gratia K ) are known to generally possess an inherentlyweak nuclear magnetic moment which barely interacts with the octahedra ( exempli gratia TeO ) in the inter-layers. In particular, the diffusion dynamics of the cations is resilient to local magneto-spin interactions in thehoneycomb layers since the weak magnetic fields originating from the large number of cations in the honeycomblayers tend to randomise and average out according to central limit theorem. This means that even thoughthe Gaussian average (mean) of the magnetic fields vanishes, (cid:104) B z ( t ) (cid:105) = , the mean-square (cid:104) B z ( t ) B z ( ) (cid:105) (cid:54) = need not vanish. Hence, the diffusion and magneto-spin properties of the cations are encoded in the mean ofthe random magnetic fields in the honeycomb layers. However, measuring these properties by applying theGaussian average over magnetic quantities is an intricate task that often proves elusive to undertake due to ascarcity of effective techniques. In 2D, the Langevin equation given in eq. (2b) is replaced by, d (cid:126) pdt = − µ ( (cid:126) n × (cid:126) v ) + q ( (cid:126) n × (cid:126) E ) + q ( (cid:126) n · (cid:126) B ) (cid:126) n , (5a)which together with eq. (4a) form the Langevin-Fick framework of equations (analogous to eq. (2) ). Notice that since the magnetic field (cid:126) B ∝ (cid:126) η is proportional to the Langevin force, its mean-square is given by thefluctuation-dissipation theorem, (cid:104) B z ( t ) B z ( ) (cid:105) = k B T µ q f ( t ) (cid:54) = with f ( t ) a function of time. Consequently,the mobility µ (related to the diffusion coefficient by the Einstein-Smoluchowski relation µ = β D ) can bedetermined from the mean-square of the local magnetic fields in the honeycomb layers through the (dynamic)Kubo-Toyabe (KT) function given by, P z ( t ) = + (cid:0) − ∆ v t (cid:1) exp ( − ∆ ν t ) , (5b)where P z ( t ) is the spin-polarisation of the particle and ∆ ν = γ (cid:104) B z ( ) (cid:105) is the decay rate of the particle with γ its gyromagnetic ratio. The KT function effectively describes the time evolution of a spin-polarised particlein zero magnetic field with a non-vanishing mean-square. This singles out particles (in the standard modelof particle physics) with a strong gyromagnetic moment as ideal for probing such weak magneto-spin anddiffusion properties since their spin-polarisation will precess according to the KT function. Notably, muon spinrotation, resonance and relaxation (abbreviated as µ + SR) is a potent measurement technique that avails thisunivocal information pertaining to alkali-ion diffusion properties of materials to electrochemists and materialscientists.
At this juncture, it is imperative to explain the rationale for the use of muons in analysis of diffusion andmagneto-spin properties of materials. Muons stand out from other members of the lepton family of elementaryparticles mainly owing to the following reasons:• Muons are abundant and are indeed a product of cosmic radiation (recall the
Aurora Borealis and
Aurora
Section 7 HIGH-PRECISION MEASUREMENT OF DIFFUSION AND MAGNETO-SPIN PROPERTIES
Page 35 eview Article: Honeycomb Layered Oxides
Australis ). Muons can also be artificially produced using spallation sources such as ISIS Neutron andMuon Source (UK), Japan Proton Accelerator Research Complex (JPARC), TRIUMF (Canada) and PaulScherrer Institute (PSI, Switzerland);• The spin configuration of muons are traceable (technically, muons are 100% spin-polarised since they areproduced via the decay of a (positive) pion at rest into a positron and an electron neutrino via the weakinteraction, which violates parity), implying that they are easy to detect via their decay productsunlike other members of the lepton elementary particles. This aspect endows µ + SR measurements withan upper edge over other resonance techniques such as nuclear magnetic resonance (NMR). In addition,muons possess a high gyromagnetic ratio ( γ µ = . MHz/T) meaning that they are very sensitive toweak magnetic fields;• Unlike electrons, muons have a finite lifetime that is appreciable; thus, µ + SR offers a unique measurementtime window that complements conventional techniques such as NMR and neutron diffraction.Detection of alkali-ion diffusion by muons first entails the embedding of muons into a sample (or muonimplantation), the sample in this case being the layered oxide material. These muons are artificially producedvia the bombardment of high energy protons onto a carbon (graphite) or beryllium target, as is schematicallyshown in
