Misfit Layered Compounds: Unique, Tunable Heterostructured Materials with Untapped Properties

Building on discoveries in graphene and two-dimensional (2D) transition metal dichalcogenides, van der Waals (VdW) layered heterostructures - stacks of such 2D materials - are being extensively explored with resulting new discoveries of novel electronic and magnetic properties in the ultrathin limit. Here we review a class of naturally occurring heterostructures - so called misfits - that combine disparate VdW layers with complex stacking. Exhibiting remarkable structural complexity and diversity of phenomena, misfits provide a platform on which to systematically explore the energetics and local bonding constraints of heterostructures and how they can be used to engineer novel quantum fabrics, electronic responsiveness, and magnetic phenomena. Like traditional classes of layered materials, they are often exfoliatable and thus also incorporatable as units in manually or robotically stacked heterostructures. Here we review the known classes of misfit structures, the tools for their single crystal and thin film synthesis, the physical properties they exhibit, and computational and characterization tools that are unraveling their complexity. Directions for future research are also discussed.


Introduction
Heterostructured materials are a class of materials that are composed of two or more component materials with generally similar crystal structures. They were initially proposed as semiconductor materials that could provide rich properties and insight into physical properties arising from interfacial interactions, and in recent years have been the subject of intense study due to their potential for advancing semiconductor manufacture as well as enabling new technologies 1-2 such as quantum fabrics. In the most general case, the distinct materials layers making up a heterostructure will have different electronic band structures, different charge carriers and concentrations, and their close proximity to each other will permit charge transfer and potentially crossover, and the two structures may be either physically or chemically bonded together. In the case of physical bonding, this typically will take the form of layering of the two materials on each other with bonding via "intermolecular" van der Waals (VdW) forces. The component materials can be anything from solid state inorganics to two-dimensional (2D) layered materials such as graphene, to polymers and organic molecules. Substantial recent interest has focused on 2D transition metal dichalcogenides, and transition metal halides, as building blocks to provide novel electronic and magnetic properties [3][4][5] . Another recent insight is that by controlling the rotationor twist anglebetween adjacent layers, new, long wavelength, periodicities can be introduced that can serve as an artificial lattice exhibit rich physics including superconductivity, and strong correlations from narrow bands [6][7] .
In this context, it is interesting to ask: what other flavors of layered materials are known and might be usable in such technologies? What new properties and richness can result from their study? This review introduces the not widely known family of layered materials known as misfits 2,[8][9] , which provide a new axis on which to explore heterostructured functionality. They provide a platform on which to couple distinct physical phenomena arising from layers of mutually distinct symmetries (typically tetragonal/cubic and trigonal/hexagonal), while in many cases retaining well-defined crystal structures (that require four or five basis vectors to describe, i.e. 4D or 5D, instead of the traditional three). One specific example is the tuning of charge density wave and superconducting transitions in TiSe2 and NbSe2 dichalcogenidesthe degree of charge transfer and increase in 2D nature of individual layers has profound effects on the observed behaviors [10][11] . This review is organized as follows: Section I defines misfits. Section II reviews the thermodynamics of formation and synthetic techniques used to prepare misfits. Section III provides a survey of known misfits and their synthesis methods. Sections IV and V describe the importance of local bonding and effective strain in governing the behavior of misfits. Section VI reviews emerging theoretical and experimental methods capable of capturing the interfacial electronic structures in misfits. Section VII details methods used to determine misfit crystallographic structures. Section VIII details electronic and magnetic phenomena exhibited by known misfits. Section IX gives near term frontiers in new quantum and related phenomena potentially exhibitable by this unique family of materials, and Section X describes future directions to advance the synthesis of misfit materials.

