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Computational Design of FlatBand Material
Nanoscale Research Letters volume 13, Article number: 63 (2018)
Abstract
Quantum mechanics states that hopping integral between local orbitals makes the energy band dispersive. However, in some special cases, there are bands with no dispersion due to quantum interference. These bands are called as flat band. Many models having flat band have been proposed, and many interesting physical properties are predicted. However, no real compound having flat band has been found yet despite the 25 years of vigorous researches. We have found that some pyrochlore oxides have quasiflat band just below the Fermi level by first principles calculation. Moreover, their valence bands are well described by a tightbinding model of pyrochlore lattice with isotropic nearest neighbor hopping integral. This model belongs to a class of Mielke model, whose ground state is known to be ferromagnetic with appropriate carrier doping and onsite repulsive Coulomb interaction. We have also performed a spinpolarized band calculation for the holedoped system from first principles and found that the ground state is ferromagnetic for some doping region. Interestingly, these compounds do not include magnetic element, such as transition metal and rareearth elements.
Background
Electronic and magnetic properties of materials are mostly determined by their energy dispersion. For example, electronic conductivity is high when the valence/conduction band has large energy dispersion. Magnetic properties arise when the band dispersion is small. Usually, the band dispersion is determined by the character of atomic wave function. Therefore, most magnetic compound contains magnetic elements, such as transition metal elements and rareearth elements. If we can synthesize a magnetic material containing no magnetic element, its impact is immeasurable.
In this paper, we propose such candidate of ferromagnet without containing magnetic element by using first principles calculation. The bandwidth composed of the orbital of nonmagnetic atom is usually small, but in some cases, its bandwidth becomes extremely small. This narrow band is called as flat band, and if the Fermi level is just on this flat band, it is possible to take a ferromagnetic ground state. There are many studies of flat band physics, which are summarized in the review articles [1, 2].
In this paper, we briefly introduce the flat band. At first, we consider a simple tightbinding (TB) Hamiltonian \( {H}_0=\varepsilon \sum \limits_i{c}_i^{+}{c}_i+\sum \limits_{i,j}{t}_{ij}{c}_i^{+}{c}_j \) (1), where c_{ i } denotes the annihilation operator on isite, ε is the onsite energy, and the hopping integral t_{ ij } is finite and isotropic (= t) only when the site i and j are in the nearest neighbor. Quantum mechanics shows that large hopping integral gives a large energy dispersion in most cases. For example, if H_{0} is defined on a simple square lattice, the obtained energy dispersion is E (k) = ε + 2t (cosk_{x} + cosk_{y}). The band width W = 8t, which is proportional to t. Since t is determined by the overlap of the atomiclike wave functions, if a band consists of s or porbitals, it becomes a broad band. In that case, the magnetic ground state is not expected because the spinaligned state loses a large kinetic energy.
However, on some specific lattices, this simple relation W ~ t does not hold. For example, if H_{0} is defined on a pyrochlore lattice, doubly degenerated dispersionless bands appear. Pyrochlore lattice is defined as the Asite sublattice of the pyrochlore structure, see Fig. 1. We can mathematically prove the emergence of this flat band, for example, see ref [3]. There are several lattices generating flat bands besides the pyrochlore lattice, for example, 2D checkerboard lattice, 2D kagome lattice, and so on [1, 2]. Interestingly, we can prove that if this flat band is halffilled, then the system has the unique ferromagnetic ground state for any positive value of intraatomic Coulomb interaction U [4]. This type of lattice which derives flat band is known as “geometric frustrated lattice” in the word of localized spin system. In fact, a series of pyrochlore oxides R_{2}Ti_{2}O_{7} (R: rareearth element) have various novel magnetic properties, such as quantum spin liquid, spin ice, and magnetic monopole [5,6,7,8,9]. More recently, almost perfect frustration was found in the frustrated dimer magnet Ba_{2}CoSi_{2}O_{6}Cl_{2} [10]. An effective theory of this compound has been constructed, and this theory can explain the curious magnetic properties in the high magnetic field [11].