Fig. 18 . The muons (in this case, positive muons (anti-muons)) are then focused using a collimatorto the sample where they bind with oxygen ions ( O − ) to form stable µ + – O − bonds. The implantation ofmuons into a honeycomb layered oxide is illustrated in Fig. 19a , where muons typically reside at the vicin-ity of oxygen ions at distances in the ranges of ∼ . Å. The implanted muons are initially static and are
Figure 18
Working principle of (anti-)muon spin relaxation ( µ + SR) as a potent tool for investigating the diffusive andmagnetic properties of target materials. The (anti-)muon is produced when a high energy proton beam is directed ontocarbon nuclei which produce (positive) pions. The (positive) pion decays into an (anti-)muon and a muon-neutrino,which subsequently decays to a positron, an anti-muon neutrino and an electron-neutrino which escape the sample.The difference in the positron counts in the forward (F) and backward (B) detectors normalised by the total count, theasymmetry function A ( t ) , gives the spin relaxation of the (anti-)muon in the sample. Section 7 HIGH-PRECISION MEASUREMENT OF DIFFUSION AND MAGNETO-SPIN PROPERTIES
Page 36 eview Article: Honeycomb Layered Oxidesable to sense the local nuclear magnetic field in the layered oxide when it is in a paramagnetic state, a be-haviour that can mathematically be expressed using a static Kubo-Toyabe (KT) function, as is also shownin
Fig. 19b . When alkali-ion diffusion occurs, the local nuclear field haphazardly fluctuates and the implantedmuons acquire a dynamic contribution in the KT function through the hopping rate of the cations; thus areable to sense the local field that is randomly fluctuating at an average rate. The mobility of alkali cationscan be increased by temperature beyond a certain critical temperature T c where the alkali cations becomemobile, thus inducing an additional fluctuation in the local mean-square magnetic field leading to a conspic-uous increase of the fluctuation (collision) rate ν → ν ( T ) , where the mean-square magnetic field is given by (cid:104) B Z ( t ) B z ( ) (cid:105) = (cid:104) B z ( ) (cid:105) exp ( − ν t ) . Consequently, the self-diffusion coefficient D self = ∑ ni = N i Z i s i ν related to thediffusion coefficient, D ( T ) by a Boltzmann factor D ( T ) = D self exp ( − E a / k B T ) = ∑ ni = N i Z i s i ν ( T ) is accurately de-termined using the µ + SR measurements by considering the collisions as a Markov process over n pathsof the cations in the 2D honeycomb lattice where N i is the number of cation sites, Z i the vacancy fraction and s i the length of the mean-free path between collisions. Sugiyama, Månsson and co-workers have pioneered the use of µ + SR measurements in the study of both themagneto-spin and alkali-ion diffusive properties in a wide swath of layered materials such as Li M O (where Figure 19
High-precision measurement of magneto-spin and diffusion properties of honeycomb layered oxides relevantto phase transitions. (a) The anti-muon implantation into a honeycomb layered oxide framework with a stoichiometriccomposition of, for instance, K Ni TeO . The anti-muon is expected to bind onto the oxygen ions located in the octahedral( TeO and NiO ) structures of the material altering the typical decay rate of the (anti-)muon. The hopping rate, ν of thediffusing potassium ( K ) cations along the honeycomb depends on their interaction with the anti-muons through theirrandom nuclear magnetic fields which alters the anti-muon decay rate ∆ ν . (b) The analysis of alkali-ion diffusion usingthe dynamical Kubo-Toyabe function, P z ( t ) which describes the relaxation of muon spin polarisation in the presenceof a particular (typically Gaussian) distribution of nuclear magnetic fields of the cations in the honeycomb layered oxidematerial. The total asymmetry function in µ + SR experiments depends on the Kubo-Toyabe function, which depends onthe decay rate of the anti-muons due to transport properties of the cations such as their hopping rate in the material. Thehopping rate in turn determines the self-diffusion coefficient of the material.