I. What is a misfit?
Misfit layered compounds are a class of heterostructured materials that consist of two different materials layered on each other, forming an ordered superstructure 2,8 . One layer is typically a cubic rock salt-type structure, for which the archetypal material is NaCl, shown in  Misfit layered compounds are thus related to other classes of heterostructured materials in that they also form large superstructures from their component materials and there can be significant modification of electronic and physical properties, as well as interactions between the component materials via interfacial processes, which can produce interesting physical phenomena [10][11][12][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] . There are significant structural distortions which further modifies the physical properties of the bulk material due to the lattice mismatch between physically dissimilar components. Controlling the level of lattice mismatch is crucial when attempting to synthesize a misfit layered compound. Too little lattice mismatch and the resulting material will not be a misfit. Too much mismatch, and the material will not be able to form. If the latter case happens, there is a significant chance of byproduct production from side reactions taking place. For example, because elemental chalcogenides and related compounds will be reactive, if the conditions are not right for misfit formation, they may form other, sometimes more complicated chalcogenides, or in other cases intermetallics may form. It is important to consider the multitude of factors that go into a synthetic plan carefully so as to produce the desired misfit compound instead of the array of undesirable potential side products.

II. Stability and Synthesis of Misfit Compounds
Many misfit compounds are grown in single crystal form under steady state conditions, such as constant pressure and at a constant temperature. This means that they must be overall thermodynamically stable compared to the relevant competing phases. For example, the reaction to synthesize the misfit (PbSe)1+δ(NbSe2): must have a negative free energy over some specific temperature range. When synthesis is done under constant pressure, this means that the Gibbs free energy ∆ = ∆ − ∆ must be less than 0. This is possible under three different scenarios: if ΔHrxn is less than 0, ΔSrxn is greater than 0, or if both of these conditions are true. Considering only local bonding schemes, ΔHrxn is expected to be greater than 0 because in the misfit compound, there are fewer bonds in ideal geometries at the lattice mismatch interface due to lattice distortion that occurs in order to relieve the physical strain generated by such lattice mismatch. However, it is possible for partial charge transfer between individual layers to partially or fully offset this. If the charge transfer mechanism is dominant, as expected in compounds such as (LaS)1.2(CrS2) 29 , then ΔHrxn is less than 0 and the misfit is enthalpically stabilized. However, if charge transfer is not dominant, then ΔHrxn is greater than 0. In this case, the misfit compound may still be thermodynamically stable above some temperature if ΔSrxn is greater than 0. When all the compounds in the reaction are solids, which is the normal situation, the sign and magnitude of ΔSrxn is dependent on the details of bonding and associated influence on vibrational energies. Here, the expectation is that at the mismatched interface, due to fewer bonds being in ideal geometries, will possess lower energy vibrational states compared to the parent materials, and therefore ΔSrxn is expected to be greater than 0. This may be the case in (PbSe)1.14(NbSe2) 16 , in which divalent lead is balanced by dianionic Se. We thus expect some misfits to be stabilized by each of the mechanisms, and identify as a future research direction testing this supposition both theoretically and experimentally in a more general and quantitative fashion than has been done to date.
There are a number of methods that have been utilized in the synthesis of misfit layered compounds. Here we will briefly elaborate on the more common methods of synthesis and factors to take into consideration when choosing a specific method. Examples of compounds synthesized with each method are contained in Table 1 below. When attempting to synthesize misfit layered compounds, it is important to remember that the material by nature contains local structural complexity and thus likely some degree of thermodynamic unfavorability, even though the materials themselves are ultimately stable [30][31][32][33] . According to the Gibbs free energy equation, naïvely used to predict the general favorability of chemical reactions, a given material must be favored in at least one of either thermodynamic enthalpy of formation or entropic gain upon formation, the latter of which increases with temperature. Empirically, many misfit compounds are stable only over a narrow temperature range, usually just before a decomposition, e.g. by melting or vaporization of one or both constituents, suggesting that entropy plays an important role. It is also likely that charge transfer between layers also contributes to stability 34 .