Besides the ferromagnetic ground state, it is theoretically suggested that flat band induces several interesting properties, such as superconductivity, quantum Hall effect, and various topological states [12,13,14]. Therefore, it is very important to find a compound that actually has a flat band. There are several theoretical attempts to realize the flat band using lithography [15] or photonic lattice [16]. Besides these mesoscopic materials, we note that a carefully designed 2D indiumphenylene organometallic framework (IPOF) shows an excellent flat band [17]. Interestingly, this flat band is topologically nontrivial and may serve a stage for hightemperature fractional quantum Hall effect. Despite these vigorous studies, the expected magnetic longrange order has not been achieved yet, probably because these attempts are limited for 2D system. There is another interesting study which has shown a longrange magnetic order invoked by an organic molecule absorbed on graphene [18]. However, the microscopic origin of this magnetic order is still unclear.
As is mentioned above, in order to make ferromagnetism appear using this flat band, it is necessary to adjust the Fermi level just on this flat band. In most pyrochlore oxides A_{2}B_{2}O_{7}, the Fermi level is on the band composed of the Bsite orbital. However, since the flat band has to be formed on the Asite sublattice (pyrochlore lattice), the Bsite ion is to be inert. Moreover, since the hopping integral needs to be isotropic, the Fermi level must be on the sorbital of the Asite.
Imposing the above conditions, we can choose the candidate of pyrochlore oxides to have the flat band at the top of the valence band:
Since the top of the valence band has the As character, Asite ion is typically (a) Tl^{1+}; (b) Sn^{2+}, Pb^{2+}; and (c) Bi^{3+}. All of these ions have the (5s)^{2} or (6s)^{2} configuration. The Bsite must be inert, so we can choose (a) Mo^{6+}, W^{6+}; (b) Nb^{5+}, Ta^{5+}; and (c) Ti^{4+}, Sn^{4+}. All of these Bsite ions have closed shell, i.e., (nd)^{0} or (np)^{0} configuration where n = 3, 4, 5.
Among the above combination, we focus on three compounds:
The compounds (b) Sn_{2}Nb_{2}O_{7} and (c) Bi_{2}Ti_{2}O_{7} have already been synthesized [19,20,21,22,23], while (a) Tl_{2}Mo_{2}O_{7} has not been reported yet. However, an analogous pyrochlore oxide Tl_{2}Ru_{2}O_{7} has been already synthesized and shows unique metalinsulator transition [24]. Since the atomic radius of Mo and Ru are similar, we expect that Tl_{2}Mo_{2}O_{7} can be synthesized in certain condition. Interestingly, both (b) Sn_{2}Nb_{2}O_{7} and (c) Bi_{2}Ti_{2}O_{7} are known to be a candidate of photocatalytic material.
We have performed a first principles calculation for these compounds. This paper is organized as follows: In the “Methods” section, the method of calculation and the crystal structures that we calculated are described. In the “Results and Discussion” section, we show the calculated results and give some discussion. Summary is described in the “Conclusions” section.
Methods
We have calculated the electronic structure of Tl_{2}Mo_{2}O_{7}, Sn_{2}Nb_{2}O_{7} and Bi_{2}Ti_{2}O_{7} from first principles. For simplicity, we assumed that they all have ideal A_{2}B_{2}O_{6}O′ pyrochlore structure. Since there are two oxygen sites, so we call them O and O′ to distinguish them. We have used a fullpotential augmented planewave (FLAPW) scheme and the exchangecorrelation potential was constructed within the general gradient approximation [25]. We used the computer program WIEN2k package [26]. The parameter RK_{max} is chosen as 7.0. The kpoint mesh is taken so that the total number of mesh in the first Brillouin zone is ~ 1000. We have also optimized the crystal structure, with fixing the space group symmetry. The crystal structure of A_{2}B_{2}O_{6}O′ is as follows: Space group Fd3m (#227), A (0,0,0), B (1/2,1/2,1/2), O (x,0,0), and O′ (1/8,1/8,1/8). For Sn_{2}Nb_{2}O_{7} and Bi_{2}Ti_{2}O_{7}, we used experimental lattice parameter. For Tl_{2}Mo_{2}O_{7}, we also optimized the lattice parameter (a) and obtained a = 10.517 Å, which is very close to the recent experimental lattice parameter for the analogous compound Tl_{2}Ru_{2}O_{7} [27]. In this structure, the only one free parameter is the position of O (= x). The convergence of atomic position is judged by the force working on each atom that is less than 1.0 mRy/a.u.