Section 7 HIGH-PRECISION MEASUREMENT OF DIFFUSION AND MAGNETO-SPIN PROPERTIES
Page 37 eview Article: Honeycomb Layered Oxides
Figure 20
High-precision measurement of diffusion and magneto-spin properties of honeycomb layered oxides relevantto phase transitions. (a) Presence of an antiferromagnetic spin ordering in K Ni TeO below K revealed by a clearoscillation in the µ + SR time spectra. (b) The onset and evolution of K-ion diffusion revealed by an exponential increasein field fluctuation rate ( = ion hopping rate) from which the activation energy ( E a ) of the diffusion process can be deter-mined. M = Ni and Co ), LiCrO , LiNi / Co / Mn / O , Li MnO and even to NaCoO . Investigationof potassium-ion ( K + ) dynamics in related layered materials is particularly unwieldy, due to the innately weaknuclear magnetic moment of potassium relative to other ions such as lithium ( Li ) and sodium ( Na ). Thisrenders it difficult to capture the dynamics of K + in layered materials using standard techniques such as NMRspectroscopy. As discussed above, the fact that spin-polarised muons possess a strong gyromagnetic moment,makes µ + SR measurements particularly ideal for capturing the dynamics of cations such as the K + with anextremely weak nuclear magnetic moment in materials. For clarity, the nuclear magnetic moment/gyromagneticratio of K ( µ [ K ] = . µ N, . MHz/T) is much smaller than for Li ( µ [ Li ] = . µ N, . MHz/T) and Na ( µ [ Na ] = . µ N, . MHz/T). The µ + SR asymmetry function time spectrum of honeycomb layered oxide K Ni TeO (or equivalently as K / Ni / Te / O ) measured below the antiferromagnetic transition temperature( K) as shown in
Fig. 19a , is shown in
Fig. 20a , where precession of the muon (spin-polarisation) occurs. Itis evident that the muon precesses due to the emergence of a spontaneous internal magnetic field, resulting ina clear oscillation of the time spectrum. This is a response that is typically observed from a muon ensemble ina magnetically ordered state (in this case, antiferromagnetic Ni spin ordering in K Ni TeO ). The dependency plot of the fluctuation rate, which is dynamically related to the hopping rate of K + withtemperature, ν ( T ) is shown in Fig. 20b . Between
K and
K, this fluctuation rate increases with tempera-ture signifying the onset of diffusive motion of K + in K Ni TeO . The hopping rate nicely obeys a trend akin toArrhenius equation ν ( T ) = ν exp ( − E Ka / k B T ) from where an activation energy commensurate to approximately E Ka (cid:39) meV is obtained. The diffusion coefficient can be calculated using the above hopping rate assump-tions to yield a diffusion coefficient value of D K ( T ) = . × − cm / s , which is an order of magnitude lowerthan that of layered materials such as LiCoO . Caution needs to be taken when interpreting the µ + SR measurement data, as muons per se can also bemobile in inactive materials. K Ni TeO indeed shows reversible K + extraction and insertion behaviour (thus,electrochemically active) at room temperature; thus, the onset of K + diffusive motion that arises at T > K isirrefutable. The feasibility of utilising µ + SR measurements to further unveil the intricacies of the dynamics ofsuch cations as K + , which tend to possess low nuclear magnetic moments, will expand the pedagogical scope ofcationic intercalation (insertion) and deintercalation (extraction) dynamics within honeycomb layered oxidesSection 7 HIGH-PRECISION MEASUREMENT OF DIFFUSION AND MAGNETO-SPIN PROPERTIES Page 38 eview Article: Honeycomb Layered Oxidesand other layered materials.Section 7 HIGH-PRECISION MEASUREMENT OF DIFFUSION AND MAGNETO-SPIN PROPERTIES
Page 39 eview Article: Honeycomb Layered Oxides
8. Summary and future challenges for honeycomb layered oxides