The simplest method of making misfit compounds is by solid state reaction. This method consists of combining the desired elements together and heating them in a furnace to cause the reaction. Elements can be added either as is or as compounds such as halides, chalcogenides or oxides depending on the situation, and inert atmosphere can be obtained easily if needed such as via the use of a sealed, evacuated quartz tube as the reaction vessel. This method is easy to apply, and can produce material relatively quickly, making this technique particularly useful for exploratory syntheses of misfits 35 . Because the technique is simple to prepare and does not require significant lengths of time, it is possible to test many different reaction scenarios at once to try and determine experimentally what temperature range the target misfit compound is stable at by, for example, pressing a single large, homogenized pellet of the desired reagents, then dividing it and heating pieces within a temperature gradient, such as that which exists naturally in box and tube furnaces. This high-throughput method cuts down significantly on the time required for exploratory synthesis to find the necessary thermodynamic conditions of formation.
Additional preparation techniques for the solid state reaction such as pellet pressing or ball milling can be used to encourage reaction by improving the contact surface area between each reagent, as well as uniformly mixing the materials together to improve the chances of generating a homogeneous product phase 35 . Because reactions take place at chemical interfaces, homogenization of the starting reagents is typically required in the general solid state reaction in order to improve the chances of all reagents reacting together at once instead of forming other materials through side reactions, particularly if more than two starting materials are being used.
However, even though there are methods for homogenizing solid powders such as those briefly mentioned above, solid state reactions still lack the level of interfacial contact that is otherwise possible for methods such as solution/flux growth or chemical vapor transport, which hampers the rate of reaction and increases the chances of side reactions leading to unwanted byproducts.
One method that is useful for growing single crystals of the layered misfit compounds is chemical vapor transport. In order to perform this technique, a multizone furnace capable of sustaining a temperature gradient is generally used. The target material is reacted with a transport agent, a volatile material that remains in the gas phase throughout the transport process, in order to obtain a gaseous intermediate complex. This gaseous intermediate travels the length of the reaction vessel, along the temperature gradient. Typically, once the gas reaches the lower temperature zone, the thermodynamic energy present is insufficient to support the formation of the intermediate, and it decomposes into the target material and the transport agent. Ideally, the target material then crystallizes in this area of the reaction vessel, while the transport agent remains in the gas phase and is now available to react with the remaining available material [35][36][37] .
As this process repeats over time, more and larger crystals of the target material form. The primary considerations for growing misfit compounds are determining the temperature at which the desired material is stable, and how to get it or its constituent sublattice components into the gas phase. The lack of well-tabulated thermodynamics of molecular gas phase species at high temperature makes it difficult to design such reactions a priori, necessitating experimental optimization. A third method that is sometimes useful for crystal growth is flux growth. Similar to the ubiquitous solution chemistry, flux growth typically uses a low-melting solid or eutectic instead of a liquid as the solvent. The goal is to dissolve all of the starting materials into the flux, facilitating reactions between them, while the flux remains inert. In this case, the selection of flux is of paramount importance. The ideal flux will be able to dissolve the chosen reagents completely, but will not actually react to form compounds with any of them. Examples of common fluxes include Pb, Te, In, halide salts such as NaCl and KCl, and eutectics of these such as a NaCl-KCl system. Eutectics may be chosen if low-melting reagents are being used in order to get the melting point of the flux lower than any of the reagents [38][39] . This method can be useful for misfits as by dissolving reagents into a molten liquid flux, the issue of getting reagents to react is lessened as they are in better contact through the liquid interface. However, excess flux must be removed in order to extract the crystals that have formed during reaction. This is typically done via one of two methods. The first is hot centrifugation where the reaction vessel is Combining these thoughts, we arrive at the synthesis design scheme given in Figure 2, which can be used to select a suitable approach for the preparation of a given set of misfit materials.

Figure 2. Schematic illustrating a simplistic thought process for determining which synthetic
technique to use for a given misfit compound. Neither every potential factor nor every potential route to each technique are accounted for here, but some of the primary considerations for each technique are provided.

