Band structure of tungsten oxide W 20 O 58 with ideal octahedra
M. M. Korshunov, I. A. Nekrasov, N. S. Pavlov, A. A. Slobodchikov
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Band structure of tungsten oxide W O with ideal octahedra M. M. Korshunov , , I. A. Nekrasov , N. S. Pavlov , A. A. Slobodchikov , Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Akademgorodok, 660036 Krasnoyarsk, Russia Institute of Electrophysics, Russian Academy of Sciences, Ural Branch, 620016 Ekaterinburg, Russia
The band structure, density of states, and the Fermi surface of a tungsten oxide WO . with idealizedcrystal structure (ideal octahedra WO creating a “square lattice”) is obtained within the density functionaltheory in the generalized gradient approximation. Because of the oxygen vacancies ordering this system isequivalent to the compound W O (Magn´eli phase), which has 78 atoms in unit cell. We show that 5 d -orbitalsof tungsten atoms located immediately around the voids in the zigzag chains of edge-sharing octahedra give thedominant contribution near the Fermi level. These particular tungsten atoms are responsible of a low-energyproperties of the system.
1. Introduction.
Superconductivity, asone of the fundamental ground states in solidstate physics, is realized sometimes in the mostunexpected cases. These are both high-temperaturesuperconducting cuprates [1], which are dielectrics inthe underdoped case, as well as pnictides and ironchalcogenides [2, 3, 4, 5], although under normalconditions iron is a ferromagnet. These systemsare unusual superconductors, i.e. having anisotropicmomentum dependence of the order parameter. Theyare related to tungsten oxides by the presence ofa partially filled d -shell. Oxygen non-stoichiometricWO − x tungsten trioxide compounds have been knownfor a long time and their structure and properties havebeen well studied [6, 7]. However, more recently, thediscovery of superconductivity in the compound WO . with T c = 80 K and with T c = 94 K when intercalatedwith lithium [8] has been reported. This was preceded bythe observation of superconductivity near the domainwalls in WO − x [9], thin films [10] and in WO withsurface deposited sodium, Na . WO [11], which ledto the prediction of the possibility of superconductivityrealization in WO − x [12].Despite the long history of research on tungstenoxides as of today there are only a few works on theband structure calculations of WO . These are workson the electronic structure of bulk samples, thin filmsand clusters [13, 14, 15, 16, 17, 18, 19, 20], the roleof oxygen vacancies [21, 22, 23, 24, 25] and cationicdoping [26, 27, 28, 29, 30, 31, 32, 33]. Calculations forMagn´eli phases with ordered oxygen vacancies, WO − x ,are described in only one work [34], which illustratethat the compounds W O , W O , W O , W O ,W O , W O and W O show metallic properties. [email protected] [email protected] The ordering of oxygen vacancies in the systemWO − x leads to the appearance of quite large unit cells,which significantly complicates its description. Andsuperconductivity is realized in the system W O containing 78 atoms in a unit cell. Tungsten atomscoordinated by oxygen atoms form octahedra which areeither corner-sharing or edge-sharing. The octahedrathemselves are distorted, and the W-O-W bondsbetween the octahedroa are distorted as well, whichleads to an additional complication of the descriptionof the electronic structure of W O .Since the foundation for building a superconductingstate theory is the band structure of the normal phase,the calculation of the last one from the first principleswill be the first step on this path. In this paper weobtained the band structure, density of states andFermi surface for the compound WO − x with idealoctahedra creating a “square lattice”, which is thefirst approximation in the description of this complexcompound.