This review provides an elaborate account of the exceptional chemistry and the physics that make honeycomblayered oxides a fledgling class of compounds. We explore the prospects that would result in myriads of com-positions expected to be reported in the coming decades. Majority of the honeycomb layered oxides reportedtypically engender Li + or Na + as resident cations. However, the further adoption of honeycomb layered oxideswith large-radii alkali ions, such as K + , Rb + and Cs + or even coinage metal ions such as Ag + , H + , Au + , Cu + , et cetera , is expected to further increase the compositional space of related compounds, unlocking manifoldfunctionalities amongst this class of materials. In fact, preliminary theoretical computations have affirmed thefeasibility of preparing honeycomb layered oxides encompassing cations such as Rb + , Cs + , Ag + , H + , Au + , Cu + , et cetera to adopting, for instance, a chemical composition of A Ni TeO , where A = Rb , Cs , Ag + , et cetera . Bythe same token, synthesis of honeycomb layered materials that encompass alkaline-earth metals such as A = Ba , Sr , et cetera has also been proposed as another avenue of augmenting the various combinations of these mate-rials. Indeed, studies pertaining compositions such as A Ru O , where A = Ba , Sr , et cetera have recently beenreported. Undoubtedly, this class of materials offers an extensive platform worthy of pursuit in the comingyears (
Fig. 3a ).To predict or interpret possible emergent features of honeycomb layered oxides, it is crucial to understandtheir unique structural frameworks and the local and bulk atomistic changes they undergo. A combination ofcrystallography techniques that include transmission electron microscopy (TEM), neutron diffraction and X-raydiffraction are expected to offer a holistic view of the arrangement of atoms within the honeycomb lattice andthe global order of atoms within the honeycomb structure of the new materials. Moreover high-resolution TEMat low temperatures, as can be availed by cryogenic microscopy, is a possible route to discern the arrangementof transition metal atoms in the honeycomb lattice at low temperatures where transitions tend to occur.
Experimental and theoretical reports on the structures of these materials has already revealed some stackingdisorders with honeycomb layered species such as Na Ni SbO , Na Zn TeO and Na Ni TeO , associating themwith emergent functionalities such as phase transitions, magnetic ground states and ionic diffusion. Although defects have been known to have both prolific and detrimental effects on honeycomb layered oxides,they still remain vastly underexplored. Nonetheless, de novo computational and experimental techniques areexpected to uncover new defect physics and chemistry that will expand their uses into multiple fields.On another front, doping offers a prospective route to availing more possibilities with a broader scopeof chemical compositions that display improved electrochemical and additional magnetic properties. Froman electrochemical perspective, doping with non-magnetic atoms such as magnesium or strontium generallyreinforces the crystalline structure by suppressing electrochemically-driven phase transitions, whilst increasingthe thermodynamic entropy of the materials. Increased entropy has added advantages that include raising theworking voltage as well as facilitating multiple redox electrochemistry during battery operations as elucidatedin
Fig. 11 and
Fig. 14 . Relating to ionic conductivity, partial doping of the transition metal atoms in thehoneycomb slab with aliovalent or isovalent atoms is a pertinent strategy to increase the ionic conductivityof honeycomb layered oxides. For instance, partial substitution of Zn + with Ga + in Na Zn TeO solid-stateelectrolyte (with a wide voltage tolerance) aids to increase the Na + -ion mobility (conductivity) due to increasedformation of Na + vacancies. Theoretical investigations done by Sau and co-workers have accentuated theprofound effect of transition metal substitution in Na M TeO ( M = Ni, Zn, Co, Cu). In contrast,doping with magnetic atoms, as shown in