III. Known Misfit Layered Compounds
There are a very large number of misfit layered compounds that have been synthesized ever since the very first recognized as such, the FeCl3/graphene (then known as graphite sheets; the full compound has also been called a "graphite intercalation compound" 61 ) misfit material studied by Cowley and Ibers in 1956. Misfit layered materials generally have a (MX)a(TY2)b type structure or derivative of such; one example of this is the MTX3 stoichiometry, which breaks down to a (MX)a(TX2)b structure. A table is presented below identifying a subset of interesting misfit layered compounds that have been successfully synthesized thus far, organized first by primary anion, then by growth method. In the case of multiple syntheses using different growth methods, the material is placed according to the chronologically first method of synthesis.

IV. Strain and Layered Materials
Strain is a well-known physical parameter applied to structures in order to modify the properties of a material. Most famously strain is generated in thin films grown on specially chosen substrates. Typically, substrates for thin film growth are chosen such that the lattice parameters of the substrate closely approximate the lattice parameters of the target film material.
If there is too much of a mismatch, the film typically will not form. However, by selecting substrates that have only a small amount of lattice mismatch, it is possible to grow thin films that are then under lattice strain in the growth direction [107][108][109][110][111] . Similar principles apply to the growth of misfit layered compounds. The formation mechanism naturally introduces strain due to the overlaying of rock salt and hexagonal substructures in the overall cell superstructure. Like with thin films on substrates, the materials being targeted for misfit formation cannot have too much lattice mismatch or they will not form. The primary difference between misfit formation and thin film growth is that in epitaxial growth, the level of absolute mismatch is generally limited to approximately 3%. However, values approaching the maximum indicate significant strain on the thin film as the structure distorts due to lattice mismatch, generic examples of which are shown in Figure 3 below. In misfits, the level of absolute lattice mismatch can be up to 15% or greater.
Complex rearrangements of local bonding between layers in misfits generate structural distortions during the actual crystal growth, which reduces the amount of structural strain and stabilizes the misfit 89,[112][113][114] . Reprinted with permission from Elsevier 154 .
The local interlayer bonding changes that occur in misfits to stabilize the structure and reduce the amount of strain on it can take a number of forms, many of which aim to distort the structure such that the two component sublattices better match each other. A few examples include structural dislocations, layer buckling, turbostratic disorder, and wholesale changes in the physical form of the material. Layer buckling refers to the "crumpling" of individual layers as a way of relieving structural strain. Instead of a perfect epitaxial relationship, layers can "buckle" or develop a wave-like structure. This is a method of reducing compressive strain and occurs via a variety of mechanisms, including but not limited to thermal expansion/contraction and physical overloading of the substrate via thin film growth 115 . In addition to improving stability, the natural buckling of the mismatched material has been exploited for purposes such as flexible electronics 116 and stretchable solar cells [117][118] because the distortion allows for stretching without destroying the material once removed from the rigid parent substrate. Turbostratic disorder, or a distribution of layer rotational positions, is another consequence of such mismatching. The presence of turbostratic disorder in misfits (called "ferecrystals") has been shown to impart significantly decreased cross-plane thermal conductivity as well, leading to some potential uses as thermoelectrics 32 .
Structural dislocations are somewhat similar to layer buckling in that the physical structure of the layered material or thin film is distorted by missing or cut-off layers of atoms, but is typically present at interfaces, where the crystal structure becomes complicated and is often not well studied at the local level, or occurs only in small areas of the material.
Dislocations often result in a local disruption of the stacking order and occur in the process of relieving misfit strain, and also frequently modulate transport and other physical properties of the material107,119.
Simulated powder X-ray diffraction patterns of misfit layered compounds which illustrate changes in diffraction due to the presence of such forms of disorder, are shown in Figure 4. Panel  The formation of nanotubes and macroscopic cylindrical shapes from bulk misfit crystal sheets is another way in which misfit layered compounds can relieve lattice mismatch strain.