Results and Discussion
Band Structure
Figure 2 shows the energy band dispersion of Tl_{2}Mo_{2}O_{7}, Sn_{2}Nb_{2}O_{7} and Bi_{2}Ti_{2}O_{7} from first principles. First, we focus on the middle panel, Sn_{2}Nb_{2}O_{7}. The obtained band dispersion well agrees with the previous studies, while the existence of the quasiflat band was not referred [19, 28]. We see that the shape of the top of the valence band (− 3~0 eV) is similar to the tightbinding model shown in Fig. 1b. This agreement is rather surprising because this model only use two parameters, ε and t. So as a first approximation, the valence band of Sn_{2}Nb_{2}O_{7} is described by a TB band consisting of the “Sns” orbitals. Here, we note that these “Sns” orbitals are the antibonding orbitals consisting of Sns and O′p orbitals. The main difference between the abinitio bands and the TB bands is the flatness of the band at the energy ~ 0 eV, which means that the hopping integrals other than the nearest neighbor Sn atoms is also needed to fit the abinitio bands precisely.
Next, we discuss the band structure of Tl_{2}Mo_{2}O_{7}, shown in the left panel of Fig. 2. We can see that the shape of the valence band of Tl_{2}Mo_{2}O_{7} is almost the same with that of Sn_{2}Nb_{2}O_{7}, indicating the existence of the flat band in Tl_{2}Mo_{2}O_{7}. However, the conduction band lowers its energy and the band gap is collapsed. The Mod band is partially occupied unlike the case of Sn_{2}Nb_{2}O_{7}, indicating that the formal ionic configuration Tl^{1+}_{2}Mo^{6+}_{2}O^{2−} _{7} is not appropriate. This result suggests that the analysis by the pointcharge model is quite effective, which suggests that A^{1+}_{2}B^{6+}_{2}O_{7} is not a stable configuration for pyrochlore oxides. The Tls flat band is entangled with the Mod band, similar to the case of an analogous pyrochlore oxide Tl_{2}Ru_{2}O_{7} [29]. A metalinsulator transition is found in Tl_{2}Ru_{2}O_{7} and its cause is ascribed to the hidden Tls flat band [30]. We can expect that this metalinsulator transition will also take place in Tl_{2}Mo_{2}O_{7} if it was synthesized.
Finally, we discuss the band structure of Bi_{2}Ti_{2}O_{7}, shown in the right panel of Fig. 2. The obtained band dispersion well agrees with the previous study [31]. Despite the different shape of the valence band between Bi_{2}Ti_{2}O_{7} and Sn_{2}Nb_{2}O_{7}, the top of the valence band of Bi_{2}Ti_{2}O_{7} is very flat in most part of the symmetry axis in the Brillouin zone. Since the shape of the band is different from that of Fig. 1, the origin of this partial quasiflat band cannot simply be found in the flat band on the pyrochlore lattice. Nevertheless, the quasiflat band and resulting high density of states (DOS) are sufficient to expect realization of ferromagnetism when doping holes. We discuss this point in the next subsection.
Ferromagnetic States
In the previous subsection, we found a quasiflat band at the top of the valence band in Sn_{2}Nb_{2}O_{7}. For Bi_{2}Ti_{2}O_{7}, we also found a partial quasiflat band. Since they are insulator, we have to introduce holes into the quasiflat band to induce ferromagnetism. In the case of the perfect flat band, any value of onsite Coulomb interaction U causes ferromagnetic ground state when the flat band is halffilled [4]. This means that even a wellextended atomic s or porbital can cause the ferromagnetic ground state. As for the case of quasiflat band, a numerical study shows that a certain large U > U_{c} can induce ferromagnetism, where U_{c} is the critical value and U_{c} has the order of the bandwidth W [32]. Since the estimation of U and U_{c} is difficult in actual compound, instead we performed a spinpolarized abinitio calculation. Considering that the band calculation has been successful for describing the ferromagnetic ground state of bcc Fe which also has a narrow band, our approach will be justified. In order to simulate hole doping, we substitute N for O′, namely we calculated Sn_{2}Nb_{2}O_{6}N and Bi_{2}Ti_{2}O_{6}N. Since this substitution reduces two electrons per primitive unit cell (one electron per formula unit), the quasiflat band becomes halffilled.