Fig. 8 , reveals fascinating magnetic behaviour that places honeycomblayered oxides amongst the exotic quantum materials.Moreover, topochemical reactions, for instance, chemical ion-exchange of Rb or silver ( Ag ) with potassium( K ) in K Ni TeO can aid build an entire host of new oxide materials with a wide swath of physicochemicalSection 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES Page 40 eview Article: Honeycomb Layered Oxidesproperties. Indeed, such a design strategy has proven effective in the synthesis of Ag Ni BiO , for instance, viatopochemical ion-exchange of Li Ni BiO . Additionally, the introduction of alkali cations with differing ionicradii makes the tuning of the distance between the honeycomb layers (interslab/interlayer distance) possible;thus presenting avenues to tune the interlayer magnetic couplings as discussed in
Fig. 10 . This guaranteesthe feasibility to not only adjust the electrochemical properties but also to tweak the physicochemical aspectssuch as the magnetic dimensionality of the honeycomb lattice. This calls for further exploratory synthesis tobe augmented with computational protocols, as schematically adumbrated in
Fig. 21 to expedite the design ofnew honeycomb layered oxide compositions.There has been significant progress in the physics entailing topological states, for which honeycomb lay-ered oxides play a pivotal role in advancing this topical field. In this review, we have discussed the Kitaevand Haldane magnetic (spin) interactions within the honeycomb lattice that offer a path to the experimen-tal realisation of Kitaev quantum spin liquid and a quantum anomalous Hall insulator (Chern insulator) re-spectively.
In addition, higher-order magnetic interactions induced by the angle betweenthe spins of the magnetic cations, introduces other interactions: mainly, the Heisenberg and asymmetric /Dzyaloshinskii-Moriya (DM) interactions.
Due to the additional angular space-time dependent degree offreedom, these interactions are considered of higher order and thus very elusive to realise without the presenceof, for instance, single-crystals of target honeycomb layered oxide materials. Irradiating circularly-polarisedoscillating electric fields on preferably single crystals within a Floquet model (theory) is a plausible route torealising DM interactions within honeycomb layered oxide materials, as has been suggested by several au-thors.
The primary significance of these interactions is the evaluation of magnetic skyrmions -quasi-particles that have been predicted to exist in certain magnetic condensed matter systems such in mag-netic thin films either as dynamic excitations or stable/metastable configurations of spin; which shows greatpromise in topological quantum computing applications.
Regarding single-crystal growth, the high thermal stability of honeycomb layered oxides, such as tellu-rates, bismuthates and antimonates, makes them suitable for crystallisation at high temperatures conducivefor their preparation. In fact, the possibility of growing single crystals in honeycomb layered oxides (suchas Na Ni TeO , Na Cu TeO , Na Cu SbO and Na Co TeO ) using high-temperature solid-state reactions hasalready been achieved. Another fascinating pursuit will be the design of thin films from honeycomb lay-ered oxide materials, either using molecular beam epitaxy (MBE), atomic laser deposition (ALD) or pulsed laserdeposition (PLD), which will aid to accurately visualise the presence of magnetic skyrmions or any emergenttopological physics that covers, inter alia , superconductivity, magneto-resistance and ferro-electricity. Moreover,an extension of the µ + SR technique, namely low energy µ + SR (LEM) offers the unique possibility to tune themuon implantation depth into samples from 5-500 nm. Thus, synthesising thin film battery cells, exempli gratia from honeycomb layered oxide materials, allows one to gain unique access to depth-resolved studies of thedynamics of the cations at and across the buried interfaces, isto es the solid-state cathode / electrolyte / anode,respectively.