Nanotube misfits can naturally form using the typical growth techniques described above in Section II of this review 89 . Two examples are (PbS)1+x(NbS2)n and (BiS)1+x(NbS2)n, prepared by Bernaerts and coworkers 87 which were prepared by chemical vapor transport. They observed the formation of the cylindrical tubes and postulate that the natural formation of a curved structure was due to lattice mismatch, with the bending axis perpendicular to the lattice vector suffering the greatest mismatch. However, this can only occur on a significant scale in thin crystalline sheets. Much as one has a harder time rolling up paper as more and more pieces are stacked, the amount of curvature decreases as the crystal thickness increases.

V. Physical and Electronic Effects of Strain on Misfit Layered Compounds
Structural strain is able to influence the properties of the film material. Similarly to the situation of applied strain, the most common examples are thin films grown on latticemismatched substrates, but it is also possible to produce strain by other methods, such as applying pressure in certain directions. For example, BaTiO3 has been studied for its desirable ferro-and piezo-electric properties, but its relatively low Tc of 130 K is sometimes too low for usable applications. However, it has been shown that compressive epitaxial in-plane strain can increase the Tc to approximately 773 K, far above room temperature, and can be further increased to greater than 1073 K when the strain is applied in conjunction with defect dipoles 121 .
Additionally, the ferroelectric thin film material CuInP2S6 has been studied under strain and suggesting that the thin film strain was the primary factor causing the piezoresponse contrast 121 .
Of particular interest is that the strained, bubble-like area formed naturally during crystal flake formation & cleavage, a potential indication that this material may be a misfit material itself. The distortion of the layering order necessary to form such a bubble is similar to the other structural distortions that misfit layered compounds undergo in order to relax the strain generated by the large amount of lattice mismatch such as layer buckling or nanotube formation.

VI. Electronic Structure
DFT is frequently used to calculate approximations of the electronic band structure of many solid state materials. By constructing supercell approximates (vida supra), DFT has been used to study the electronic structure of misfit heterostructures. However, such an approach has considerable difficulty in dealing with interfaces where the adjacent layers lack coincident structural periodicity, such as the interfaces typically found in misfit compounds [127][128][129] .
Mismatched interface theory, or MINT, is a recent process developed by Gerber and coworkers in 2020 130 for the purpose of understanding the electronic structure of layered materials at interfaces. Developed specifically for analyzing layered heterostructures, it is relatively simple to implement as it actually uses the standard DFT approximations and functionals. However, the main difference is that it combines the two primary theoretical methods of treating with aperiodic structures, the "cluster" and "supercell", and uses them together to generate the periodic structure required to extract properties, Figure 6. The cluster method generates large clusters of aperiodic material to make an overall periodic structure, whereas the supercell method combines multiple unit cells into a single, much larger cell to get to an overall periodic structure 130 . Additionally, the methods used in MINT are dependent on electronic structure "nearsightedness"; that is, the local electronic properties of the material are dependent primarily on the local electronic structure, and the structural influence decreases rapidly with increasing distance from the point or area of interest. In the context of misfit layered compounds, this means that observable changes in electronic properties would only occur near the rock salt/hexagonal structure interface.   The MINT sandwich method also has a couple of other significant advantages over using traditional DFT methods to calculate the physical properties of misfit compounds. As mentioned previously, one of the primary techniques for dealing with incommensurate crystal structures such as misfit compounds is to use the "supercell" approach, which takes many unit cells and creates a much larger cell that is then approximately commensurate. However, because the cell is so large, computations take much longer as the entirety of the large supercell must be accounted for when extracting the properties. As seen in Figure 7, the crystal structure required by the MINT sandwich method is not extremely large, reducing the computational load and increasing the throughput of the method. Furthermore, because of the very large structure required to approximate periodicity in the traditional supercell approach, extraneous misfit strain is generated because of the many repeating layers required to generate the superstructure. In the MINT sandwich method for this example compound, because it starts with NbSe2 without any material in the interlayer gap and then extracts properties while the LaSe layer grows, extraneous strain is not introduced into the structure which improves the accuracy of the calculated properties 131 .