Figure 3 shows the DOS curve for Sn_{2}Nb_{2}O_{6}N and Bi_{2}Ti_{2}O_{6}N. The abovementioned quasiflat band forms a sharp peak just around the Fermi level. We can see that both compounds become halfmetallic, i.e., the spin state of the electron with energy E = E_{F} (Fermi energy) is fully polarized. The total magnetic moment M is 2.00 μ_{B} per primitive unit cell for both compounds, also indicating that the conduction electrons are fully spinpolarized. The exchange splitting between upspin and downspin band is ~ 0.3 eV for Sn_{2}Nb_{2}O_{6}N and ~ 0.4 eV for Bi_{2}Ti_{2}O_{6}N. These values are much smaller than the exchange splitting in bcc Fe, ~ 2 eV. Since the exchange splitting is approximately determined by the atomic wave function [33], the dband has larger exchange splitting than the s or pband. Nevertheless, since Sn_{2}Nb_{2}O_{6}N and Bi_{2}Ti_{2}O_{6}N have very small bandwidth, the exchange splitting exceeds the bandwidth and halfmetallic ground state realizes.
Conclusions
In this paper, we have shown a guiding principle to design flatband compound. According to this principle, we chose three pyrochlore oxides and investigated their electronic structure by first principles study. Combined with a tightbinding analysis, we found that some compounds actually have quasiflat band. We also found that hole doping toward these compounds leads to the ferromagnetic ground state, despite these compounds do not contain magnetic element. These findings will be a large step to realize not only a flatband system in a compound, but also a ferromagnet without including magnetic element.
Abbreviations
 DOS:

Density of states
 FLAPW:

Fullpotential augmented plane wave
 TB:

Tightbinding
References
 1.
Derzhko O, Richter J, Maksymenko M (2015) Strongly correlated flatband systems: The route from Heisenberg spins to Hubbard electrons. Int J Mod Phys B29:1530007
 2.
Liu Z, Liu F, Wu YS (2014) Exotic electronic states in the world of flat bands: From theory to material. Chin Phys B23:077308
 3.
Bergmann DL, Wu C, Balents L (2008) Band touching from realspace topology in frustrated hopping models. Phys Rev B78:125104
 4.
A. Mielke, J. Phys. A: Math. G E N. 24 (1991) L73, Ferromagnetism in the Hubbard model on line graphs and further considerations. ibid. 24 (1991) 3311, Exact ground states for the Hubbard model on the Kagome lattice. ibid. 25 (1992) 4335
 5.
Ramirez AP (1994) Strongly Geometrically Frustrated Magnets. Ann Rev Mater Sci 24:453
 6.
Canals B, Racroix C (1998) Pyrochlore Antiferromagnet: A ThreeDimensional Quantum Spin Liquid. Phys Rev Lett 80:2993
 7.
Ramirez AP, Hayashi A, Cava RJ, Siddharthan RB, Shastry S (1999) Zero Point Entropy in ‘Spin Ice’. Nature 399:333
 8.
Castelnovo C, Moessner R, Sondhi S (2007) Magnetic Monopoles in Spin Ice. Nature 451:42
 9.
Kadowaki H et al (2009) Observation of Magnetic Monopoles in Spin Ice. J Phys Soc Jpn 78:103706
 10.
Tanaka H et al (2014) Delocalization of Surface Dirac Electrons in Disordered Weak Topological Insulators. J Phys Soc Jpn 83:103706
 11.
Richter J, Krupnitska O, Bbaliha V, Krokhmalskii T, Derzhko O (2018) Thermodynamic properties of Ba_{2}CoSi_{2}O_{6}Cl_{2} in strong magnetic field: Realization of flatband physics in a highly frustrated quantum magnet. arXiv: 1709.10356. https://doi.org/10.1103/PhysRevB.97.024405.
 12.
Kobayashi K, Okumura M, Yamada S, Machida M, Aoki H (2016) Superconductivity in repulsively interacting fermions on a diamond chain: Flatbandinduced pairing. Phys Rev B94:214501
 13.
Wang Y, Yao H, Gu Z, Gong C, Sheng DN (2012) NonAbelian Quantum Hall Effect in Topological Flat Band. Phys Rev Lett 108:126805
 14.
Hatsugai Y, Maruyama I (2011) Z_{Q} topological invariants for Polyacetylene, Kagome and Pyrochlore lattices. Europhys Lett 95:20003
 15.
Tadjine A, Allan G, Delerue C (2016) From lattice Hamiltonians to tunable band structures by lithographic design. Phys Rev B94:075441
 16.
Ruostekoski J (2009) Optical Kagome Lattice for Ultracold Atoms with Nearest Neighbor Interactions. Phys Rev Lett 103:080406
 17.
Liu Z, Wang ZF, Mei JW, Wu YS, Liu F (2013) Flat Chern Band in a TwoDimensional Organometallic Framework. Phys Rev Lett 110:106804
 18.