A plethora of unprecedented amazing phenomena may also be found when honeycomb layered oxidesare subjected under high-pressure (stress) conditions. This has, amongst other things, the effect of makinghigher-order interactions finite and thus non-negligible. In particular, exerting pressure perpendicular to thehoneycomb slabs bring into play 3D interactions that may have been otherwise negligible. Experimentally, Na Cu TeO shows new bond coordination (dimerisation of Cu bonds) at high pressure, leading to a changein the magnetic properties technically referred to as magnetic phase transitions. Generally, high pressureexerted in these layered oxides can introduce defects or microstructure evolutions that may show great potentialfor novel functional materials. Although the global topology of honeycomb layered materials is robust againstlocal defects, whenever these defects are related to topological invariants ( exempli gratia
Berry’s phase,
Section 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES
Page 41 eview Article: Honeycomb Layered Oxidesthey will affect global properties of the material such as phase transitions, as exemplified in
Fig. 15 , Fig. 16 and
Fig. 17 . Phase transition phenomena inherent in these classes of honeycomb layered oxide materialswill certainly necessitate the use of spectroscopic techniques such as muon spin relaxation ( µ + SR), as well ascomputational and theoretical techniques as displayed in
Fig. 21 and
Fig. 22 respectively, to discern the natureof the spin interactions innate at high-pressure regimes. Moreover, resolution at the atomic-scale of relatedfunctional materials when subjected to ultra-high pressure will attract tremendous research interests in thecoming years.The heightened interest in oxide materials based on honeycomb layers is expected to spearhead the designof a new generation of materials that promise to make remarkable contributions in the fields of energy, elec-tronic devices, catalysis, and will ultimately benefit the scientific community in a broad swath of fields in thecoming decades, as can be envisaged in
Fig. 23 . Recent reports are also emerging on honeycomb layered ox-ides as photocatalysts, optical materials, superfast ionic conductors, and so forth.
A grand challenge with most of these materials lies in their handling. Particularly for honeycomb layered ox-ides comprising alkali ions with large radii such as potassium and rubidium, handling demands the presenceof a controlled atmosphere ( videlicet , storage in argon-purged glove boxes) as they are sensitive to moisture
Figure 21
Computational design techniques that typically can be applied to simulate various physicochemicalproperties of honeycomb layered oxide frameworks. The schematic illustrates the potential of using these computationaltechniques for the designing of new chemical compositions of honeycomb layered oxide materials.
Section 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES
Page 42 eview Article: Honeycomb Layered Oxides(hygroscopic) and air. Future work should also focus on the improvement of the stability of related honeycomblayered oxides, for instance, when exposed to air; to enable handling and mass production of these materialsin ambient conditions. Their instability can be contained and controlled, for example, by tuning their chemicalcomposition. Partial substitution of the constituent transition metal atoms is a possible route, as has been notedwhen Na Ni SbO is partially substituted even with a minuscule amount with Mg , Mn or even Ru . Onanother front, the use of multiple transition metals in equivalent amounts has also been presented as a newroute for the design of stable layered oxides (often referred to as ‘high-entropy oxides’) with unique physico-chemical properties.
Although this concept presents new possibilities for the design and application ofhoneycomb layered oxides, it is still in infancy with a lot of growth potential.In brief, partial substitution also induces a change in the phase transitions observed when alkali cationsare electrochemically extracted, as is the case when they are used as battery materials. Hygroscopicity presentsanother avenue for tuning the interslab distance and editing electrochemical profiles in some materials bringingforth several advantages such as superconductive phase transitions, as has been noted in layered
NaCoO whenhydrated. Honeycomb layered oxides (particularly for compositions incorporating pnictogen or chalcogen atoms that
Figure 22
Schematic of the rich science anticipated in honeycomb layered oxide materials relating to topol-ogy, curvature, geodesics, Chern-Simons theory, Liouville’s equation, Brownian motion and Bose-Einstein conden-sates.