Electron microscopy techniques, such as high resolution transmission electron microscopy (HRTEM), have been used in the study of misfit layered compounds and contribute a valuable aspect in that these techniques are able to help visualize the crystal structure of the material in a form that is relatively easy to understand. HRTEM is a variant technique of transmission electron microscopy (TEM) that allows for the direct visualization of structures on the atomic scale and is based on the interference behavior of electron waves, using the waveparticle duality. The electron microscope is able to record the amplitude of electron wave interference resulting in contrast images. It is important to realize, however, that aberrations of the imaging lenses used in the microscope will affect the image and because the technique produces contrast images, the images are not necessarily 1:1 reflections of the actual structure.
Care must be taken to interpret the image and assign atomic positions appropriately. However, the technique is very powerful and can assist greatly in verifying the nature of the misfit structure, as shown in Figure 9 and 10 below 84 .   30, 4, 1373-1378. 84 Copyright 2022 American Chemical Society.

VII. Crystallography of Misfit Structures
One of the biggest challenges in analyzing misfit layered compounds is the question of determining the precise crystal structure. Understanding the nature of the crystal structure is crucial to correlating it with the material's properties. In the case of misfit compounds, common techniques used for less complex crystals such as powder and single crystal X-ray diffraction, as well as high resolution electron microscopy and electron diffraction, are also applicable.
One method that attempts to fully describe a misfit crystal structure is to solve using superspace groups in either four or five dimensions (4D or 5D). These groups use four or more crystallographic basis vectors to describe the crystal lattice instead of the traditional three, depending on the number of incommensurate lattice directions. For example, if a misfit compound were incommensurate in the a direction but not in the b or c directions, a (3+1)D superspace group would be needed to fully describe the crystal structure. A list of (3+1)D superspace groups has been published previously 134 and can be understood in terms similar to that of the well-known conventional space groups. For example, the superspace group symmetry possessed by a material such as (Sr2TlO3)(CoO2)1.17 135 or Ba2TiGe2O8 136 can be determined by derivation from diffraction patterns observed via single crystal X-ray diffraction, high resolution transmission electron microscopy, and single crystal neutron diffraction, among other techniques.
However, single crystal X-ray diffraction patterns of incommensurate misfit compounds are more difficult to correctly analyze. First, individual sublattices in a misfit material can often result in partially overlapping reflections. Further, individual sublattices can also interact with each other due to their close proximity and produce weak, but distinct diffraction peaks that do not match with either sublattice. These weak reflections are one example of satellite reflections, a type of diffraction that occurs due to X-ray diffraction off of something that is not the primary lattice of interest 137 . These satellite reflections can help determine that the structure being looked at is actually a misfit compound, as they generally appear in modulated, incommensurate structures due to X-ray diffraction off of multiple sublattices. As an example, in a (3+1)D crystal structure, the required crystal planes have the indices hklm. If the component sublattices are commensurate in the a and b directions, but not the c direction, then the hk00 planes are common to both, and one lattice will have hk0m planes while the other will have hkl0 planes. In the case of true satellite reflections, they will have indices hklm where every index is nonzero, meaning that they contain diffraction from the incommensurate planes. We anticipate that these issues, particularly regarding integration of overlapping reflections, can be remedied by changing the analysis approach: rather than integrating and scaling to obtain a set of hklm/intensity pairs that are then used during model refinement, methods similar to whole pattern fitting, as routinely used for one dimensional (powder) diffraction fitting, and often used for 3D/4D neutron scattering datasets avoid the requirement of disentangling overlapping reflections entirely.