Garcia M et al (2013) Longrange magnetic order in a purely organic 2D layer absorbed on epitaxial graphene. Nature Phys 9:268
 19.
Hosogi Y, Shimodaira Y, Kato H, Kobayashi H, Kudo A (2008) Role of Sn^{2+} in the Band Structure of SnM_{2}O_{6} and Sn_{2}M_{2}O_{7}(M = Nb and Ta) and Their Photocatalytic Properties. Chem Mater 20:1299
 20.
Aiura Y et al (2017) Disappearance of Localized Valence Band Maximum of Ternary Tin Oxide with Pyrochlore Structure, Sn_{2}Nb_{2}O_{7}. J Phys Chem C121:9480
 21.
Kikuchi N et al (2017) Carrier generation in a ptype oxide semiconductor: Sn_{2}(Nb_{2−x}Ta_{x})O_{7}. Phys Rev Mater 1:021601R
 22.
Henderson SJ, Shebanova O, Hektor AL, McMillan PF, Miller MT (2007) Structural Variations in PyrochloreStructured Bi_{2}Hf_{2}O_{7}, Bi_{2}Ti_{2}O_{7} and Bi_{2}Hf_{2x}Ti_{x}O_{7} Solid Solutions as a Function of Composition and Temperature by Neutron and Xray Diffraction and Raman Spectroscopy. Chem Mater 19:1712
 23.
Krasnov AG, Shein IR, Piir IV (2017) Experimental investigation and ab initio calculation of the properties of Sc, Indoped bismuth titanates with the pyrochlore type structure. Phys Solid State 59:483
 24.
Takeda T et al (1998) HighPressure Synthesis, Crystal Structure, and Metal–Semiconductor Transitions in the Tl_{2}Ru_{2}O_{7−δ} Pyrochlore. J Solid State Chem 140:182
 25.
Perdew JP, Burke K, Ernzerhof M (1996) Generalized Gradient Approximation Made Simple. Phys Rev Lett 77:3865
 26.
Blaha P, Schwarz K, Madsen GKH, Kvasnicka D, Luitz J (2001) WIEN2k, an augmented plane wave + local orbitals program for calculating crystal properties. Vienna University of Technology, Wien
 27.
Mori D et al (2006) Singlecrystal growth of Tl_{2}Ru_{2}O_{7} pyrochlore using highpressure and flux method. J Solid State Chem 179:935
 28.
Velikokhatnyi OI, Kumta PM (2010) Abinitio Study of Doped Tin Niobates and Tantalates as Prospective Catalyst Supports for Water Electrolysis. ECS Trans 28:37
 29.
Ishii F, Oguchi T (2011) Energie electrostatique des reseaux pyrochlore. J Phys Soc Jpn 69:526
 30.
Fujimoto S (2003) Electronic Band Structure of the Pyrochlore Ruthenium Oxides A_{2}Ru_{2}O_{7} (A= Bi, Tl and Y). Phys Rev B67:235102
 31.
Hinojosa BB, Nino JC, Asthagiri A (2008) Metalinsulator transition, spin gap generation, and charge ordering in geometrically frustrated electron systems. Phys Rev B77:104123
 32.
Kusakabe K, Aoki H (1994) Firstprinciples study of cubic Bi pyrochlores. Physica B194196:217
 33.
Janak JF (1977) Metallic ferromagnetism in the twoband Hubbard model. Phys Rev 16:255
 34.
Yanase A (1995) Uniform susceptibilities of metallic elements. Fortran program for space group TSPACE (in Japanese). Shokabo, Tokyo
Acknowledgements
We thank H. Aoki and H. Eisaki for fruitful discussions. This work was partially supported by the KAKENHI (Grant No. JP26400379) from Japan Society for the Promotion of Science (JSPS).
Funding
This work was partially supported by the KAKENHI (Grant No. JP26400379) from Japan Society for the Promotion of Science (JSPS).
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IH is the main contributor; TY performed part of the calculation and has critically read and revised the manuscript. KK performed part of the calculation and has critically read and revised the manuscript. All authors read and approved the final manuscript.
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Hase, I., Yanagisawa, T. & Kawashima, K. Computational Design of FlatBand Material. Nanoscale Res Lett 13, 63 (2018). https://doi.org/10.1186/s116710182464y
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Keywords
 Flat band
 Ferromagnetism
 Pyrochlore
 Electronic structure