2. Structure and calculation results. W O belongs to the family of oxides with the Magn´elistructure and the general formula W n O n − [6]. Spacegroup is P / m : b , lattice parameters are a = 12 . ˚A, b = 3 . ˚A, c = 23 . ˚A, β = 95 ◦ [35]. The crystalstructure consists of WO octahedra which are eithercorner-sharing or edge-sharing in the (100) plane.As stated earlier in the compound W O the WO octahedra are distorted and the O-O bond length rangesfrom 2.63 to 2.72 ˚A. In order to model the idealizedcrystal structure, all octahedra were made ideal withaverage oxygen-oxygen distance equal to 2.68 ˚A. In thiscase, the bases of all ideal octahedra form a “squarelattice”. Fig. 1 shows the supercell × × for theidealized crystal structure W O .To calculate the band structure, the density ofstates (DOS) and the Fermi surface we use the density Письма в ЖЭТФ M. M. Korshunov, I. A. Nekrasov, N. S. Pavlov, A. A. Slobodchikov
W17W12W19 W15 W18W14 W16W1 W17W12W19 W15 W18W14 W16W1 W17W12W19 W15 W18W14 W16W1 W17W12W19 W15 W18W14 W16W1
Fig. 1. Idealized crystal structure of the W O supercell ( × × ). functional theory (DFT) with all-electron full-potentiallinearized augmented-plane wave method (FP-LAPW)implemented via the Elk code [37] together with thegeneralized gradient approximation (GGA) [36]. For theself-consistent ground state calculation we used a × × k -points grid in an irreducible Brillouin zone makingsure that the results are almost indistinguishable fromthose for a × × grid. E n e r gy , e V W1,12,14-19
Total W-5d O-2p
Fig. 2. Total DOS for W O with idealized crystalstructure (left), DOS for tungsten atoms W1,12,14-19(center), DOS for oxygen atoms (right) in a wide energyrange. Zero corresponds to the Fermi level. The density of states in the wide energy range areshown in Fig. 2. The top of the valence band from − . eV to − . eV is formed mainly by O-2 p states.In the region from − . eV to − . eV we see stronghybridization of W-5 d and O-2 p states.In the stoichiometric compound WO tungsten W has a d configuration, that is, an empty 5 d -shell and,therefore, a completely filled O-2 p shell, being a banddielectric. The oxygen deficit in WO − x leads to electrondoping and a finite conductivity value [7]. This is clearlyseen in our calculation as well, where the W-5 d statesare almost empty and form the conduction band. Atthe Fermi level we can see only a low-intensity tailcoming from W-5 d states (see Fig. 2), which are filledwith electrons due to an oxygen deficit compared to thestoichiometric composition of WO .We would like to emphasize the presence of flatbands at the Fermi level in the direction A − E andnear it, as well as in the direction Γ − A , which areshown in Fig. 3(a). For the bands structure we usedhighly symmetric k -points and corresponding directionsgenerated with the SeeK-path [38] tool. You can seethem in Fig. 3(b).To demonstrate which states form these flatbands Fig. 3(a) shows the bands structure with thecontributions of individual atoms in the vicinity ofthe Fermi level. One can see that the flat bands areformed by 5 d -states of W1,12,14-19 tungsten atoms,which are arranged around the voids in zigzag chainsof edge-sharing octahedra (see Fig. 1). Located directlyaround the voids atoms W14,19 and W16,17 give the Письма в ЖЭТФ and structure of tungsten oxide W O with ideal octahedra d states of W1,12,14-19 tungsten provide about 70%to the value of the total state density at the Fermilevel. Note that some “chaotic” bands structure inthe Γ − A direction, visible above the Fermi level, isnothing but multiple crossing of bands, resulting fromthe rather small volume of the Brillouin zone and thelarge number of atoms in the unit cell, split betweenthemselves by small hybridization interaction.(a) -1-0.5 0 0.5 1 Γ ZD B Γ A E Z C2 Y2 Γ E ne r g y , e V W16,17W15,18W14,19W1,12 (b) a* Fig. 3. (a) — Band structure for W O with idealizedcrystal srtucture near the Fermi level. Color denote thecontribution of individual tungsten atoms W1,12,14-19.Zero corresponds to the Fermi level. (b) — Brillouinzone for idealized crystal srtucture W O .Fig. 4. Fermi surface for idealized crystal srtuctureW O . Fig. 4 shows the Fermi surface for W O withan idealized crystal structure. The corresponding Fermisurface contains six sheets. The sheets located near the Γ -point (yellow and red) are clearly three-dimensional,while the other sheets are quasi-two-dimensional. Notethat the flat bands at the Fermi level in the A − E direction form rather large two-dimensional hole pocketsat the corners of the Brillouin zone.
3. Conclusion.
We have studied the compoundW O with an idealized crystal structure via thefirst-principles DFT-GGA calculation. Despite the largenumber of atoms (seventy eight) in the unit cell the maincontribution to the states near the Fermi level originatefrom the 5 d -orbitals of tungsten. These atoms arelocated directly around the voids in the zigzag pattern ofedge-sharing octahedra. Thus, it is this zigzag pattern,disordering the ideal “checkerboard” arrangement ofoctahedra, are responsible for conductivity and othereffects related to the states near the Fermi surface. Onthe one hand, we are dealing with a complex crystalstructure with a huge unit cell, which is caused by thedisordered arrangement of some tungsten atoms, and onthe other hand, the 5 d -orbitals of these atoms determinethe low-energy physics of the compound WO − x .We would like to thank S.G. Ovchinnikov andM.V. Sadovskii for useful discussions. This work wassupported by RFBR and Government of KrasnoyarskTerritory and Krasnoyarsk Regional Fund of Scienceto the Research Projects “Electronic correlation effectsand multiorbital physics in iron-based materials andcuprates” grant No. 19-42-240007 (MMK), by RFBRgrants No. 18-02-00281, 20-02-00011 (IAN, NSP, AAS),by the President of Russia grant for young scientists No.MK-1683.2019.2 (NSP and AAS). The computationswere performed at “URAN” supercomputer of theInstitute of Mathematics and Mechanics of the RASUral Branch.Список литературы
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