Single crystal X-ray diffraction is one of the most common techniques used and can provide enough information to solve the crystal structure using a superspace formalism.
However, electron diffraction, neutron diffraction and high resolution transmission electron microscopy (HRTEM) have all been used to provide complementary information and potential confirmation of the predicted crystal structure, HRTEM in particular as it allows for visualization of the crystal structure at the atomic level.
A number of different misfit compounds have had their structures determined via a combination of the above techniques. For example, as previously mentioned 135 , (Sr2TlO3)(CoO2)1.17 has had its structure described by monoclinic sublattices that are incommensurate in the b direction via the combination of single crystal X-ray diffraction and electron diffraction, the latter collected via transmission electron microscopy. Additionally, there is a misfit compound, (SbS1-xSex)1.16(Nb1.036S2), which has been studied via a combination of three techniques: single crystal X-ray diffraction, selected area electron diffraction (SAED), and HRTEM 138 . This material was only partially indexed in X-ray diffraction due to significant complexity of the diffraction patterns. Four different sets of unique reflections were identified, two each belonging to individual sublattices, and there was significant diffuse scattering along the stacking direction. This added to the difficulty of fully solving the structure from the X-ray diffraction, and only the Nb1.036S2 layer could be solved successfully using one reflection set.
SAED and HRTEM were performed in order to help resolve the other SbS1-xSex layer as well as confirm the stacking order, respectively. The need for all three of these techniques to more accurately illustrate the true structure of the material, plus the use of a superspace approach to fully describe the crystal structure, underscores the difficulty of studying misfit materials because in order to understand where misfit properties may arise from, it is important to first understand the crystal structure.

VIII. Known Physical Phenomena
A number of interesting physical phenomena, such as superconductivity and complex magnetism, have already been realized in a variety of misfit layered compounds, summarized in Table 2, showcasing their potential in the next generation of devices and encouraging further study as a robust and expansive field.

IX. Frontiers in Physics
Because misfit layered compounds are generally composed of two separate sublattices that interface with each other through a limited contact surface area, it should be possible to generate physical properties in each sublattice that normally are not able to coexist in the same material. However, it is difficult to predict with total certainty which phenomena will be able to coexist in these materials, because the key for multiple orthogonal properties to exist in a single misfit compound is for the rock salt and hexagonal/trigonal layers to host at least one property each. If one layer is found to be simply a passive "observer" capable only of charge transfer, it will be unlikely to possess multiple orthogonal phenomena.
One canonical and likely difficult example to realize in solid state physics is the combination of ferromagnetism and superconductivity. Superconducting materials are defined partially by their complete expulsion of internal magnetic fields below the transition temperature  140 . This process can take place if the non-superconducting part of the interface is anything except an insulator, and the metal/superconductor interface has also been constructed by sandwiching a Pb or Al film electrode between two Au electrodes which served as the metallic part of the interface. From these experiments, it is not so much of a stretch to imagine that this interface could be recreated in a single misfit material via the layering of a metallic sublattice on a superconductor sublattice.
There is also the possible discovery of half-metallicity in a misfit material.
Halfmetallicity refers to a material that, at its Fermi level, exhibits metallic behavior in one spin direction and exhibits a band gap in the other spin direction. This half-metallic behavior has been confirmed in the monolayer FeCl2 material 141 . Further studies have been done attempting to realize half-metallicity in other layered transition metal halides such as TiCl3 and VCl3 142 .
However, to date it has not yet been realized. Where the FeCl3/C misfit comes in is the inherent strain present in the material from the lattice mismatch, which will modify the overall electronic properties, potentially allowing for realization of the half-metallic electronic structure.
Additionally, the lattice strain may produce further property modifications, for example of the magnetic behavior, which could also be of significant interest to the community. Finally, graphene is known to be an electron donor 143 , which could serve as a means of charge transfer to realize the half-metallic behavior prediction experimentally.