Section 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES
Page 43 eview Article: Honeycomb Layered Oxideshave been delved in this review) can serve as high-voltage cathode materials for rechargeable batteries, assummarised in
Fig. 12 , exhibiting theoretically high capacities. A challenge is their safe and stable operationat high-voltage regimes; warranting the adoption of stable electrolytes that can tolerate high-voltage batteryoperation. Ionic liquids, which consist of organic or inorganic anions and organic cations, manifest a plenitudeof desirable properties. Paramount amongst them is their low flammability, good chemical stability and excel-lent thermal stability.
In particular, the inherently large voltage tolerance makes ionic liquids propitiouswhen matched to high-voltage layered cathodes during battery operation. Stable performance of high-voltagelayered cathode materials using piperidinium-based ionic liquids has been shown; likewise, assessmentof high-voltage honeycomb layered oxides using stable electrolytes (such as ionic liquids) is a plausible routefor harnessing their high electrochemical performance. A schematic list of the choice of stable electrolytes,especially ionic liquids, for honeycomb layered oxide cathode materials is furnished in
Fig. 11 . On anothernote, exotic redox chemistry can be manifested in honeycomb layered oxides. For instance, Li FeSbO is cur-rently amongst model materials to study oxygen anion redox chemistry; a topical area in battery researchnowadays. Much room still exists in the search for related honeycomb layered oxides.
Figure 23
Diversity of honeycomb structures in various realms of science and technology. The schematic highlights thefuture perspectives of honeycomb layered oxides that can be envisioned such as superconductivity, phase transitions,photocalysis, thermochromism, spin dynamics, photovoltaics and optics.
Section 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES
Page 44 eview Article: Honeycomb Layered OxidesAt this juncture it begs the question: quo vadis , honeycomb layered oxides? The rich electrochemical,magnetic, electronic, topological and catalytic properties generally innate in layered materials, indubitablypresent a conducive springboard to break new ground of unchartered quantum phenomena and the coexistentelectronic behaviour in two-dimensional (2D) systems. It is our expectation that this will unlock unimaginableapplications in the frontier fields of computing, quantum materials and internet-of-things (IoT).Finally, the vexing question of why magnetic atoms in the slabs of these layered oxides conveniently alignin a honeycomb architecture, to our knowledge, remains to be addressed; an attestation that the landscape ofhoneycomb layered oxide materials still remains broad and uncharted, moving forward into this new age ofavant-garde innovation. An eminent mathematician has elegantly posited a solution to why bees prefer thehoneycomb architecture in what now is emerging as ‘the Honeycomb conjecture’.
Presumably, it is througha review of the materials found in nature that we can glean insights for future design in this universe ofhoneycomb layered oxide materials.Section 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES
Page 45 eview Article: Honeycomb Layered Oxides
Acknowledgements
This work was conducted under the auspices of the National Institute of Advanced Industrial Science Technology(AIST), Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 19 K15685) and JapanPrize Foundation. Part of this research was also supported by the European Commission through a MarieSkłodowska-Curie Action and the Swedish Research Council - VR (Dnr. 2014-6426, 2016-06955 and 2017-05078), the Carl Tryggers Foundation for Scientific Research as well as the Swedish Foundation for StrategicResearch (SSF) within the Swedish national graduate school in neutron scattering (SwedNess). The authorswould also like to thank Dr Minami Kato, Dr Kohei Tada, Dr Ola Kenji Forslund, Dr Elisabetta Nocerino, DrKonstantinos Papadopoulos, Mr Anton Zubayer and Dr Keigo Kubota for fruitful discussions on this manuscript.We acknowledge that this review paper was proofread and edited through the support provided by Edfluentservices. T. M. gratefully acknowledges Natsumi Ishii and Rei Ishii for the unwavering support in conductingthis work.Section 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES
Page 46 eview Article: Honeycomb Layered Oxides
Conflicts of interest
There are no conflicts to declare.Section 8 SUMMARY AND FUTURE CHALLENGES FOR HONEYCOMB LAYERED OXIDES
Page 47 eview Article: Honeycomb Layered Oxides
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