One final area of potential interest is the combination of the obstructed atomic insulator with a more standard electronic structure. The obstructed atomic insulator (OAI) is unique in that it is a topologically nontrivial insulating material, but the band representations deduced from irreducible representations are not derived from occupied Wyckoff positions; that is to say, charge centers in the insulator are located at positions not occupied by atoms in the crystal structure [144][145][146] . A recent study 146 146 . What is of interest in the context of misfit layered compounds is that a number of 2D exfoliable forms of the group 14 elements exist in the form of silicene, germanene and most famously graphene, and graphene has been combined previously with FeCl3 as noted above to form a misfit layered compound. Due to the layered nature of misfit compounds, it may be possible to access the OAI surface states of these elements and combine them with a larger variety of materials with different phenomena such as metallic behavior, superconductors or even potentially half-metallic states in the case of FeCl3/graphene to investigate the effects each has on the overall electronic structure and what physical properties may arise.
Overall, there are a number of potential new research areas where misfit layered compounds may exhibit novel combinations of physical phenomena and could lead to new discoveries. In Table 3 below, we present a list of some new frontiers in physics that may warrant further study.
Standard Electronic phases(Insulator/Semiconductor/Metal)/Superconductor Table 3. Summary table of a selected list of potential frontiers in physical phenomena that may be realized in misfit layered compounds.

X. Frontiers in Misfit Synthesis
Misfit layered structures have come a long way since they were first made by layering graphene and FeCl3 in 1956 67 but there are significant avenues that have yet to be explored.
Chief among these is the potential synthesis of mixed-anion misfit layered compounds. Up to this point, misfit materials have used the same anion in both of the sublattice materials, typically sulfur, selenium, or oxygen. However, it is conceivable that two different anions could be used in the synthesis, as long as the appropriate sublattices, the rock salt and the hexagonal, can still be formed. For example, a rock salt sulfide such as LaS could be combined with a transition metal diselenide such as NbSe2 to form a misfit layered compound. Changing the anion for one of the sublattices is likely to change the electronic band structure of the sublattice, which will in turn influence the band structure of the whole material. This can potentially produce desirable physical phenomena such as metallic or superconducting behavior. Additionally, the concept of Os0.55Cl2 is a recent layered material that displays significant structural defects with a large number of vacancies at the metal site as well as a complete lack of magnetic ordering down to 0.4 K, which also suggests a potential spin liquid state 149  example, the FeCl3/graphite system, in the context of battery electrodes [152][153] . Additionally, intercalation of alkali metals such as potassium and lithium have already been attempted for certain misfit compounds such as (SnSe)1.16(NbSe2) and (PbSe)1.14(NbSe2) for use as battery anodes in order to improve potassium storage and overall battery cycling performance, and these materials have been shown to hold on to the alkali metal more efficiently 16 .
Finally, while misfit layered compounds are already under inherent strain due to their nature as mismatched sublattices, there is potential for these materials to be deposited as thin films on solid substrates such as oriented silicon, gallium arsenide (GaAs), or lanthanum aluminum oxide (LaAlO3), among others. Depending on the choice of substrate, a secondary lattice mismatch can be generated between the overall misfit crystal structure and the substrate, causing additional strain on the misfit structure and likely modifying its electronic properties further. A significant number of misfit compounds have been grown on (100) oriented silicon substrates 20-27,32,33,41-60 which generates a new potential angle of investigation for misfit layered compounds via the misfit-substrate interface.

Conclusion
Misfits are a fascinating class of naturally occurring heterostructures that, compared to more traditional crystalline materials families, have been underexplored to date due to challenges in characterizing structures and correlating with emergent physical properties. With the recent theoretical and experimental advances capable of investigating the structure and electronic and magnetic properties of heterostructures with mismatched alignment between layers, misfits provide an avenue to greatly expand the repertoire of 2D and heterostructure phenomena.